<%BANNER%>

Optimization Methods in Financial Engineering

Permanent Link: http://ufdc.ufl.edu/UFE0021770/00001

Material Information

Title: Optimization Methods in Financial Engineering
Physical Description: 1 online resource (126 p.)
Language: english
Creator: Sarykalin, Sergey Vlad
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: capm, deviation, omega, option, trading, vwap
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Our study developed novel approaches to solving and analyzing challenging problems of financial engineering including options pricing, market forecasting, and portfolio optimization. We also make connections of the portfolio theory with general deviation measures to classical portfolio and asset pricing theories. We consider a problem faced by traders whose performance is evaluated using the VWAP benchmark. Efficient trading market orders include predicting future volume distributions. Several forecasting algorithms based on CVaR-regression were developed for this purpose. Next, we consider assumption-free algorithm for pricing European Options in incomplete markets. A non-self-financing option replication strategy was modelled on a discrete grid in the space of time and the stock price. The algorithm was populated by historical sample paths adjusted to current volatility. Hedging error over the lifetime of the option was minimized subject to constraints on the hedging strategy. The output of the algorithm consists of the option price and the hedging strategy defined by the grid variables. Another considered problem was optimization of the Omega function. Hedge funds often use the Omega function to rank portfolios. We show that maximizing Omega function of a portfolio under positively homogeneous constraints can be reduced to linear programming. Finally, we look at the portfolio theory with general deviation measures from the perspective of the classical asset pricing theory. We derive pricing form of generalized CAPM relations and stochastic discount factors corresponding to deviation measures. We suggest methods for calibrating deviation measures using market data and discuss the possibility of restoring risk preferences from market data in the framework of the general portfolio theory.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Sergey Vlad Sarykalin.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Uryasev, Stanislav.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021770:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021770/00001

Material Information

Title: Optimization Methods in Financial Engineering
Physical Description: 1 online resource (126 p.)
Language: english
Creator: Sarykalin, Sergey Vlad
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: capm, deviation, omega, option, trading, vwap
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Our study developed novel approaches to solving and analyzing challenging problems of financial engineering including options pricing, market forecasting, and portfolio optimization. We also make connections of the portfolio theory with general deviation measures to classical portfolio and asset pricing theories. We consider a problem faced by traders whose performance is evaluated using the VWAP benchmark. Efficient trading market orders include predicting future volume distributions. Several forecasting algorithms based on CVaR-regression were developed for this purpose. Next, we consider assumption-free algorithm for pricing European Options in incomplete markets. A non-self-financing option replication strategy was modelled on a discrete grid in the space of time and the stock price. The algorithm was populated by historical sample paths adjusted to current volatility. Hedging error over the lifetime of the option was minimized subject to constraints on the hedging strategy. The output of the algorithm consists of the option price and the hedging strategy defined by the grid variables. Another considered problem was optimization of the Omega function. Hedge funds often use the Omega function to rank portfolios. We show that maximizing Omega function of a portfolio under positively homogeneous constraints can be reduced to linear programming. Finally, we look at the portfolio theory with general deviation measures from the perspective of the classical asset pricing theory. We derive pricing form of generalized CAPM relations and stochastic discount factors corresponding to deviation measures. We suggest methods for calibrating deviation measures using market data and discuss the possibility of restoring risk preferences from market data in the framework of the general portfolio theory.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Sergey Vlad Sarykalin.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Uryasev, Stanislav.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021770:00001


This item has the following downloads:


Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E20101117_AAAACK INGEST_TIME 2010-11-17T20:59:49Z PACKAGE UFE0021770_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
FILE SIZE 1053954 DFID F20101117_AABMFO ORIGIN DEPOSITOR PATH sarykalin_s_Page_028.tif GLOBAL false PRESERVATION BIT MESSAGE_DIGEST ALGORITHM MD5
71a873384639e3f71e92738aac1156e2
SHA-1
a386ed0ddb75ced4f0b2ff6a6cef6cae0c9791da
51123 F20101117_AABLZU sarykalin_s_Page_118.jpg
af5cf1a03bf20e29c51e2fb9760f3f15
009542193f9735e450a34ea14188911e6ce17947
25271604 F20101117_AABMGD sarykalin_s_Page_048.tif
32bd47b80686af26a0b99fdbcf14da4a
258aac7469f0ac4b0c35271f1ee3368b8e4e94e0
F20101117_AABMFP sarykalin_s_Page_029.tif
70049ad37ccd2142dc3d33f82421733c
2de63284f1872fa5d63c9662f92459d55e890d68
62202 F20101117_AABLZV sarykalin_s_Page_119.jpg
2efbc866b46530f26c4e351b5bfde14e
baaa45b6e10a5a3e350cef98f2cf3127eeca2fde
F20101117_AABMGE sarykalin_s_Page_049.tif
4803de6cb97b070da21e6b360a1b0178
2b99b62446396300089bb48ca2b5144db6537072
33686606 F20101117_AABMFQ sarykalin_s_Page_030.tif
89d5642beb5bdf461f0134e10b77029f
067a9f29a8725b6954e7487abfd223c5fe7c286c
50858 F20101117_AABLZW sarykalin_s_Page_120.jpg
b281dcdf0574c79a2dd3d1e4bf8f16c8
b5d3b2465a707a079d9e5ff4d07a3b2fa07d0790
F20101117_AABMFR sarykalin_s_Page_031.tif
8a23214dff1501ec7b8c59fedf7fabb1
8e4f5a8b59d68b946aa76c95d0d4c61a2aeb0780
45900 F20101117_AABLZX sarykalin_s_Page_121.jpg
fe3d102e813999b0900096cade464409
083ab4729cee893393a9570244a1936d3d6be55f
F20101117_AABMGF sarykalin_s_Page_050.tif
32def7eda5c652592249a613396f78d8
70765744125e5b379aa0b4deeb51ca922246dbe3
F20101117_AABMFS sarykalin_s_Page_033.tif
145d5d936b9fbef344ce30e4efe62d05
ff4f910d4ce156020b26975bbe224cb8f0c9f2bc
75314 F20101117_AABLZY sarykalin_s_Page_122.jpg
7bae3a3155627cada9cae70751ffa3ea
8c11414839691cff6f68624288092684f7e9d33d
F20101117_AABMGG sarykalin_s_Page_051.tif
551616c52b71b391e126ebfb371163e7
f5afc18a4a213da51883c2021092a531fcdb725c
F20101117_AABMFT sarykalin_s_Page_034.tif
261e38ddf2e8b644463c5ecb9ffdeb69
5fff38470dd4922a880fb60072fe61d7d6cbfea5
F20101117_AABMGH sarykalin_s_Page_052.tif
d7c50d00ddbf76d104d5d7154eb39669
d141d054c6713a8d1cd29d82b44b2e4808ee4117
F20101117_AABMFU sarykalin_s_Page_036.tif
407c4e4bab8448428d97531aa889062b
55e9a23ec034cc6157c1f64a3b7234c12a0e5a9e
83417 F20101117_AABLZZ sarykalin_s_Page_123.jpg
86200b77cd59e1d78007bf4dd00c55d9
746b17eea2a5923226ead7cf18e2f06135a55724
F20101117_AABMGI sarykalin_s_Page_053.tif
87b8d7e1fdca5bc143f7baabdc4e0dde
c3cccbf2d5f1825def3edc232e7f948d9c79ce0f
F20101117_AABMFV sarykalin_s_Page_038.tif
a09037d1693db37e61c9552a8def5975
7f0f9cd21dad690dbcf50dfdc74219f2b6f5e85e
F20101117_AABMGJ sarykalin_s_Page_054.tif
b86ea449d67591f12d10d9d26404c342
f494a405f06049938b8b2449c47bb8e8286a3b0f
F20101117_AABMFW sarykalin_s_Page_039.tif
0027da3ea79c28e1528706ff9b34c998
c45f46d214c682862fa8cc593450c19e57d95afc
F20101117_AABMGK sarykalin_s_Page_055.tif
65c6d3a2d1791dfeed84f07f0eaa1189
a7a53045b9239f0088256b26a3ba539a64b014d8
F20101117_AABMFX sarykalin_s_Page_040.tif
bff6b3a423354eee8389e5d0bb366b70
3eaaa238ef174de145523c38b4e1022f6bef0901
F20101117_AABMHA sarykalin_s_Page_076.tif
2aad287d5d47cbd33e778d6c0dc06a59
a972dfa1c0a19e768188c0427c184f11085c6aa9
F20101117_AABMGL sarykalin_s_Page_056.tif
cdc8470bc0bbdb68c9ff3720e6ea6203
8cb20fb0839f3b561e3d63c00477e1eb39671cd0
F20101117_AABMFY sarykalin_s_Page_042.tif
7f6fcc07abc666d6e745e17ce5fe9777
8eb2aa6b9703886be2bdbf69fd4ad13b11386584
F20101117_AABMHB sarykalin_s_Page_077.tif
da7f6a20b551994b9c17480d5f297afb
d81657cef0e54ac0f0accb22629166755b4c1582
F20101117_AABMGM sarykalin_s_Page_059.tif
aeba428b7c667696b6119b0d0984bbf9
d823d91d09bf857e2bb5aba15282ca3f6534955b
F20101117_AABMFZ sarykalin_s_Page_043.tif
adf20ecd5853bb20125f55ddf007fe47
7c7898a1800b618088203d33dbcdd9db77a48b1e
F20101117_AABMHC sarykalin_s_Page_078.tif
5dd4dc90e565c88c7d7cf631baed57e6
38b0dbdb657862de674467c1f3e69d4ecd1c7864
F20101117_AABMGN sarykalin_s_Page_061.tif
d676c40e9169503d03a5d08c7ce4a1e9
85ade0075207316874c4f5450f8d902fee9daaae
F20101117_AABMHD sarykalin_s_Page_079.tif
968ed790c5814530c0e0fc4305edb42e
93a616cd1c356f69f1d99e6c5fdb2b044698def2
F20101117_AABMGO sarykalin_s_Page_062.tif
3deaab6ce5ffea25d703ea07405eed6b
5c8303adbb2d813407a17927fea12be260bbf296
F20101117_AABMHE sarykalin_s_Page_080.tif
080b59b653f1181896053ad265c063c5
7ef39543adbe4963836b212acb803529a42837a2
F20101117_AABMGP sarykalin_s_Page_063.tif
d10509ca938f566bcc3fe119e9de6bcc
4c237f5c77f045e676842eaa46485032a41f869f
F20101117_AABMHF sarykalin_s_Page_081.tif
3491182db1960ad13d2256ee45ed3721
7f6c4ed84eac859c721dbf67ca5a05832915102e
F20101117_AABMGQ sarykalin_s_Page_064.tif
dcb63d30c800043a906f3ff7707f4918
2fa8001595c00bef3e6731674b809a762e5c9244
F20101117_AABMGR sarykalin_s_Page_065.tif
596b1a4c660f43a2ea6ec219b0d561e2
84c9ac0d7832205920b34f2328e1f75514820197
F20101117_AABMHG sarykalin_s_Page_082.tif
064bd7eaa8676e61f2b93b4bbf84c02b
f5d0dd912592b909ffa5234f49c3bf5a157ebf9e
F20101117_AABMGS sarykalin_s_Page_066.tif
82ff8a0876b122d926def24bdfd965a2
6aa41d8f4b98b785b19c39777aaf835cb493f2c8
F20101117_AABMHH sarykalin_s_Page_083.tif
efbcdfed96b7db3aa4974ca33560ad21
a5623d75751d715d72f1b4481104cef602cb70f0
F20101117_AABMGT sarykalin_s_Page_067.tif
41085a0a7d425bb319fdf279368b2498
0064448d1b41987fd08bcf2dadc50f9d5165ef7a
F20101117_AABMHI sarykalin_s_Page_084.tif
a62580b4b35469c44a87c6c2615bf908
0f9aad911a730372bd3f088501262c2785f9c956
F20101117_AABMGU sarykalin_s_Page_070.tif
cb7cc6b454ba8eafe6730aa721c8b3b2
23b702e6abd6c1779f31782e654fd39ba27c2fbc
F20101117_AABMHJ sarykalin_s_Page_086.tif
c6669d0ba133493ad36bd6d2c1e92e74
60c26bb2bc9f6443ff940ca42c77c16e04af2fe0
F20101117_AABMGV sarykalin_s_Page_071.tif
5a5013186cd2c6aa4036a053d3263800
0366c812abb64d898591ed35d09c7d57f8950c0e
F20101117_AABMHK sarykalin_s_Page_087.tif
73c8250142bfdb00f170c521d7355a2f
373d27b1a395dc3dfdb63ddc2f983215d195e29a
F20101117_AABMGW sarykalin_s_Page_072.tif
0680e0d33ecab06e3ecfc81bda06810c
9399963ec542cfa0be862cceef5412c2cc2609a4
F20101117_AABMIA sarykalin_s_Page_105.tif
ab1f45b451e35bc38cea3929aeac1732
6fd930ba656ddd7a83ac14294f4ee3de506f16c4
F20101117_AABMHL sarykalin_s_Page_088.tif
e0d3d6185c3f79c3fc6142de493b98d6
af030ca9ae7a8c20de4196868ba0bea37cb6f5ae
F20101117_AABMGX sarykalin_s_Page_073.tif
e26032077ef718c5180bfa56b90aab64
797770c2beaec8519064b7ec212492efcc9f71ac
F20101117_AABMIB sarykalin_s_Page_107.tif
f182a404675529c809e803ac4567901b
b9b92ae71144ae2f2521338f784d6aa87b4df29d
F20101117_AABMHM sarykalin_s_Page_089.tif
1c9d21cb845f167cabb2b76a08deb1b7
4cc513b3df689b2872d566c06d6f06a0d2769743
F20101117_AABMGY sarykalin_s_Page_074.tif
178c4b9f3346e8b3762dabc4cd1427d6
7db2bc8902223a5f547b9d36010357575bb5769a
F20101117_AABMIC sarykalin_s_Page_108.tif
f7801047315fc409de4f0e8d43ba0b8f
078d4a25d0950909436da21e6e0512b9084e4fe2
F20101117_AABMHN sarykalin_s_Page_090.tif
a599c468af375bda13fb1edc99ac2700
1bbfdc72174c9e66ca3dcb08c5755cbb05775017
F20101117_AABMGZ sarykalin_s_Page_075.tif
57c236d15e4b29dd12f61dc8a2382ba0
c5a2fd47a55bcceb0c2a84239035639a83c75e12
F20101117_AABMID sarykalin_s_Page_109.tif
91585fa13d3ab5706e2cfe2db885c34a
770cb5041f19bed7943386706392e80e1c75190b
F20101117_AABMHO sarykalin_s_Page_091.tif
b3f81ec6bb847eb7b5614de5b6f6b3f1
3396e09e62a3936ec9f61e821567ba0db099da5c
F20101117_AABMIE sarykalin_s_Page_110.tif
9c54cc350078ca08ad9bf542bd7b6d44
2c2c9b8590bf73c1b587511c64da258c8252756f
F20101117_AABMHP sarykalin_s_Page_092.tif
bf883c3c87662ec0201566dc8d979397
27d8bfcfa10356d24f682bdef6ff8fd44fc7c4ad
F20101117_AABMHQ sarykalin_s_Page_093.tif
673764a278490077c366c7dbda762103
b624da65587060c441114db992ae8e25bea50a02
F20101117_AABMIF sarykalin_s_Page_112.tif
1b86731c73b509b13109c589aaeb8c03
f66ec290fc99a7cbf8b4107a0ba7916b11e35dc2
F20101117_AABMHR sarykalin_s_Page_094.tif
11b532fafc3b1072e1c24218372dcdc2
59d086166ad1fc93b5a2a7417d36136a34ddd3e3
F20101117_AABMIG sarykalin_s_Page_113.tif
c6b3f009a863d8b3c12ccf0cc6d06255
9a1197b90433c9fefd14430f126e14715c9ffc9b
F20101117_AABMHS sarykalin_s_Page_095.tif
7325544cbd628c69384abfca21a66793
cd1027a7dd60088b9d8b93c62ebdd83590406837
F20101117_AABMHT sarykalin_s_Page_096.tif
de56d7261af924494a4b9ea5a90d825b
cc7641cf785861158ca10e7a75ed6c663be7d470
F20101117_AABMIH sarykalin_s_Page_114.tif
c94714badf881ac873c89f450d2edb66
b3a67fae9b540e3daf8b2b80a0a8a635346fb572
F20101117_AABMHU sarykalin_s_Page_097.tif
48f1a751785f4111b41f62b772018c0a
63192f2990de1efeffb691af0cbeae66a5dd5a81
F20101117_AABMII sarykalin_s_Page_115.tif
231e7e1e9c00c62b99addbc2efae79e4
d31ba47cb2227fdef3d2caebd2406594a9c240e4
F20101117_AABMHV sarykalin_s_Page_099.tif
3cdc01f9909ac411d21cd66b872d0179
f913edd510662cffc95689f934651f7e1efe57db
F20101117_AABMIJ sarykalin_s_Page_116.tif
f996c37d1b686ca63e6088ca133b4dd7
0c29f9e083080f71176ebacca7fa5a9bef09f7da
F20101117_AABMHW sarykalin_s_Page_100.tif
549a8595692cd56b472d1769e9f88e45
1ad2b04135a5e3bde6e84e970827f56a1e1e0f44
F20101117_AABMIK sarykalin_s_Page_117.tif
8b997478f32145006c5e6af698bdfddf
818c2a0eeabad3d958221c9f2222395d0497d6ca
F20101117_AABMHX sarykalin_s_Page_101.tif
3ed2785c591b3d9d8c5767b57886d61e
28d99607e552b90481aa5ae45214c5fff9098650
39829 F20101117_AABMJA sarykalin_s_Page_007.pro
7213dc9a75179fc4a6ca602f17957946
5dc0337989c2f59ffe7de19eedb081f908c9296e
F20101117_AABMIL sarykalin_s_Page_118.tif
17022dcb0ac99f579f70a46cb1dc3f12
9c246776e7debd4697c4eb5f9b3c1e81096f3968
F20101117_AABMHY sarykalin_s_Page_102.tif
bf84d04512fbb9b88c9c3fa244479379
d947486365a6cf0b7323042ca96e7967200fa88b
38842 F20101117_AABMJB sarykalin_s_Page_008.pro
a41c0ae67d5bf62b66f9ea82504d9a28
500ec589c558e35875864ca58f4c5c196db7e343
F20101117_AABMIM sarykalin_s_Page_119.tif
dcf186e94dbbe54c3ea4fec2f8947047
dd9fa9a07eb2b5aa2c18ba9b5ed0453a570dde83
F20101117_AABMHZ sarykalin_s_Page_104.tif
18549ae7a1e5293a427956277af457b8
4beb799b7ef3af83e13f3c6305193ac7b9bb6bb4
45170 F20101117_AABMJC sarykalin_s_Page_009.pro
9be71d7f5f747c46a10b911da91fe80d
f05c7197879f883d37f64ef3a46762c5ece894af
F20101117_AABMIN sarykalin_s_Page_120.tif
b75440419450a8776fa654bec6169469
7a76db61bdc9fa57b14918bfa9ff20963bbf8a56
12169 F20101117_AABMJD sarykalin_s_Page_010.pro
31586032a20ef4217eb64db759be7a74
fa9fd1317eed057b51327e7b38881bf536bb6323
F20101117_AABMIO sarykalin_s_Page_121.tif
15dbf57f523fb66bc5c89938cae5ef65
772a3d598733a2e9250a87c894b07a3af13e03cc
55829 F20101117_AABMJE sarykalin_s_Page_011.pro
33d3ed1cb122ab1cadeed739693adf26
b49009a1c4d666874d9b87f170166f429c7c0da0
F20101117_AABMIP sarykalin_s_Page_122.tif
c2feceb8b40f01f9a108fa884c85b30b
9df66d019f2fb875f1bdee68e647106805e2babd
46939 F20101117_AABMJF sarykalin_s_Page_012.pro
67ebd24b67f1f8749e7f36c9d3f09ae2
4cc2cd9b3ee2c053cd5080f0910827e0dad9a4ed
F20101117_AABMIQ sarykalin_s_Page_123.tif
5adadfe573072896159f2653006cb1de
4366e59f7e4c1eb4a98ae9e4d76c202a2d037519
47681 F20101117_AABMJG sarykalin_s_Page_013.pro
8557d57d4311a6139108ea4774cc5f72
0337365bd0dc4b5b6aec467a6f8c8ae32b532010
F20101117_AABMIR sarykalin_s_Page_124.tif
5a9bfce621f81446a45d35da89b90376
d97dd4145851072c3da647df3ad41707d0282183
55459 F20101117_AABMJH sarykalin_s_Page_014.pro
ddb93d5e5b4f0f51a3d7ef4b653e88b0
3dcd2253d289062a0e62be51d25f683d9b84fba7
F20101117_AABMIS sarykalin_s_Page_125.tif
4dbd7ad22f6dc1bc70294d0d8fe032aa
df7d45fccd6a12a4746fbf7dae9b370127fd695b
F20101117_AABMIT sarykalin_s_Page_126.tif
e914b517f015f229481e664af5790f9a
ac404686927f091659da6215b1530f1ece9d3649
39355 F20101117_AABMJI sarykalin_s_Page_015.pro
a8ba905659742d6237b7cd546c00abc3
c99dfb8a2ce8a259a3b840ba9791763a78ec43a6
7450 F20101117_AABMIU sarykalin_s_Page_001.pro
10eae06c37dbd7e99fcd23efab6b4862
c294bbed982ff897a8a26909e45c9046a19e01f0
46576 F20101117_AABMJJ sarykalin_s_Page_016.pro
17d26d194401b4f0ff921ba52c6b15ee
26c49ac9612c5c4d4f2781a4a8e86ee2edcc2605
968 F20101117_AABMIV sarykalin_s_Page_002.pro
0d82d5d13cef649613471a4f32a43e80
9adb0026d91dcb414451a9773bbd4570a4652bef
38869 F20101117_AABMJK sarykalin_s_Page_017.pro
feecbee47d5e468d07b0d511dc3899e1
26df0cce8c221f57c26b689e8d52f9f68e3b76d0
654 F20101117_AABMIW sarykalin_s_Page_003.pro
f2179a76ccb3a8cad83d787c42af9875
71263bb3e1761ab97288ac5393c173cdc1209b18
44195 F20101117_AABMKA sarykalin_s_Page_038.pro
e10a3cd4ad7e8f2e14f02b93e90e7956
0dee5d168eb21405c09246e60abc498463b6cc9f
35733 F20101117_AABMJL sarykalin_s_Page_018.pro
40cfb9599748295e7a8941c1562b5ea0
b6fdf8a6c79411de87a4c9059ac3cbee0959b31a
14060 F20101117_AABMIX sarykalin_s_Page_004.pro
7a47572b6f86d36804fef76adf709286
39f5cf9f6a6bca9de92c8f1e31c40f72ed254ea9
42578 F20101117_AABMKB sarykalin_s_Page_040.pro
4b2eacffd8c338cab4a0e994b7ae6632
7914501e5d2b08ab3d50d679d257f0567699371b
38775 F20101117_AABMJM sarykalin_s_Page_019.pro
c061e5a226a31f6d67dd95d00cd4611a
1e4fd92e6191bf8855213073477c6dc0e206583b
63167 F20101117_AABMIY sarykalin_s_Page_005.pro
45f66213ec8b7257206d415843304c4d
b1a4a840fb33108d5a734f71cc5fc527a80bb476
43270 F20101117_AABMKC sarykalin_s_Page_041.pro
c7efb7fab589ff216c5281ec1ffa33ef
cbd9469d3752390a93e904301301f2119a5e05c3
31907 F20101117_AABMJN sarykalin_s_Page_020.pro
ae3055ddf3e1bea0fddb851e458d563a
029de923db5f97249b9b51178e5ddd2ed40c2e11
79210 F20101117_AABMIZ sarykalin_s_Page_006.pro
4ce897205860d0c38781c31998953a59
4b79223fe791d581af20d1e42314d1cbdb57f459
43867 F20101117_AABMKD sarykalin_s_Page_042.pro
e36868708f32af9ee12f1d2cf52afb4e
8ff942b620c73f668afb3ac6e32eb0a9213d7202
35343 F20101117_AABMJO sarykalin_s_Page_021.pro
beb59d6c13a3cbf5bd30029ee7ef93d4
217e92f2e2a4b31318845b9f09aa26bcb137db5e
28948 F20101117_AABMKE sarykalin_s_Page_043.pro
4a80849001cb523fa3646e5e71186f3b
daf4fa617d8ac718d33b318feca6c7af742e8aa4
30730 F20101117_AABMJP sarykalin_s_Page_022.pro
f0f8de9e9f632e939d67307294f711a7
65d72ffb5552b65e880d42d3ecc7685a694f58d3
48773 F20101117_AABMKF sarykalin_s_Page_044.pro
c48257bc410a36b5d4ce4ca32349415f
eb28c3a0593655bb8c4da3d1fbbb3528c9872715
50414 F20101117_AABMJQ sarykalin_s_Page_023.pro
be4b4c7bdc272968e103d127af12a3ca
4efd3e9e74957d3d299f311b28354a22503819d4
58688 F20101117_AABMKG sarykalin_s_Page_045.pro
1570e98da507bbed58f15e2db10f9332
c7c6a84c9ed2dedb881cb7c6d992f517b8066f69
31888 F20101117_AABMJR sarykalin_s_Page_024.pro
963070949f4beeba7cdb6a503b678595
785054b48cc1981884e37dcadf6a9ecfbbf4926f
29053 F20101117_AABMKH sarykalin_s_Page_046.pro
0f318ca8e98ca41e75931a6814d30476
3805fd1e85a2c94f33703938685c439e1806e6a7
40159 F20101117_AABMJS sarykalin_s_Page_025.pro
208566fbed59a61f4572723efb56b344
cea232c0635d30baa06fa85dd100c7d30ef44823
34922 F20101117_AABMKI sarykalin_s_Page_047.pro
33891dd012b4166db576bfb91e28baba
f3deb6257cf962e09f77be29f5fbcd078a0ffdba
54538 F20101117_AABMJT sarykalin_s_Page_026.pro
3ffe48bf1c70527dd19c106218fa402a
47ea3b25d1d1178b1e3be0f5fa023ca19df739cb
34380 F20101117_AABMJU sarykalin_s_Page_029.pro
1182f1bbe320412a87fbfcbf5e330c45
3623bbce158b71c442de3515e9a2ac7648bfa6ed
54287 F20101117_AABMKJ sarykalin_s_Page_049.pro
98f2904dffb3f0c6712d5f3da6bbe66b
cf8cbb87bf3aabf83795d483ef12507ee5bee2eb
5487 F20101117_AABMJV sarykalin_s_Page_031.pro
1e2d3788836236f4e66ad90efa992ec2
f1849a3a844d4f976802af698c5a1d21444a07d3
43159 F20101117_AABMKK sarykalin_s_Page_050.pro
3152b46ddd3069fe6a72ca6e04ef31b8
825b880252fad87d8965394086bac128f3ab71f8
57676 F20101117_AABMJW sarykalin_s_Page_033.pro
ea2d5ec987d0155215624d37b806c6ba
3f40dbb385da0d5c38db52c42e7d2705e9d5a488
43200 F20101117_AABMKL sarykalin_s_Page_051.pro
669b3d0a9e69de26d2930589b112ac15
574a7457e07104ca23e8461c8ecfc3535ed18bd1
59661 F20101117_AABMJX sarykalin_s_Page_034.pro
cc7faf3253cfce358da2097acefe92d7
219e877f38645c636a9cadf502482f62300c7ce4
41550 F20101117_AABMLA sarykalin_s_Page_068.pro
fe8e243d8020e940e256f7e703f62a23
e3aeb2c3e1dc2d92025994a7fb04577d8f30fe21
41640 F20101117_AABMKM sarykalin_s_Page_052.pro
d5a86581057f064683246d74c9a7fea0
08f916383df901f020921749b877ed10af1fba8f
61764 F20101117_AABMJY sarykalin_s_Page_035.pro
15ad07ec33adc9d386555bd347546c45
d4dfce52dc0c73cc62a16deaf572cdc12706733b
31057 F20101117_AABMLB sarykalin_s_Page_069.pro
25370b3d6ae9a92ec8927d7e8f9a6942
324c5d338ec11a49537bd12cdcaad4af465278b3
33940 F20101117_AABMKN sarykalin_s_Page_053.pro
65c9fa191e42ae120ce283e0f3aed503
21f6b423a44efab26bb6dbd4a7ac7a6eda66e3ce
49460 F20101117_AABMJZ sarykalin_s_Page_036.pro
decd95a841fc0a1332a0b6efeb058bdf
7662ff8bd7da74324350d325a742f5dd87f11ef2
54091 F20101117_AABMLC sarykalin_s_Page_070.pro
1d923dd59e9b41bef653f3e0b11b9ce9
317b71fd6587c23a2fc7ad7a1fc456740cab3fdc
43581 F20101117_AABMKO sarykalin_s_Page_055.pro
92cdae094d01b68bcce4ea903ff206e9
03ef9498dc2361d457d9f658f1abcdde6c637b89
54108 F20101117_AABMLD sarykalin_s_Page_071.pro
881321566cf3b7ddbd34f961a0934f08
f96549d0a2ead6b6d0bdf41606a0048e5afffce7
38854 F20101117_AABMKP sarykalin_s_Page_056.pro
790ee2adc4473ccd890a55fbdd8d46a8
b9e72a9843beef7b4a1fbdb67fb8cef2695a9030
54304 F20101117_AABMLE sarykalin_s_Page_072.pro
afaa81400bc57ee8fa9b345f2c0a4e46
545d5558faa5999f5d3b68162178383fe93a5a84
47790 F20101117_AABMKQ sarykalin_s_Page_057.pro
be4d107fd84899ef44b856206320c53b
36748ea84afbf2e529cd83cbaaacf96a2dc338d4
57965 F20101117_AABMLF sarykalin_s_Page_073.pro
32fe60f89f3d2abad30f2713f939bee3
eef5785751da16e58600af81f4002df044ed798a
41694 F20101117_AABMKR sarykalin_s_Page_058.pro
bc4527df7767aafa6680ed3357ef6172
abfe92329f39ff0bf99ca9c95c494c9e943f1e0d
57946 F20101117_AABMLG sarykalin_s_Page_074.pro
7d76a2318fdc82694e198724a3a6afdd
826956346eada04a2956b4596b75c262713cf3d3
56516 F20101117_AABMKS sarykalin_s_Page_059.pro
ce7765fd0e799c4810906ac85eb75a9d
e4151c192b094fcf8de046e03bf83758c814b476
33761 F20101117_AABMLH sarykalin_s_Page_075.pro
17d91222850433005689381b24bc10c3
987c8eaaa5496e5515285e18e5e4dd1a6db263fb
60175 F20101117_AABMKT sarykalin_s_Page_060.pro
10959485070917282de76895d66712ba
285fcb8cf898911e2f9474533c92a7c289a71355
30966 F20101117_AABMLI sarykalin_s_Page_076.pro
1c9fe7fe4da28d20ae7d1ef00e9357a2
262d0d653052b4127f5ef27a59148f538b2a246b
55205 F20101117_AABMKU sarykalin_s_Page_061.pro
c6da7a228dce350c9a567ac258ca61a0
4f13ee2ee2d0cf14b1b210721345ea3af4540dfe
24274 F20101117_AABMLJ sarykalin_s_Page_077.pro
9c890a21ed26f6958633cb797ecf3f32
db046e700225ca043f9784377dff7d054bc435d0
51078 F20101117_AABMKV sarykalin_s_Page_062.pro
dba77e10b65f89b58c15b4df647e1da7
8181cb67c67c0782e377359625f7594b42feea29
35241 F20101117_AABMKW sarykalin_s_Page_063.pro
fb05168689e58bb939f31178a0735c72
9faa857aef0801e70e40283315d4e4f79abaa6ca
45758 F20101117_AABMLK sarykalin_s_Page_078.pro
b8215c653d9edac6562cb77b2d673d7c
018146df529788a2ca4b404f0f6e6fb0f4d98eca
32207 F20101117_AABMKX sarykalin_s_Page_064.pro
98580047f3b62a010c5443a04704f3eb
cf7e6454347bb224804b3ea6f9a8981e1dfc80d3
36118 F20101117_AABMMA sarykalin_s_Page_095.pro
90131463b607a8497b9be51351df024a
c462662d9f0f009dc442a7d3fa8f04c5cfb31b60
33861 F20101117_AABMLL sarykalin_s_Page_079.pro
26d88411708112509b3025b8919c6281
f215c403b6faf5321d248fb2bca4ed969ac662fd
22139 F20101117_AABMKY sarykalin_s_Page_066.pro
f360763c09718e49af377e3fe27ed328
85dd5b7c65fc210e842dad869bb4506b16d9c5d4
43964 F20101117_AABMMB sarykalin_s_Page_097.pro
f6258d1731aaef7c1dd20dce7845490c
380abd12e91801cb452babe4c38400cfd7b5bf07
37619 F20101117_AABMLM sarykalin_s_Page_080.pro
eff542c98af76a80be4f48d4d8702bde
d85f0c7742115e7b80ec1522e843d37447ce17ee
17506 F20101117_AABMKZ sarykalin_s_Page_067.pro
e41e618379d7131912bbd3b9c7b883f9
424bbc6935d5acd1c4e8d7f8bb8c874ef4adc5c1
50063 F20101117_AABMMC sarykalin_s_Page_098.pro
e7491ace363d247d2086595a3f9eb2d7
d2f8329f265a3f466cce6ad7694aa3f9feb1cec3
40800 F20101117_AABMLN sarykalin_s_Page_081.pro
f4e10c10477e9a1ed87322ee7bc82c53
bb862f3edcca7ceea2114350b86e9c2a2a68c478
49222 F20101117_AABMMD sarykalin_s_Page_099.pro
92904ff76e594e3a2debd8b156918ed2
f98b856424f98dfa1a41b05ebcc32a89a6158224
F20101117_AABMLO sarykalin_s_Page_083.pro
4c985431397f3db7a0f37b5ce86f038c
91f1d38ffca2e03f85a0754d02797e0b8e420977
33004 F20101117_AABMME sarykalin_s_Page_102.pro
dbefc29c8d84ce67089790b212a8abaa
5f4e20bc58d220bdb888f2fa201949be96dd7cdd
45738 F20101117_AABMLP sarykalin_s_Page_084.pro
456aca9c97c890dce7083131838cb8ed
86fa7c75045486ebfb78cf30002ae1ac166be0c8
48052 F20101117_AABMMF sarykalin_s_Page_103.pro
fbb317324f2ee1424a740107bd2ceaa4
ebc8f8988bd35538ea80c4cc66f5de48a03a563a
45039 F20101117_AABMLQ sarykalin_s_Page_085.pro
109b2b8fe803c3a79d0b7cc8fd402440
6146384c756c9baa764a16997cc063a565a17df4
20126 F20101117_AABMMG sarykalin_s_Page_104.pro
3c115b1dd042f6a656b1134e2c5f7a1a
3b66d0630d5af5558e2a0e32415e5f9a0b5ca8a8
27887 F20101117_AABMLR sarykalin_s_Page_086.pro
dc11536789a3ebf40d012787fa2a191c
89036c057b1eb9aafb09b8dc3e481eb1d3ba24ac
27488 F20101117_AABMMH sarykalin_s_Page_105.pro
7f3fa214b45991f839d0803d87b8c711
1cb3c4e23464882e7fb0d3881ffd5be0083965a8
34492 F20101117_AABMLS sarykalin_s_Page_087.pro
c95052dc8917dd22b69a4f50f2820fc1
7358165a612d3bd1e2ba7b509a6650ff2f8c2bfa
56754 F20101117_AABMMI sarykalin_s_Page_106.pro
f4fe7a67a3da53cb84ced24899b460b3
92187eeef64f1bf7c0cdfb8bf0f4d02415d7ee90
22677 F20101117_AABMLT sarykalin_s_Page_088.pro
5c93202e487c1d2074175dd6f2cb90fc
0f3498b2ef116bc14cf653126e598a88d9fc4a4d
60826 F20101117_AABMMJ sarykalin_s_Page_107.pro
af30f8c3016d77e39a0e897c9c19a843
33d82e4ecc0bf77c21a113afbdf63ce313f69a6d
48412 F20101117_AABMLU sarykalin_s_Page_089.pro
ba2825014fa3747234436d0b58cecdfc
e5c5dabc2ff0f2fc50d853e960392877bd0fb760
59060 F20101117_AABMMK sarykalin_s_Page_108.pro
5b6486666bc86e53914265b923b73fc3
b97a962f10e15a5164a95106fa191fb42e2a5ada
41339 F20101117_AABMLV sarykalin_s_Page_090.pro
7eb0e4735b87d02955891d133ac6485c
8143110afd2c04eec3b3e239de9ab7cf174c92e2
55910 F20101117_AABMLW sarykalin_s_Page_091.pro
9acba685b24f55ceec27fbbc8f935070
747d2a43912e1689ea0201bdb5256a5774b2f76d
91 F20101117_AABMNA sarykalin_s_Page_002.txt
4b344184f2f4b2d41252057b46f522eb
6cc6a1e80149d73dcd98c70a1fac18c58439dc6e
46923 F20101117_AABMML sarykalin_s_Page_110.pro
ec9fb29ad22cf5d99030421409d0b327
631dcb9cf05428b2974199460e90de4a12eaa13c
41599 F20101117_AABMLX sarykalin_s_Page_092.pro
eb6e764b98e1946c6411d23a6c0ea116
683ecb049fc0f1330f22cf95df2d97b19d6dba24
85 F20101117_AABMNB sarykalin_s_Page_003.txt
7579ff36c2265cd26e241129bd047745
fedf45c16230d8e4bceb7fc890dbee743ad79987
34819 F20101117_AABMMM sarykalin_s_Page_112.pro
764884686b9e3dba639b60600cfefdbf
d3adc313353af521d02fec6c6678f7e55fbcb9cc
31316 F20101117_AABMLY sarykalin_s_Page_093.pro
844aebf55363e1f2dfb9b0dbf6224496
a8070c67b7489cac0b9d72b1ae0c1c59ff472a50
40869 F20101117_AABMMN sarykalin_s_Page_113.pro
458482cf03f339700c27a95471f850cc
5e540c32769822eedbf77cba9254be350e3210f9
20982 F20101117_AABMLZ sarykalin_s_Page_094.pro
75ed45b8c82dd8d1bba3ce224ca0c6b3
d37ceed9646d1d32f9a9c3f71d76c9a27b977fd8
612 F20101117_AABMNC sarykalin_s_Page_004.txt
1ed8f5cffbd6b7bfa2ab1bf27ea74071
ae8d7e108b2dafe186a5d6f888fe1921fb9caa1b
48781 F20101117_AABMMO sarykalin_s_Page_114.pro
35aea57433a7a199542d15c022ee6a4e
60ff0f4976291e681b913ff0b09c899403f2a33d
2778 F20101117_AABMND sarykalin_s_Page_005.txt
e5bcf213c8a0ea0957fa5eb7874a2003
b1b8f9389090b628fa9a22562eb24fc5f9a9f601
57449 F20101117_AABMMP sarykalin_s_Page_115.pro
f22dcf6fff931f9477d9acb4cfbafd66
e2a425070a13f6376f9ea0d650cf8387ceafb584
3383 F20101117_AABMNE sarykalin_s_Page_006.txt
aee78bebe2d2fca13469a0c015f7321b
01a58da0b61c905f91f1fa356fd835a8379f7611
55034 F20101117_AABMMQ sarykalin_s_Page_116.pro
0131106ef6fb3f104ea01f6b8c0a3ab0
24313fbe59d9229736e5422ea79f0d57dcd8ecd4
1694 F20101117_AABMNF sarykalin_s_Page_007.txt
d9b0be790f15e61c7ff59658970431c7
9739d426d8d2ba5dfbc16a376155bd7724ea47a3
37801 F20101117_AABMMR sarykalin_s_Page_117.pro
aede0a6503db3e861ece9702b5438b1b
5952db9871dc9e3d286a75c1c24aff702d0f5946
1658 F20101117_AABMNG sarykalin_s_Page_008.txt
6fec36de2c39f1ebb9ca4ee8fd0eceb0
a6d90ee6a64a425a590f2a9a487844b280be48ff
27728 F20101117_AABMMS sarykalin_s_Page_118.pro
a9cebff59728867a0cc1485556d92ae9
4ce36faa7fb65d79402778cc541feebc76f02c8b
1972 F20101117_AABMNH sarykalin_s_Page_009.txt
9cba6120c34932737c7ea0f6cf6a1a91
e6f88cdd5badee3edea92ef495b751b46befcabf
37679 F20101117_AABMMT sarykalin_s_Page_119.pro
fbbbfc5f2a11508bee06ad1ee419a66a
01bb9811f87a1c97f4073db25ec5b8892a9923a8
491 F20101117_AABMNI sarykalin_s_Page_010.txt
6baaaec77c2a208a52714e957e11725a
6d2199a7beb98c698be2f40bc3887c5dfd7e39cf
30056 F20101117_AABMMU sarykalin_s_Page_120.pro
b4fdfb1c8b22f2bb6f43143d9d0c9484
3ae680ca547e8925bbd7704b3397d71813e02b42
2267 F20101117_AABMNJ sarykalin_s_Page_011.txt
d364c44423790e468c01a6479930cea7
7bb52a7e3fae009b87404eeb59ffc424a1e19778
55111 F20101117_AABMMV sarykalin_s_Page_122.pro
8cdcb9cd56d32f3253cdb22a0a1d1069
dc8fd878a81c4d3b3c18405f675377d8ff726485
1870 F20101117_AABMNK sarykalin_s_Page_012.txt
c13677a967e385387b82a06a8c88381b
c0209baedfc875dd1cc55e12fdd356aaa4e86dd9
60191 F20101117_AABMMW sarykalin_s_Page_123.pro
6a872435b1f839f373e440b3eabcb5ad
9127eb8a8e029d482d95be61919cca441c14c026
1983 F20101117_AABMNL sarykalin_s_Page_013.txt
8fe2dfcf1238523b9360e42b7167dec6
0604eb12b6d331c20356eba9779814a4f9cdc1cf
57156 F20101117_AABMMX sarykalin_s_Page_125.pro
fc058b3d0cb122b02181fc02cfc956d5
4c81bb63b244bd0f35bdf3a00dfb4004eab15524
396 F20101117_AABMOA sarykalin_s_Page_031.txt
6baf9543b136e2d77b13955d7fb27ba0
101c71bb19f54f2ff76f045dd6c9ff1a333fe154
13562 F20101117_AABMMY sarykalin_s_Page_126.pro
f9d12e0c1d65f1d9b18f0d4499ff1567
af90f5778abe8b37024c11890e55a6164a1fcd3a
2266 F20101117_AABMOB sarykalin_s_Page_033.txt
df630a432072c5794a260f94a0a9db30
6b17991295ff10c0287ff5c4ea9ca88e6afd98a9
2189 F20101117_AABMNM sarykalin_s_Page_014.txt
2d38a235ad037b323386af3662b0c3f5
1db9f1bf0c698f4addacaf8072da8c252bc58aa4
422 F20101117_AABMMZ sarykalin_s_Page_001.txt
4f846ff23d9ad65a95af89cabe5df11c
d300265bd5053e246e9c482b6876d73af4da92f5
2340 F20101117_AABMOC sarykalin_s_Page_034.txt
750e452785172b1ebef981e8adde2ce7
d92b3c55b98652be74921da8c6c41ac17fc7ba3b
1967 F20101117_AABMNN sarykalin_s_Page_015.txt
9f3254b29abd0965b962193bc7a7506d
ddf116afe2ff364abd548ac6111c12093aa8dc2f
2422 F20101117_AABMOD sarykalin_s_Page_035.txt
cac5ff48d98d25940ee2ac1517a1d8fa
70671c57221f459eebe280a2c40753e0463d7f91
1896 F20101117_AABMNO sarykalin_s_Page_017.txt
00500039c767bd60e0fb8f52caf3c25b
cbbcebb64da8839f0f4ce1548ecaebfdef62abeb
1971 F20101117_AABMOE sarykalin_s_Page_036.txt
5c0603c0d31ec6c77d059f5d54cc4648
01037d49d93fa43f079c0d824606885b8b7bc85b
1628 F20101117_AABMNP sarykalin_s_Page_018.txt
7ab01755d9eb43e0b62b1924e4fff523
98a89b8c9d4a747ccdce719a11d2e9a9ed778d71
1798 F20101117_AABMOF sarykalin_s_Page_037.txt
63329bae85f75c89cae04c3e80d8cf3f
41e235f2d60ebef8d15f7214c942851d114e5814
1688 F20101117_AABMNQ sarykalin_s_Page_019.txt
1ba2a308a0e5a4f51c3b2c224eec54cf
e66a86b4fc36b6d427a2fb81595a24bf1538ce93
1986 F20101117_AABMOG sarykalin_s_Page_038.txt
9a5704b04edf941118a18afdff6d606f
bdb262b724afd11f1de2c69d10dde9c7fd4d6845
1649 F20101117_AABMNR sarykalin_s_Page_020.txt
1a2117155797ffce693a59cf371ba8aa
524e2c3f1d4d6f1f573649274875b08001c20701
1848 F20101117_AABMOH sarykalin_s_Page_039.txt
2b202a1f122766d2482d7a7b5b8b17c5
70a3d6e92924ea4755d01c7321130d84cb0bc434
F20101117_AABMNS sarykalin_s_Page_021.txt
fe60661a6a513f8a2adc397058cf30f4
0bf155da772284d49abee0d006ce8e43dd88096a
2013 F20101117_AABMOI sarykalin_s_Page_040.txt
18fd7665327f8141ff105bb44d9c94f3
1e205bb8f5129b9c19d1d2af585ca9be869b7847
1621 F20101117_AABMNT sarykalin_s_Page_022.txt
cc58108a78625b2c9fab7c9e48a4e7ff
a8e0a446161afb3c1f3e653197a47bb95f6204fc
1985 F20101117_AABMOJ sarykalin_s_Page_041.txt
deb0bcf212e5477d4b27992346ca3865
19b716d2b073ff3851d0348c4438eba5038dc279
2061 F20101117_AABMNU sarykalin_s_Page_023.txt
e9c70a1881d066efd66391c7749c99ce
ee0369bceacfd9ea485a3350ad69836d3fbd4dbc
1838 F20101117_AABMOK sarykalin_s_Page_042.txt
af82de1c8c883bee6923a41fb1fabdd3
fc89fe38fbd500bfec8aa7ead69d2532e5c2b0a3
1949 F20101117_AABMNV sarykalin_s_Page_024.txt
23ce5089f9505859142ead79ebb8e734
ad23a603d68e5a37d46a34b8fc2023e29c978bfe
1447 F20101117_AABMOL sarykalin_s_Page_043.txt
79cada20548b78ad406bba548c5f095c
2c851b0939c6ffe8a4d898d44a5e51448d2c72d5
1886 F20101117_AABMNW sarykalin_s_Page_025.txt
600b946a6d2182dfd6f53dd09a792273
1d2dfe0feed5450c2039489dd561911505080798
1773 F20101117_AABMPA sarykalin_s_Page_058.txt
4cad36c8d2ebdf642639c23c61d50336
a86b546a380d9ca519fa0bcac5768688b3ce942c
2047 F20101117_AABMOM sarykalin_s_Page_044.txt
38606437d56bbc53cc35d7077913f6bc
e4ac562f6ffc7a7e4425fa1e415d349a365cd27c
2271 F20101117_AABMNX sarykalin_s_Page_026.txt
9449cc779f8b70711245029d64dbe7f2
4ba6e5a56dd9fc8f496bbf8548314dd855244bf3
2241 F20101117_AABMPB sarykalin_s_Page_059.txt
66037bb2fbae3239bff3fb67dd9ade13
629284b01c7e496fc1a6175cfa36e3ef23d9bf82
1040 F20101117_AABMNY sarykalin_s_Page_027.txt
f32a526052dd397526f4e1d843680021
fbcc4b3a16183288153e344b44aa850db02cc8db
2362 F20101117_AABMPC sarykalin_s_Page_060.txt
604a85493535aea84ad73c7c20fb12ba
68959976d8f9cd3dafa6908d0b03c4798ec2e924
2304 F20101117_AABMON sarykalin_s_Page_045.txt
339c1677badde50afa0a4c90488c9eb1
5a7da583aca173a33dead815e709face34950a26
1687 F20101117_AABMNZ sarykalin_s_Page_028.txt
a18954cec5e57b32091358dfd19acd6e
81a0e3af063ab9abbbb62cfdda656a236a538b27
2185 F20101117_AABMPD sarykalin_s_Page_061.txt
58c6de8e6696ef3d31694918f1057138
dbd75f51380e596a4f5c24b8c326399e0246a74f
1515 F20101117_AABMOO sarykalin_s_Page_046.txt
3952fdb9e7d48a8f76063fd55c34536a
f3951fe6a7bbc5b12cb826dee5c6a553c54e5ae0
2034 F20101117_AABMPE sarykalin_s_Page_062.txt
ddb74c0bf5a3ebf92daac99d1822dd3a
5b9ee00b5e09281a788416a4ef16c3a088cb4d2d
1675 F20101117_AABMOP sarykalin_s_Page_047.txt
a397c51cf15f9b979f67bafd3da1806c
1441f76dc6e83c779fa88d87d72b93db16d13255
1430 F20101117_AABMPF sarykalin_s_Page_063.txt
901fc063c7b546e37cbc4a7aa995d96f
939c12e47ff453b7613b40ac88e9037912c02cdb
1820 F20101117_AABMOQ sarykalin_s_Page_048.txt
06b1043ef9d0fb8d10bc264d35a1d788
69437855ab1ec25bdbf003455677004138846d78
1315 F20101117_AABMPG sarykalin_s_Page_064.txt
4ac658080c5eda1accdba8467c6fd881
ad398396877d416259b1637aad824a23138cf2a2
2175 F20101117_AABMOR sarykalin_s_Page_049.txt
cfaaae90b58b66af4837aa8621adfe0d
f34dab195686266b27718f98562b9fd23726de00
1433 F20101117_AABMPH sarykalin_s_Page_065.txt
0a90d65ef2d151cf4f81fbfe9ffd06f9
7b65536502fe9fa751bf665d8193971934cd609e
F20101117_AABMOS sarykalin_s_Page_050.txt
0fb48424eceb3bd9821e39318cb68d76
1ab9a7394baf103ad8cfdef997724262e50a2e5a
1193 F20101117_AABMPI sarykalin_s_Page_066.txt
2401592696896d8c71b47e8c528a4a18
678f7df2ce567903c16b16238525a3964ed52a74
1900 F20101117_AABMOT sarykalin_s_Page_051.txt
1b1d122dee6ecb336f111acd1c97d200
123a928079a39513d1d2df08a8449bd5dd08aea5
747 F20101117_AABMPJ sarykalin_s_Page_067.txt
acef07bb88ead65d35563deeda61f31b
41353e819530ddab7a03dcb7ab292833563cc211
1907 F20101117_AABMOU sarykalin_s_Page_052.txt
d579a87584c94187e77434e776771bf1
81e00ed374c1985ee5aa03ef6c07bf6ad918e86c
1860 F20101117_AABMPK sarykalin_s_Page_068.txt
9dfa47591f168893646aceb747e854aa
d8447e5b66ebf08a0e3b398b100d5bec11bb5071
1490 F20101117_AABMOV sarykalin_s_Page_053.txt
25dace5170dd77814cf438304512bece
0d61b57931329cf075c6cad72db06cd813da5d91
1528 F20101117_AABMPL sarykalin_s_Page_069.txt
98bb14294d2949d32e64397f18c5d887
b77fcc4f3330d8c3fab04389b4130103e90ce1e3
1646 F20101117_AABMOW sarykalin_s_Page_054.txt
1237762574a0e11e4869d20f75662b5f
df14d3df56fbfa9dc3821568bc6f50077267f008
2528 F20101117_AABMPM sarykalin_s_Page_071.txt
40ffb69b3f884e2729938c1fc35342cc
c0a7bf65d24317de562b588d54163180003f8bd2
1790 F20101117_AABMOX sarykalin_s_Page_055.txt
9aa0eada39b048f000d96c524c695821
c36c1cc448e35e13d7b6b51fd18e6207dd8b0e32
2104 F20101117_AABMQA sarykalin_s_Page_085.txt
cc9e7d002db6ea1e3b327d02ac62c11a
06912d729982cfb11dff85e6c1f9e229d8c7d0da
2376 F20101117_AABMPN sarykalin_s_Page_072.txt
8975927ada9c5e0f0b1b9cce3401ff7e
5605dcfd25256df1027221645a97faa8567b2c76
1910 F20101117_AABMOY sarykalin_s_Page_056.txt
11180dfb53f22b9bc0ac22349032a994
12dbd16b4bbef3ec28337cd23afb181ff492cd35
1610 F20101117_AABMQB sarykalin_s_Page_086.txt
c1d3a3d20f1933f64f9781c250536a37
fdc9c5fc5c26e6618e0a85e36741efcbc44710cd
1989 F20101117_AABMOZ sarykalin_s_Page_057.txt
f8698c75747e8f462ed47fa11a5335fa
be79c50d7955e2be83dfeb2865c6afd92ae3605e
1726 F20101117_AABMQC sarykalin_s_Page_087.txt
d1b43a21b8da549f61e3401c8f61edbd
2216328077b90d35ca11a52b40175c4a5b6463a3
F20101117_AABMPO sarykalin_s_Page_073.txt
fda6cf36a1600bcb7ed87ae19ce38d66
cd054053bc67c933712497efc45f19d210444ba0
972 F20101117_AABMQD sarykalin_s_Page_088.txt
32e245d13de0ed02cefab90961409556
4673b1ba4bcb8202c68c4cd0497344f8327fc1f1
2279 F20101117_AABMPP sarykalin_s_Page_074.txt
8b59cca1e486bb3a8fdade4a6a46e02f
ec0024185d528cdd9dc0a48cc16523a64b3fe0f3
2020 F20101117_AABMQE sarykalin_s_Page_089.txt
ff2f1c740e0a28102c86c0c3f55b6e77
bac47e52e95e9a9449331943b712c6e5a97cbfd1
1686 F20101117_AABMPQ sarykalin_s_Page_075.txt
c72e63aa2e2f1cac295040750c838fc1
2ff2c48fd15a75a603b884e0e4859b33a01800b6
1840 F20101117_AABMQF sarykalin_s_Page_090.txt
83513e9295f2aae11b117ff90e4f15f1
91ed27750e5182a0c817ee93fff0306656511625
1547 F20101117_AABMPR sarykalin_s_Page_076.txt
cb1012e8e8408bd2c5b7ce41ddd0cec0
ecd1cb563ed16bb080da6f47c6dc464747f49a8b
2210 F20101117_AABMQG sarykalin_s_Page_091.txt
b0305bee458f64db81d36b1700267cf8
722e6ea77b26a00188a577283fb4920b50cc06ae
1184 F20101117_AABMPS sarykalin_s_Page_077.txt
d890a4427e998b721e94b70cbdfea5ad
1c8cbe3ad6d5880fe590fbea18fd1ab6f65ed081
1863 F20101117_AABMQH sarykalin_s_Page_092.txt
39784bcba9b891938dc5241ebebe22e7
e7d91ad3e7b4681a0e2c6e700eea7534cd55a792
1993 F20101117_AABMPT sarykalin_s_Page_078.txt
7cec726164e6fd9def50136ade02bb2b
037fcecb2927de4e66d5e148353c1e15097f2e44
1605 F20101117_AABMQI sarykalin_s_Page_093.txt
8f46a9cb6b5db38a4fc0d03026b37729
c4d3944fc9929a2317abe8f01300b3f9251776c6
1740 F20101117_AABMPU sarykalin_s_Page_079.txt
4808578327b84478ecd46cc87eef5f04
50ea0019ed47ea44e34922b1a71ad5ea2922afde
1494 F20101117_AABMQJ sarykalin_s_Page_094.txt
5656b87ece65d2faffafbc8dc2cc78ac
f2a82eee3b557360958de1db145c7a5c29183b21
1888 F20101117_AABMPV sarykalin_s_Page_080.txt
175e12bba3a782912e7c09e50fe3dffc
2bbe9b5ff1b37e7d6384dad36d2bc158ed523d88
1976 F20101117_AABMQK sarykalin_s_Page_095.txt
323ca65a48d923d4126f6989f3e5bec5
cb7ab5491c5ffe30cf509078bfc4049b89c7b268
1752 F20101117_AABMPW sarykalin_s_Page_081.txt
6667a93243abc32794b87f7838b76dd5
72cfc0871bf88e9269c7b6a79ede8fb36b5892f4
2144 F20101117_AABMQL sarykalin_s_Page_096.txt
a0efc7285d18c2564c8c05defcef8e5d
facfe79c4227945e0ab01b21a0963a9132a4583e
2129 F20101117_AABMPX sarykalin_s_Page_082.txt
cedbaae1298c9b918280eec2bbd677b9
36eda8ca0a92eb0b781524952fc2595a89261fa0
2063 F20101117_AABMRA sarykalin_s_Page_113.txt
ed62902aca0a29a59a84bfadc9165e04
226316402f1ea178feed3bacf59beeac53380c5c
2051 F20101117_AABMQM sarykalin_s_Page_098.txt
43bf4635c466ae37db9d372858679862
aefa0fb2ff69557b89d1f10f35016d8b4a1c814e
1936 F20101117_AABMPY sarykalin_s_Page_083.txt
64a148f0e710a3f87c0931aa49388f4c
a5420c9d6ccdc0f62d0adc495a4fe3926afecd97
2148 F20101117_AABMRB sarykalin_s_Page_114.txt
b8ce6a7596777b14a3dd1db05e2c733c
0336f0f4137a6f5f5d82c1e382f87cf6c0e6402c
2002 F20101117_AABMQN sarykalin_s_Page_099.txt
fb0e41f6b2d32446ff32a82cc10b9802
ca7d8fdcdb5d69f66664df0aa7a18551d82a3988
F20101117_AABMPZ sarykalin_s_Page_084.txt
0181ac0c880d65f8321a63913b18aee8
b7d7b19ae30e9f2d909069db5f5d594465ba6061
2348 F20101117_AABMRC sarykalin_s_Page_115.txt
969879004e9361e66eecb7acbde52452
969970721d95f3f7bdf72d2fbb5ad877a0a6e7e2
1705 F20101117_AABMQO sarykalin_s_Page_100.txt
0068f297b1281b55a0b0b2a5a17b18d4
87ca963aabf0d527e714ec71ec8dbbc6f808a9e2
1757 F20101117_AABMRD sarykalin_s_Page_117.txt
380dce5ec047a55954e37caaa7f54a28
2d97dbf81841f6cead5043da417626d2d1b66256
1222 F20101117_AABMRE sarykalin_s_Page_118.txt
8b7839eca8aa0e807eecae80067e232b
0aa0c12a1e6f48252d75d0fe892639ac76cdd340
1729 F20101117_AABMQP sarykalin_s_Page_101.txt
56701c7f4248d0e6fa302bd43b135403
dbf8805afb9cd19510ca4b5580228280cb92f2f5
1747 F20101117_AABMRF sarykalin_s_Page_119.txt
03e967932ae48a5e9b2b386db1387222
ea94b5a4bddfe607bf722188be45b4508fc8c1ce
1516 F20101117_AABMQQ sarykalin_s_Page_102.txt
b5d3c983d60cb770d28e6fb3aca88967
94ea134356f4a86414819b46dbab3cc0f3752386
1512 F20101117_AABMRG sarykalin_s_Page_120.txt
c6064cb7b94b4dbeb17abb7c6cd79603
3590ee0050206892e734a26eee3c577030042d40
F20101117_AABMQR sarykalin_s_Page_103.txt
90fa0d6701ebc54a0e7b66079806275b
6e6c477c71c8ff2ba2f400f70637da453b7ddee8
1419 F20101117_AABMRH sarykalin_s_Page_121.txt
86b8f5ee319c296d6aab42068018fa0c
1ab912c019968e663f9508382b41ddd6f76056e7
1024 F20101117_AABMQS sarykalin_s_Page_104.txt
5a3cc8b2fe5d3f21f1fd85777190c9e0
7141d9815da1f566fecd98b013b7d340e44c0ff2
2335 F20101117_AABMRI sarykalin_s_Page_123.txt
6a306fae3a3f55ee0c5a6942e1224409
4e8efc51058c7611144680f4a3493e6cfa7333a1
1631 F20101117_AABMQT sarykalin_s_Page_105.txt
4af596e35cd452eb26bb420af3f7a0b1
9b241158db230a083954e43885ca5bd8d57facbf
2054 F20101117_AABMRJ sarykalin_s_Page_124.txt
45111fca31ce55c27a0531ea6dd8b26d
c8b6e188a3af5876382dbea8a61ba1f9cebe56d2
F20101117_AABMQU sarykalin_s_Page_106.txt
03a185464391e03014d69b439ece0bbe
cc6891829f261f36604f14612a9aded4d7e4952b
2217 F20101117_AABMRK sarykalin_s_Page_125.txt
e0200f11cf8bd7bf20ac7e9b285c0fd7
76e43e5cff0882ba7b7f02f8611163e858354a3e
2387 F20101117_AABMQV sarykalin_s_Page_107.txt
6c43aae1bac22183c7c9d746c39555d6
cdffff6c5e275f986fa513e4e85639711270f7c0
575 F20101117_AABMRL sarykalin_s_Page_126.txt
af0fbb09e387b79279316d9d0dd3c18d
3f3c3a441c4e32da160058f0b8f619e4e1a92de6
2322 F20101117_AABMQW sarykalin_s_Page_108.txt
edb453d87b49acc1cf9eccabb01aea1a
464ba004c28bd23e061c469ed96d94f3ba35514b
2136 F20101117_AABMRM sarykalin_s_Page_001thm.jpg
c299208a3840a85c78fb391fd2b45d21
7d2c1a1322182d54934783552f3dcf579209c650
2031 F20101117_AABMQX sarykalin_s_Page_109.txt
2ddaaa4f1db7fe597715f53661441dc9
b9142cd9aec3cf2cc1d206ddfa6e06a7ade932f1
16572 F20101117_AABMSA sarykalin_s_Page_008.QC.jpg
1780c8cd397c495b6a3e9c21dc676f8e
d61836c10c7c6cb011767fca980272ea605a7a63
825430 F20101117_AABMRN sarykalin_s.pdf
0f7dfd8e9dbf71bac53b77489381a17b
26037ee74ebfdf660bb8adac0c9a7b79eb1f2885
2001 F20101117_AABMQY sarykalin_s_Page_110.txt
03251563ecd618e2c8112139ebf39ed8
6872f4d51b98533e9766bf203fbfd06bbb0f0bb7
4326 F20101117_AABMSB sarykalin_s_Page_008thm.jpg
6913a05ea9552c8ffe374de34db3ed6f
8d9f6d5fc8e91f04bf7b5436c7604f9526edc8a1
6731 F20101117_AABMRO sarykalin_s_Page_001.QC.jpg
822a5942a0c4a7bee04c290e712ce43e
8713e1799ba5d3836ee25431b71d22aeedb6e82f
1912 F20101117_AABMQZ sarykalin_s_Page_111.txt
ae1d4e9754894b72622b5f3467a6a135
29ab5a442d7055a046f75c924ff5706c105b8204
19703 F20101117_AABMSC sarykalin_s_Page_009.QC.jpg
7abcf449ab68655883e6488b74535ff1
4d906d2123f86e0da55fd821e7846e027bd1b888
3245 F20101117_AABMRP sarykalin_s_Page_002.QC.jpg
fa2645a797fc8444f0460f3c986e207a
12e7987bdc5a3986d360f7962f57c67d57f411cb
5344 F20101117_AABMSD sarykalin_s_Page_009thm.jpg
2d7a93d4c24ce45f4cb8e5c59c408d3b
2e41a3578eddb7f539cce1afc31e07810f6900d1
7170 F20101117_AABMSE sarykalin_s_Page_010.QC.jpg
f378062b0259bea86ec2a21d461eadac
9893000e27bf3f4859e992f762bafc726074a2fa
1359 F20101117_AABMRQ sarykalin_s_Page_002thm.jpg
dcfc2cba739596dfa051cf96dccef560
08eb95c767784d2b28045805040f75afd5613e9c
2327 F20101117_AABMSF sarykalin_s_Page_010thm.jpg
5a9e56cfb6ddde9bdce08e1b41eadb51
6ab07e9f85652b9adb977278d84b99d41f26fb63
2943 F20101117_AABMRR sarykalin_s_Page_003.QC.jpg
d05a78b1651f464b262ddaab71e6db78
cf6555a10a336c0215312afc15d7b06e5b646362
23390 F20101117_AABMSG sarykalin_s_Page_011.QC.jpg
4ddd47b152c88ab5221fea22e3b6d902
d7cde35920d73f31cd668bd7eb8e89f6911983b9
1331 F20101117_AABMRS sarykalin_s_Page_003thm.jpg
546f504907686db39a8987015d58569e
b082a1525b18ab55ad80d5ad9280521ae077da88
6030 F20101117_AABMSH sarykalin_s_Page_011thm.jpg
0c50285c0e9fcd3e86f3230a97b928bc
e92c0472487446cbaeffb253fca7f030f5bbcf89
8570 F20101117_AABMRT sarykalin_s_Page_004.QC.jpg
64070f72daaf5b3e0aef3f890cb7eab2
97e40c23703cb7680a1a53a6544104d31591b22e
20263 F20101117_AABMSI sarykalin_s_Page_012.QC.jpg
b330f4c57412286e90ed563a798b4359
f86bc61890086e2a1c384456efc7cae4520188fd
2586 F20101117_AABMRU sarykalin_s_Page_004thm.jpg
ef6a6a4e62d8a621c1e02b35721569b3
33c9247f499a477389e279a73d772067581b34ef
5325 F20101117_AABMSJ sarykalin_s_Page_012thm.jpg
b5a004d75c9742905dc501c53ed0d791
26fe095a59e36027a7c7c3324f5e299477913321
21437 F20101117_AABMRV sarykalin_s_Page_005.QC.jpg
0ddb1a3f48f86256f1318739c107b2c3
07c7ea49aa7a8ec635823127b772397b0060e2e3
21771 F20101117_AABMSK sarykalin_s_Page_013.QC.jpg
c0ec2e317b36a90bea024d573a6b94b9
239706b52bfff0768436b33ca74e75240f6bf907
5202 F20101117_AABMRW sarykalin_s_Page_005thm.jpg
a60527c95d25b0d35af829101bbaa2dd
fca3dce41464937f90bb11a5dfc0ad04e1847017
5556 F20101117_AABMSL sarykalin_s_Page_013thm.jpg
10c62c0e12de60097853589dae4de6f8
f88441bf89e9629742a899820490d5c36fae0778
26045 F20101117_AABMRX sarykalin_s_Page_006.QC.jpg
10e14b9aea5f153e7596a43f6ae54ad9
dfc872c63c0c14b6644ec0026e9cb433d5abc216
23637 F20101117_AABMTA sarykalin_s_Page_023.QC.jpg
e7549bdffa676bd071a0661ef93be8b3
8f3b957e1e31664bd1fed1ec92a4e7321f9ac207
24375 F20101117_AABMSM sarykalin_s_Page_014.QC.jpg
4e0e2e0bfe9bc72cba9984116c5a8cc1
c221c0c99f08544efe7981d991958a87dba4b349
6247 F20101117_AABMRY sarykalin_s_Page_006thm.jpg
108f192df909dd456b55feb905546449
2685a2df1245002c0931143879dc58fd300f31f8
6207 F20101117_AABMTB sarykalin_s_Page_023thm.jpg
f896a5619ee7cb2881cc0917ee7de4e3
8eac72a65454777632ada0785b13617939e5c7a0
6045 F20101117_AABMSN sarykalin_s_Page_014thm.jpg
27fd4da845ef8b660c58034356c4a4ff
8a6364660108b350015d027fc5f28e6b4e33a202
4325 F20101117_AABMRZ sarykalin_s_Page_007thm.jpg
f99e2f01e0457f8389dcd7288010b58d
1c8d376f6d4c67a9eced7756fab2f3dd2f415c0d
14970 F20101117_AABMTC sarykalin_s_Page_024.QC.jpg
e4f09cee9b050f4db13f4aab4746e843
06bd547a58555fbffeb0189b9906fb28dabb0b6c
17197 F20101117_AABMSO sarykalin_s_Page_015.QC.jpg
cff1f5994d28bd55bd83ed8cbff5f8c9
15308127970f0316ef8fd5ef6a784adbe3cb28e0
4648 F20101117_AABMTD sarykalin_s_Page_024thm.jpg
eb4204e5a3b718255775f0643d339f9e
704f043485798f9d383bfdfe23b66260b9b2c0f8
4971 F20101117_AABMSP sarykalin_s_Page_015thm.jpg
8821793319c660c76ec6c6b938fbb358
bf5483975cbd13f3dfb4655ab38a1326e5da7a49
19427 F20101117_AABMTE sarykalin_s_Page_025.QC.jpg
05f259a7a93d40eb7448872275523894
625ea442b03ef6cd096318429c7d09a17b52994f
21866 F20101117_AABMSQ sarykalin_s_Page_016.QC.jpg
51a120273822fdcb60d02d951bfc9401
785ba7b2ace8f2a896a8fe2d0119a0ca79e0f13b
5367 F20101117_AABMTF sarykalin_s_Page_025thm.jpg
7b0cb7b68b9ece70b060a60f18504821
af9f2cb5592c66c86856d6a0f9db296947d2366f
25758 F20101117_AABMTG sarykalin_s_Page_026.QC.jpg
863150089fdeafb7607978f74b630014
833998f15330d910ccb83901da2ed7e0b9904b65
6064 F20101117_AABMSR sarykalin_s_Page_016thm.jpg
2147108f931b181364b94b52b43f156f
db66061a48eda76991bb69f363b89fb09fa68e93
6716 F20101117_AABMTH sarykalin_s_Page_026thm.jpg
da4b020d77a3406033b055978858382e
e2b71c6dd75e44a8de604924fdbbc1bef99ad780
18599 F20101117_AABMSS sarykalin_s_Page_017.QC.jpg
3c0e4887d21d026c8b8abbb97fe7553d
0eddab9c4c3259ad89375a122c63f8b8aab1fa58
13944 F20101117_AABMTI sarykalin_s_Page_027.QC.jpg
5e5fb97ab8bede4ad39ebba6837926df
be29e7e4e2c0164e32e71a5770642f952e320664
5182 F20101117_AABMST sarykalin_s_Page_017thm.jpg
75a7a47b85502c8058b2f746f4f47671
5d8ef7744a4490007f3b1f7fbad77c9ff4a96579
3887 F20101117_AABMTJ sarykalin_s_Page_027thm.jpg
1e033c068e834cecbb83cecfd4f3d4ba
a707d3043821bb559c3d1c2b0d4e8695b8494b7c
16894 F20101117_AABMSU sarykalin_s_Page_018.QC.jpg
08dc4ed886cec5d6d64995d6e085921c
717977f9f6b017199ee7c445f5fa17da7cbad505
14045 F20101117_AABMTK sarykalin_s_Page_028.QC.jpg
2387265ef04a8b6b104f95eaa94a9df7
9c5af8642777ca116c8c20b08e5808ad30f37731
5088 F20101117_AABMSV sarykalin_s_Page_018thm.jpg
f32100a89abb9ef5302b28aac2324b82
373c86b88e2518ee36c060f6c7e0766eeacc7254
3994 F20101117_AABMTL sarykalin_s_Page_028thm.jpg
ea4b339df56647eb46e660147ebfd46a
067e601133740d770b3519a6aae4a44055ca79c4
15161 F20101117_AABMSW sarykalin_s_Page_020.QC.jpg
5c5c61a98449d5930cd3e3853c0bb66e
1c2587d0034208f9b4143dbf36be95f1ce7c1065
21468 F20101117_AABMUA sarykalin_s_Page_036.QC.jpg
ae88735d61e337b5362c91358727f2e9
e1c66f5d3ca3e2f9d48c5ea2b49f7cddb30bff58
16422 F20101117_AABMTM sarykalin_s_Page_029.QC.jpg
3b505eb00ed01debdc5e9d681dc3572d
164e0a02111a667513d68ade8a2cec714ed99269
17904 F20101117_AABMSX sarykalin_s_Page_021.QC.jpg
e1fbda5f092ec7a94710b083c643cc62
9a2e7ad9857843ff0779b8e5030768d2a9996273
5390 F20101117_AABMUB sarykalin_s_Page_036thm.jpg
2c77b6f02177561101c0ec951b5b6656
bc5a26609dfe1233358f1578ced3ae6e48476424
4651 F20101117_AABMTN sarykalin_s_Page_029thm.jpg
47956bbff59242480d2f6caf63accbd8
5323cde88661c87453b6315c8ab21dba01bec31c
5106 F20101117_AABMSY sarykalin_s_Page_021thm.jpg
04d5ca40e4ca411c30b710494fba2562
9d6618dff16ab5d6d4c0738b8d733e516b0eb68a
20396 F20101117_AABMUC sarykalin_s_Page_037.QC.jpg
1a43305c0e58f18bad621b4e519954bf
9ae3e7fa1b51839918b3706c34ccc7005e1b40f6
31535 F20101117_AABMTO sarykalin_s_Page_030.QC.jpg
e34c1993922123a781a963bf55478a4e
678c9575e906b9b13c138d589a26d4575ba1c98a
16796 F20101117_AABMSZ sarykalin_s_Page_022.QC.jpg
8178f61f6cd17729dc49d61e3b2fdea8
71d9822c40fdfecc9ae4a150c40eda1ac1c3ecc7
5634 F20101117_AABMUD sarykalin_s_Page_037thm.jpg
ee383651fba6519ac67dc3c6730c9dff
1eae23d17f2f1d50b3e53cac0b8e67a9218483d0
8984 F20101117_AABMTP sarykalin_s_Page_030thm.jpg
d272427a697cb71e89a349cdd30f25d0
371e918f2e2ff396d541d16c875e186f250f5c47
19334 F20101117_AABMUE sarykalin_s_Page_038.QC.jpg
17ba88628fd62741d96d70e81f8adb44
5b2229e0ffc25c3110cb1bac3d7cdc09534269df
7772 F20101117_AABMTQ sarykalin_s_Page_031.QC.jpg
9c88966f8edb9a87a81c2af758f0b57f
30055e3809eddb59d141de22b18bf0490dbdf587
5162 F20101117_AABMUF sarykalin_s_Page_038thm.jpg
fb99db882e3513345842e6eefa8d041a
6b7bd77750d6955bb249c78050588b3c4722878a
2543 F20101117_AABMTR sarykalin_s_Page_031thm.jpg
af5901c10f80ebd4b0408b570fdfc14c
800c7920eb3b8eff606c24abe7e7949f6c14974b
16251 F20101117_AABNAA sarykalin_s_Page_118.QC.jpg
faebab0b2a21f614589fb36f906591d4
7cedb3fbabeb1cf29d14ece8ce6bc742104464b0
18111 F20101117_AABMUG sarykalin_s_Page_039.QC.jpg
ef93f3501a100d763cea27a575013fb7
59f03107bce4edc4b2a36da6833fcb7f02c057a3
4890 F20101117_AABNAB sarykalin_s_Page_118thm.jpg
f894798cade8d90076a2b9faa60b1558
4cde5815c2adf6bb0b5a8dce54fb14042d62b89b
5119 F20101117_AABMUH sarykalin_s_Page_039thm.jpg
ad1c1375e69cf950543b1938e9441e2b
57f5f393bb39185103c37982b5ee52876b520059
23105 F20101117_AABMTS sarykalin_s_Page_032.QC.jpg
73a7f6aa063792baf2fff4fc12383ffb
c858de0d9ca6c9497020d3f1ebe4c72859c341ed
19961 F20101117_AABNAC sarykalin_s_Page_119.QC.jpg
d3a2dfb1524ec075a20c377f40968019
aa636306989ba2d07246b7c91d26ebb1d3b22e2f
20606 F20101117_AABMUI sarykalin_s_Page_040.QC.jpg
658b0782835f6a68f08a559c886f7c6d
f7170d238db7696a3d27815cadb134cb05a2a789
5839 F20101117_AABMTT sarykalin_s_Page_032thm.jpg
7b33dbea22f42b861aec3d3f21c6fcd5
3a9cbb3f4c6f7e084775fed6555de499813b89f0
5504 F20101117_AABNAD sarykalin_s_Page_119thm.jpg
e4126615180b6086ce1faab43f1480f6
96b9c51e3fd37229e09365ed84fe4e7f58a52921
5584 F20101117_AABMUJ sarykalin_s_Page_040thm.jpg
ae793741260393ea506debb9e2b6d0e7
6eac3db0c09bf939478dde52a0e8e1a01de701ea
25174 F20101117_AABMTU sarykalin_s_Page_033.QC.jpg
49657976dd4c6883451d756c2821c225
3d7c7bd3f08b11e3a5d44480cd7c5edfcfdf73d4
15958 F20101117_AABNAE sarykalin_s_Page_120.QC.jpg
fc38bc49e6fff40a62d2c21c48d62f4b
8a0c1af5de3d8a74d847f0207c6cdd3ddffc1c22
20758 F20101117_AABMUK sarykalin_s_Page_041.QC.jpg
ab0d7b4dc19ab8af057df3695a63c920
9a41cc10629c765224d198bdcaa919aa73c8b448
6378 F20101117_AABMTV sarykalin_s_Page_033thm.jpg
4954b65551e338c9890251034980fb79
79a0ccb9a05b51a09dd3d90771557d5328f7dec5
4882 F20101117_AABNAF sarykalin_s_Page_120thm.jpg
60d5ca067546fb7b6b4b20e3122720b7
25866c0f55a29b708911fae914c100b482d7fa65
5505 F20101117_AABMUL sarykalin_s_Page_041thm.jpg
f33691c2fdc01ddd1fa64dda5834f9b5
c024344163dfe12ba90afea9dbba5ebfaa09d7f2
24916 F20101117_AABMTW sarykalin_s_Page_034.QC.jpg
65bb62facf940a78ce151a53d21cd819
a0a9fa0c063748c3247737ffe4694a877af50df3
14505 F20101117_AABNAG sarykalin_s_Page_121.QC.jpg
592c9a782dea6faf5ea14728653975f8
8b417774207824623761f9a9df7dc77b408cc223
21001 F20101117_AABMUM sarykalin_s_Page_042.QC.jpg
bc010f7eb63e69c5c4cfcd96569276a9
c9eaf9996fb69f720dbd16cf3073cfdc29fb1e80
6301 F20101117_AABMTX sarykalin_s_Page_034thm.jpg
6c5cdced7085ea62fdd53969c5e120b4
72b9a8d2859f5bff0ab4710e38148f7ed32c3c4e
23393 F20101117_AABMVA sarykalin_s_Page_049.QC.jpg
a0b3361a4c7473c9064905db2677d58c
d33bd3729f42279d13466e6d347ad0a36993855a
4296 F20101117_AABNAH sarykalin_s_Page_121thm.jpg
cd5d36b177890a8af7de05656dc993db
750efab57b32cd22a63e7acdd8f2f901497b8161
5726 F20101117_AABMUN sarykalin_s_Page_042thm.jpg
d693bbc349023c67d1240a1eb1139b59
9155b1f43fa41ed5d76b4d57863e6404cc8d9e4c
26390 F20101117_AABMTY sarykalin_s_Page_035.QC.jpg
bdf1a4a25fdcaf4759515e91a6a0437d
e3c82feaf0944532ef6af3885adb5ed808d10264
6022 F20101117_AABMVB sarykalin_s_Page_049thm.jpg
02f9558fed0d6ce1742175f367301b5e
b17e18d521ea1153ccef4eb9ab1524176b4c9bb5
22446 F20101117_AABNAI sarykalin_s_Page_122.QC.jpg
804905d162a5614c941d627d9153778e
8ec2244ccbf68bcfb028e447463e36d178e7b21f
14171 F20101117_AABMUO sarykalin_s_Page_043.QC.jpg
998f6d27ed320a83713a2c97aa66152b
3e6fdc6a53f23ffb3361b95589dc9e0352e3888f
6461 F20101117_AABMTZ sarykalin_s_Page_035thm.jpg
67ceae2778a3a46a65b8f18720a24c17
d1d46fc20018fc53dcfda61569747135c70ff060
19411 F20101117_AABMVC sarykalin_s_Page_050.QC.jpg
7468ade37f37f295c46f6c4753106b90
e1de75493dec387a2ae6ac6ce18f0aed5362c8ef
5892 F20101117_AABNAJ sarykalin_s_Page_122thm.jpg
09a3a95a5cb80fcd5d2a322976e8af7e
ef53eabdf5b50a75a9dbc9e5a260896229803281
4384 F20101117_AABMUP sarykalin_s_Page_043thm.jpg
5c63f28ec7b6ff9eabf3462463a195a6
861e6fd3a0436054d07278662e9cd139e3c20101
5384 F20101117_AABMVD sarykalin_s_Page_050thm.jpg
fc26e024f00245f919a89ddda442fd1b
2c7d5518101ca625ee3dcbbf1446cc78707eb418
23924 F20101117_AABNAK sarykalin_s_Page_123.QC.jpg
8a9aab99349613310320242b8304ca3c
bc6a0684b81dc9cbc56c0fe56ebac8a08d322de3
23596 F20101117_AABMUQ sarykalin_s_Page_044.QC.jpg
80bf95d7fc99abc42293087cf63c740a
88d8f5753656ca81167dc41a82495d111dd42c56
19616 F20101117_AABMVE sarykalin_s_Page_051.QC.jpg
23b64571e631643f3210d26df6d127aa
a5d7d524f24207570f2312db19de0d33a7d4758b
6199 F20101117_AABNAL sarykalin_s_Page_123thm.jpg
39c22e61941be8d56888c9ae07a7db85
d1ff6751e41fe9d966897ba56857ce62d9ebb817
6349 F20101117_AABMUR sarykalin_s_Page_044thm.jpg
3683d70cf7e07341ea3d4f3052714c80
15b053220c6672483a263fb388b150fd5b233cf9
5361 F20101117_AABMVF sarykalin_s_Page_051thm.jpg
405ff6390dc29da3e98b80e4362d398b
e0af7060ade297be8a54a79b97069745f2ae08f3
22234 F20101117_AABNAM sarykalin_s_Page_124.QC.jpg
0d0b12a743422478b1c73e105995ef13
75a94fd56f0f113b51c9a403f72a1e4f21daac38
27740 F20101117_AABMUS sarykalin_s_Page_045.QC.jpg
f559ced6337450ad64ed5b2f944ee784
38433688cc05f28da886eb71800654c3e0fb3c61
19626 F20101117_AABMVG sarykalin_s_Page_052.QC.jpg
987dd92e3016ce18efb9e091d44af29a
b64402cc0a55c159905de948f199a130a019a612
6112 F20101117_AABNAN sarykalin_s_Page_124thm.jpg
aca1af685594e1f35830b787111d41aa
fd42d0769ad63e3b878fa5f79379a51e52bce8f2
5362 F20101117_AABMVH sarykalin_s_Page_052thm.jpg
15961b3913fc0b2cd93ef0f5ae9efc69
ca66a3a00dca3ceca5a95d892041b07eef93e250
22610 F20101117_AABNAO sarykalin_s_Page_125.QC.jpg
e8f4e3d95ba0f91feb21e0a299d7dde0
73358dbe83fe7fb53389fc1f3b25b1de98c123e2
6899 F20101117_AABMUT sarykalin_s_Page_045thm.jpg
8e120a5e5eb6706da8c8385a5aae8297
3954ac4ecbfb4751e7268a5b16ce4d9f1fef7119
16205 F20101117_AABMVI sarykalin_s_Page_053.QC.jpg
2f5b7496edb122da4e3af47216498adb
acba83c0e96a745b5aab1ead96435aaeb62b80cd
6053 F20101117_AABNAP sarykalin_s_Page_125thm.jpg
9b36652e47c17b7d94c2976c9150d963
6c8af9dcef625d2714d9c5b15705683888bd177c
15295 F20101117_AABMUU sarykalin_s_Page_046.QC.jpg
d6c98d9a45bfb68aa6cbcffe3e0b98fb
23ef93748420081314de0b4a9cbf602be51c6f52
4508 F20101117_AABMVJ sarykalin_s_Page_053thm.jpg
5338cf9fab27f8d0e590e06c7b633f2b
02eaa23d27beae335a617708d443c62313840676
8184 F20101117_AABNAQ sarykalin_s_Page_126.QC.jpg
5c0bd994750eb53ce717e3cdbd221275
ab1e99ceb1899dcda671d76f7d03647dc2499ad0
4839 F20101117_AABMUV sarykalin_s_Page_046thm.jpg
7acc2b58a4e9c5c963bc521c04115329
4a46fd3b0e0abfb35f2911e11db3b3dd99307825
15212 F20101117_AABMVK sarykalin_s_Page_054.QC.jpg
b5c6fc9c8f278ebf032d780acdc99ad1
57ba6de64aa84427ffd08055e3b83b6a2d49e8e2
2490 F20101117_AABNAR sarykalin_s_Page_126thm.jpg
378dc28edcbaf029d5856939f381c45a
816b1f15f6723cb032ea960fb04825df3fc4694d
18425 F20101117_AABMUW sarykalin_s_Page_047.QC.jpg
6e1bca1077510b2342d5439ed85696f6
7a22de98a96e3721c430684a92997a3277ea55a6
4383 F20101117_AABMVL sarykalin_s_Page_054thm.jpg
2abbfbeeaceea3403dd943a455e7792c
4db86d98472e77cfad1e6f1d92080842ffbf1a88
147341 F20101117_AABNAS UFE0021770_00001.mets FULL
ba9302419a310ebb9c10022f362cce91
e779942f35bb4bba3439038b9277dfacf0c44e4a
22325 F20101117_AABMWA sarykalin_s_Page_062.QC.jpg
0ed8161899c7493a9f02a28955d613d4
49fe35f507fe0a3e4adc7ed08f9620104455adc2
20294 F20101117_AABMVM sarykalin_s_Page_055.QC.jpg
8cccff39b20bd81d8d80af490f3eec1c
e543162e3bfa21fb6be004ef28747b6699af4f8f
5100 F20101117_AABMUX sarykalin_s_Page_047thm.jpg
73349567b48e81366f2c79dd21823a68
ad2fa68bfcf799ae0a8df2858c6c60c7a4c57023
5628 F20101117_AABMWB sarykalin_s_Page_062thm.jpg
a1020f12b37d4ce73238b04f4cb6432e
d5c67c5a388cf7f5a59088a6398850753fd37a06
5635 F20101117_AABMVN sarykalin_s_Page_055thm.jpg
4003943c6d6f50b985543f75938e83a0
9cee9b7676157f3ca5d499d60c0ab002de3b2b86
21813 F20101117_AABMUY sarykalin_s_Page_048.QC.jpg
fc011d15b2c2d9675700169c52628ccf
2871a00bd29fc088369c04160e9de52911f8e9a1
2200 F20101117_AABLTA sarykalin_s_Page_116.txt
068c341b28bd73c112a299d92f2d11f0
d0991a22690608d12cd4a32f4ba26081c9cce94c
16277 F20101117_AABMWC sarykalin_s_Page_063.QC.jpg
95a9ee6eb0224e161d79df736e4fb875
0cd0940ddf444db39f2b926241cc661bf51d343e
17540 F20101117_AABMVO sarykalin_s_Page_056.QC.jpg
05163cc6bed8664aef1b87c84b97c053
f909fb087311d59a16c632b7394a5e3312434f8f
5637 F20101117_AABMUZ sarykalin_s_Page_048thm.jpg
f5e1cae67f251000dfb5f0889cb37f84
27b35d3938409ef226d72a139c418b569225149b
5848 F20101117_AABLTB sarykalin_s_Page_110thm.jpg
da65093d76ba7d79fc94eed58682bafd
ff39bc518de4e00a7c4596dee29c86aed20fa494
4280 F20101117_AABMWD sarykalin_s_Page_063thm.jpg
26236dca35027e04f0039fe6154987db
153623852844358658a625137ee05f01f7e46144
4853 F20101117_AABMVP sarykalin_s_Page_056thm.jpg
2b30e2f77fc3311eb002a359b6196f60
3ca2abd12574c505fafaf3d24c7a9ff466553424
30555 F20101117_AABLTC sarykalin_s_Page_121.pro
8a890811c65d84ffe1b2d377576efd43
6a76893be48feae40860297ae603f61761ed3968
19462 F20101117_AABMWE sarykalin_s_Page_064.QC.jpg
6e3d1158294b52052986b87c7c82cbd1
9ddf3b065448938587268f70a40d629d0f4ed699
23715 F20101117_AABMVQ sarykalin_s_Page_057.QC.jpg
1f8daadf4063f73be09e449b2a5e0dcb
32fcf85c882cdfd773c268ff317fcd6759e3ac45
51839 F20101117_AABLTD sarykalin_s_Page_096.pro
a699b821575d91c54dea199c03622c06
5df06e9c30c038e439190a1002e392db9534a1f9
5553 F20101117_AABMWF sarykalin_s_Page_064thm.jpg
d9d2f7ccceab73f9014b1324c27a8974
68066e335f28f8643874381973fc126f0bddfe96
6109 F20101117_AABMVR sarykalin_s_Page_057thm.jpg
e882a906274eb8b4ae633977f4f009de
1efb25f76f3650f6274900a444da9e1b8887e7f7
F20101117_AABLTE sarykalin_s_Page_026.tif
7133e7fde46477843585e83ae6fd44b6
70b89f928884476c2795d537f4937b214494d80d
18666 F20101117_AABMWG sarykalin_s_Page_065.QC.jpg
086ed94e24b0ae9603b9847a6afb7306
1301fe5ebbaa179ec2f02d0adea0c994595f57bd
18754 F20101117_AABMVS sarykalin_s_Page_058.QC.jpg
616c27045b726f3a6b13eedfa4fa07eb
382d312d1161df298a6aba414938d39c0389c003
F20101117_AABLTF sarykalin_s_Page_047.tif
6ab063666998d63c5210111a8f7446d1
7ff4b3f96a2e535388b3acdeab5b59bd0fe639a3
5426 F20101117_AABMWH sarykalin_s_Page_065thm.jpg
4b2af098b926828fd0e1dc1200d04ac0
944e3b246a66306c9af90f2fa2835c2939145259
5116 F20101117_AABMVT sarykalin_s_Page_058thm.jpg
a943b99c0eeab07db8ad865b117c5bcb
c45e90071f4ce97bce33f4a582483d1bc5e9d064
5420 F20101117_AABLTG sarykalin_s_Page_019thm.jpg
20e3a9764e0d8ac6a48fe93cb73563c2
775d4138dae5608f018af621554c8ed0a3c1e1ce
15112 F20101117_AABMWI sarykalin_s_Page_066.QC.jpg
6e2c57e916639321be1f0af8f8d8bb8a
0df8df7bb0edff658739868ba1c3f70aa50c4378
1642 F20101117_AABLTH sarykalin_s_Page_029.txt
8bb56b1496f88ced8b8c70089f100f5d
3b3e75db814954834c924ba9baa42e2371f0b734
4696 F20101117_AABMWJ sarykalin_s_Page_066thm.jpg
988a06f912c2af5e7695e11f3a746807
306dbb07f4a0355b787eb5addebd4b8c5db23dfb
67523 F20101117_AABLSS sarykalin_s_Page_028.jp2
b2b91e9212a53fa1af27ea0cb8f74e2b
cdc959d6ebecf0dc6c946f0bf81d481235c7b4e6
26375 F20101117_AABMVU sarykalin_s_Page_059.QC.jpg
72841418aa1f5ef7ebbedcb34c66ef21
b3c345ca2132256ebf6cfc9a90efe4e9f230517e
18990 F20101117_AABLTI sarykalin_s_Page_071.QC.jpg
62b3eb6cc379126cecd5810c42398d65
39fc6824be615751f9c6cd9117d97268a9c0f9f2
14691 F20101117_AABMWK sarykalin_s_Page_067.QC.jpg
05c374d0f47c129e863559ab14c26d09
0aa2e49155242d42f496e82a81c2fe0780619ccf
2000 F20101117_AABLST sarykalin_s_Page_097.txt
28dd326a7bdb634e596ce29b91e93ed0
873b405f1dee6f6e496193bc413d95240a0480cf
6906 F20101117_AABMVV sarykalin_s_Page_059thm.jpg
1203dca8fb526a5362ca1a03fe19c59c
afef9aa2f9c9da6c127d3f0f4c37d3dd282d9da8
67757 F20101117_AABLTJ sarykalin_s_Page_041.jpg
b2b80e7284e124453b251990d6217b71
b9343507a9adefee08521bcab2a4471aac08e901
4844 F20101117_AABMWL sarykalin_s_Page_067thm.jpg
e5d8a27dae917abe00c0debad48c8b59
3ca5fd4c159a61454d6391b8ff16d27413b8f46e
F20101117_AABLSU sarykalin_s_Page_014.tif
8b27ad0ec928c2cb08b3cf0876b6c9b1
aa818f77d6ddb4138073560a995f29a2ee7aafde
28320 F20101117_AABMVW sarykalin_s_Page_060.QC.jpg
663fe60874fdf4a97fe3b046c18dd87b
f95b159f9282d8aaa7e7575cea92ee2c7e865cc5
4612 F20101117_AABMXA sarykalin_s_Page_075thm.jpg
648d44f860ca84bdf71d419c2cd19fc5
59fca126d73e2eb0110ea4be7a1b573d1c2f0d4b
24937 F20101117_AABLTK sarykalin_s_Page_027.pro
ba71982abd8b5af5571330bda3f5ad83
d2612d3c10624a33293ee369968ba3c29f28cf2c
15851 F20101117_AABMWM sarykalin_s_Page_068.QC.jpg
b8b3f11f8310e8895b899e9a9c4ece93
fecfd20d386edb92378493b94c96408dfe169e6c
61247 F20101117_AABLSV sarykalin_s_Page_007.jpg
3b4122b9948b51eadee9e1788c868817
40fff3bded399437b7db7d5cdb6f8e7c3e1283fb
6916 F20101117_AABMVX sarykalin_s_Page_060thm.jpg
07261f1cbec54f15c5f22dd9d262f1de
d7abfda241ea3fd59dad534cac76b8e40965bdca
16115 F20101117_AABMXB sarykalin_s_Page_076.QC.jpg
a640a6f8ab361b10894f7500003c7069
038de8ce454e1abaaab3f6c7cea6bcb0e821cc7f
33449 F20101117_AABLTL sarykalin_s_Page_054.pro
e1ec657dadf3bc40ac1b2911a5dd3cf7
fc1d6c55a411c7620f2bc678582a71c4c65aab84
4248 F20101117_AABMWN sarykalin_s_Page_068thm.jpg
199f839e2d7ea7a2c1b0d379681f2702
02046bc31b497e1009c7f8dde0f99130d851b7ed
F20101117_AABLSW sarykalin_s_Page_085.tif
50268e6ac8aa4adcc00b2c0a3f1948e4
5dc009733ec35101b453f04fb8364b53cd2da70e
26957 F20101117_AABMVY sarykalin_s_Page_061.QC.jpg
724a2d78ae4896f845b4339579a1e867
058996a825be92f836ae08aa7a0bd7763365c341
F20101117_AABMXC sarykalin_s_Page_076thm.jpg
55701f83bb84281d559f9719f9e1136e
c3cbc442f00d129f3286f7e3bf5bd5cd75e49ebd
F20101117_AABLTM sarykalin_s_Page_041.tif
8222c4cfa40af01e5fb7c4b2af674c28
e138834ee39517b6aa8b632a4f3d533d78078873
13183 F20101117_AABMWO sarykalin_s_Page_069.QC.jpg
a2d35165c7028c9551b34ba1d5996c47
71bb7e49b35f2707a9d135a61e26c0d341169b79
F20101117_AABLSX sarykalin_s_Page_068.tif
7963df4233f5f1e4f93498d801af1a42
0d009c49e573ceb047030f00af4ceda84b500bad
6757 F20101117_AABMVZ sarykalin_s_Page_061thm.jpg
ab5068f83c2e8291e83680fd00f249c4
69a7aecea4664e164c80b908c638e40b3bf99367
37926 F20101117_AABLUA sarykalin_s_Page_100.pro
b4be7dbb6bb51ddb2addafed34c651e6
0d45415185e2d61bafabc2b67d327db196a9642a
11807 F20101117_AABMXD sarykalin_s_Page_077.QC.jpg
c009669c31d2b7bb2ca839e66f24e593
73a437048d36834da4747fd25daff0d1dfb9fa22
43917 F20101117_AABLTN sarykalin_s_Page_048.pro
c49f086568565088e1831fcc76552d3e
42ff55acf34746e48d3d9a421a24e633543bf886
3664 F20101117_AABMWP sarykalin_s_Page_069thm.jpg
7e1d4fe84d6dff700d1021b6d2d395d9
e132242af413f99c0d8ff40aad2ac6e04962e7a9
49162 F20101117_AABLSY sarykalin_s_Page_082.pro
3b23d0d32b406ecd96ba32be468cc421
245e69a5f3137123bfce4260e734ddf8aa0ce36c
12121 F20101117_AABLUB sarykalin_s_Page_104.QC.jpg
05a26eee76e057d9e9ca5f3b429460a0
f5acc206d85923615dff8f07f7ddd67f6248a970
3536 F20101117_AABMXE sarykalin_s_Page_077thm.jpg
b429bbc0b78402925b54f5cb6f6a47c6
b14089354a57b7dd89d71d6f10ccb2d876844386
43359 F20101117_AABLTO sarykalin_s_Page_037.pro
befbe25059b25540f0bc3f4353408da9
2988d0e4c3cd234cf57cb9cc26cf8ef59f853c73
19053 F20101117_AABMWQ sarykalin_s_Page_070.QC.jpg
cb50d7c4d3cbf234311d5c8135aea77d
4f51fa08351ec41a2bb9830ed455b1945f60f42f
851515 F20101117_AABLSZ sarykalin_s_Page_065.jp2
3e16b4a3e5a4833f7ee912b14c9141c7
ff7a55c2f45ce09577b637cc91f0b924474ccea2
51235 F20101117_AABLUC sarykalin_s_Page_109.pro
75ad808274d2c07e84caf5a7ab9cca99
b5d301bc123a6980550362c8a09bc77b0d720f4e
20769 F20101117_AABMXF sarykalin_s_Page_078.QC.jpg
e42c88f492a45fc3d98428efe296d8b1
2be2ce50b08740151a897ad0730c02639c025721
F20101117_AABLTP sarykalin_s_Page_032.tif
1335f1b2d422e3fff7ae68f52c22da74
720fed2e0df904d5806e7bf60c8d30bd49aa0282
4825 F20101117_AABMWR sarykalin_s_Page_070thm.jpg
2157871732b75a369634660d0b64d669
32bc2c430a2f2acc80cfba8bd1f9ad2a39fbb7a0
F20101117_AABLUD sarykalin_s_Page_035.tif
2c77978e4293065a626b7dba14dd7455
2a648f3bfe82d2a74a710928eba3311b86fdf47f
5679 F20101117_AABMXG sarykalin_s_Page_078thm.jpg
3dc01bdfc8178de12ce6f798ebcad021
e25cf6c350d6acfb1990aff424e6ed7e4003ed89
F20101117_AABLTQ sarykalin_s_Page_001.tif
7cf4f2cd2808b952c7e9db78fd4566cf
d67c5ead0c356e497617a5b8da8b98b9b6f26bfd
4820 F20101117_AABMWS sarykalin_s_Page_071thm.jpg
64517b5d7ea9bad81a10a82f7ad54b5f
ab9017eeb04cc303c45c4f7964a8cc2565b8bed3
34405 F20101117_AABLUE sarykalin_s_Page_004.jp2
33eeb3774d9da90cacdfa93259dede10
a426890764ede6d70f940933e80e94e60a5d50e1
15492 F20101117_AABMXH sarykalin_s_Page_079.QC.jpg
69703a69f72ef926399dec63263a9c8f
b4c18591900e9923ca8258add24dc4b60d46a34e
128262 F20101117_AABLTR sarykalin_s_Page_035.jp2
060245387a87c745f751af6d5cbbaeee
d07dcb1689a636a7a2120d75c7d70dd0d9f3bbcd
20889 F20101117_AABMWT sarykalin_s_Page_072.QC.jpg
82c8781fdb30e9a8642b0ef176b8455e
996b8619fe65aca28c6c96d67ed5c4e7d2991b13
F20101117_AABLUF sarykalin_s_Page_106.tif
3e3b8e9bb774a953e14e32e8c2849e44
6db20117a12a1b36db9bfdd9998c097cff4610e6
4396 F20101117_AABMXI sarykalin_s_Page_079thm.jpg
74c0deb3187aaec1057599b992c0af2a
dd1033f958287acf510010066382581a2ffaf77d
68963 F20101117_AABLTS sarykalin_s_Page_048.jpg
aa460cc3021c9ea402c5f78ddb9698ef
98d7cd3a7446226839dd19400b06cbff348d6933
5805 F20101117_AABMWU sarykalin_s_Page_072thm.jpg
ceffdc7e558552bd4018e426ec196ab4
f778337ec6c4c67e5d4eedf17cbebcb038e13d38
72907 F20101117_AABMAA sarykalin_s_Page_124.jpg
e8b0a87eaf1db5e25f94d0d93c394f54
e542cc19cff054b9de0cf153080f1568b2fca4db
124558 F20101117_AABLUG sarykalin_s_Page_033.jp2
35a56d14b48f593c3d8923dc0eeb9a89
75c3d13c6238430285ceb78a4a09da32156178e3
18270 F20101117_AABMXJ sarykalin_s_Page_080.QC.jpg
b943cefd55b1f650ad1643cb034b5665
16a9d19e4c4b621514430d8427c6b5d67aab10d9
77553 F20101117_AABMAB sarykalin_s_Page_125.jpg
230d2796cbd7741e28836074e1857424
029de274595a20b8aea213307a0935b9a996acaf
35203 F20101117_AABLUH sarykalin_s_Page_065.pro
3f9c5f08ea7de328c0a75d9167abf543
0176a645a02b2437dac1bf084775bee8436a2dce
18907 F20101117_AABMXK sarykalin_s_Page_081.QC.jpg
fb6280fe933ffba248324ad4fd336b17
668e22467369f898a0ba37ff6392770c6dcf9795
790257 F20101117_AABLTT sarykalin_s_Page_017.jp2
7a2eb3364d0a71bc5e6a375a1209035b
208df4b717de2517840df665faa6c91b128a0329
24776 F20101117_AABMWV sarykalin_s_Page_073.QC.jpg
381e0f338d33fb30a97f8782ed04d694
b95f94c631a83a4865d48ad3d06f5b9f5f86ab9a
25822 F20101117_AABMAC sarykalin_s_Page_126.jpg
66d47a9e636f6ee1c6f6f17f7496864d
9468251b41305418006d650d6432c5504bc438c8
F20101117_AABLUI sarykalin_s_Page_111.tif
5788d6697b20cd029469d39bb83d8035
3eae9d4f691e4813d5ddc21df98bb115fd746a25
5097 F20101117_AABMXL sarykalin_s_Page_081thm.jpg
850f41d43c795eafa22fba55add80306
8b7eddcb8e027b17acd5749e16696985ff277f2a
40371 F20101117_AABLTU sarykalin_s_Page_111.pro
8722ff5a123855db81dc2ecd3729a039
a90a2f460fd6cf10c0646d9e7d2e32e241b91f3a
6250 F20101117_AABMWW sarykalin_s_Page_073thm.jpg
0598914d4db5e171b31f0a7f496fa468
4eb1f715473c3fe5b5b808f9630f1a05c2436874
22220 F20101117_AABMAD sarykalin_s_Page_001.jp2
d185dbd733e5855c31c88a1e8a571d60
ce394f8a7a81255b3c52f0ec924562286d679739
F20101117_AABLUJ sarykalin_s_Page_069.tif
937f6326cc78650d760a96c4cf3c9042
a9f7b175f89dcf16ae47ca5452f20a3259ed061a
22183 F20101117_AABMYA sarykalin_s_Page_089.QC.jpg
033e91340d00c47898d9c35e7c6541b9
a428e8fa31ac4aa947b658ba619cc6ae07f5803e
20794 F20101117_AABMXM sarykalin_s_Page_082.QC.jpg
ce428d9f98a585ff2d8cdcdeb069ceb3
3e42ce191f9ee30234b6a0cb04ae3879c87db626
14782 F20101117_AABLTV sarykalin_s_Page_093.QC.jpg
721550de759c975492825121bbf51829
308a0494d139628cc8870faf4f897be501d58f08
24697 F20101117_AABMWX sarykalin_s_Page_074.QC.jpg
5b8ac0ed98794a2c56797b75dfcaf40f
b40050a989535aaf47cd1b601723b6126d921549
5740 F20101117_AABMAE sarykalin_s_Page_002.jp2
14eb8f24766c4765735284cb5d01d6af
b5269a451ef75c1f0c6a90ec9b352fbbd580445a
41901 F20101117_AABLUK sarykalin_s_Page_039.pro
cfd5b233d98b1c4e2fae085ec8b252dc
dacef46995197974da972630cf2ca1a5a45972fd
5903 F20101117_AABMYB sarykalin_s_Page_089thm.jpg
be801cb6bfefadd4a4507383a7753fb8
0ba6631df33ddd489828fb90d906abbbf34e5f0f
5837 F20101117_AABMXN sarykalin_s_Page_082thm.jpg
c8df9581ecf5fec54e47df55169785e6
0de4fed62e11031a0dbd201d420e87299eaaff25
6166 F20101117_AABMWY sarykalin_s_Page_074thm.jpg
a2350583cd4002a70ec6c34049e7b565
4f013403df1d9c412fa5d7226e7566952c8af337
4561 F20101117_AABLTW sarykalin_s_Page_020thm.jpg
c8acd9c42d33e624c8926e0aee5fc2a7
46d8c00b0181b5a03e6b24c2768742f90324ff41
4444 F20101117_AABMAF sarykalin_s_Page_003.jp2
3899726587df8037a2ab4d78184a257d
54b6082c9d806d736dd6accf78ef60a7f2b4767c
16958 F20101117_AABLUL sarykalin_s_Page_007.QC.jpg
22d8649636a4b90a6f4db45c69ecf33f
df8eca4caf59a427b4cdc30710adb6096c977f36
20288 F20101117_AABMYC sarykalin_s_Page_090.QC.jpg
e4b09a9e83547cb179655a7cc6cc1ef5
b87b2c70d88557634016ee3eec375cad04764a78
21342 F20101117_AABMXO sarykalin_s_Page_083.QC.jpg
05bcde88de0a2933caed11e8eb0ebd4d
49211da9304d7d7f8566123cd4e92742e8eb98f1
15765 F20101117_AABMWZ sarykalin_s_Page_075.QC.jpg
344eae2d6b8ed93f5a0902b68764d736
bd4634fb16f44ebf4b147519bd0852a23520a6fc
118722 F20101117_AABLTX sarykalin_s_Page_115.jp2
50205d06513ce9895e1d992a9899b625
8ae55397c1c406407366e8318164007ddbfb9bd7
1051982 F20101117_AABMAG sarykalin_s_Page_005.jp2
792a0ee52973bc1fd0c692c08b3fbd68
a31d332b24a7d9c8af11f4ae2856fcd6f0fbd6b6
4719 F20101117_AABLVA sarykalin_s_Page_022thm.jpg
5424d21b035f376e3111e76267aabd88
6382f60e583f6a7d15492729744441bc5c1d4a8c
F20101117_AABLUM sarykalin_s_Page_098.tif
99120bf731b5f8b8c6b38d08176d7ccb
712915b6a9170595e9b3ef85095514a7cb38c44a
5412 F20101117_AABMYD sarykalin_s_Page_090thm.jpg
ea547259d3fd818f26b7fcb18d2621bc
3a33d854b478463f6fbff10d2fe3550d3448bfb2
5864 F20101117_AABMXP sarykalin_s_Page_083thm.jpg
998fac8b2ebbb705568da37bcb1f887d
bbe77dee99914730f5c3a93a4fc662eb341a8777
41223 F20101117_AABLTY sarykalin_s_Page_101.pro
8e722e8d8618801971973b08ea78bdf3
32d2319331158d81a3855127cd9d683c9c28cdff
1051985 F20101117_AABMAH sarykalin_s_Page_006.jp2
72124ad8d486a14c5ed3347ab820f8e7
61d41b3d9c035220607eafed0bc4b7a980e97537
53956 F20101117_AABLVB sarykalin_s_Page_029.jpg
1540c314d162b71bc4582c1b28449093
26e1e2ae5f6f1dc0ec00bbb697dd6bc4582c7ff8
2531 F20101117_AABLUN sarykalin_s_Page_070.txt
da1cef03cf6942a9749b7393e686b338
922aed7442ebe01fb7373e2ee3cbfa73ef5da17a
26826 F20101117_AABMYE sarykalin_s_Page_091.QC.jpg
0cc3d46291207a268e21a6cafd4e8088
575a3102c9aa707b8d11b3eea96fa25a93b710ba
23254 F20101117_AABMXQ sarykalin_s_Page_084.QC.jpg
35b5a4adea580f006854a74b1a6a9bee
0b56bc799c03d6279472ede67f3f6226308266b1
4631 F20101117_AABLTZ sarykalin_s_Page_093thm.jpg
bd78fe497dc52a817b23f71f82920cb0
cfab280100ca87fe15680a2d78cfa871cda31a96
1051976 F20101117_AABMAI sarykalin_s_Page_007.jp2
6821afb0f3e6a8e6cf642e2e38e4a1dc
246ba9b8792bbff569f994aa8bdfbea1c9a18d7d
F20101117_AABLVC sarykalin_s_Page_037.tif
c152e84403e9d3c8b2a6657b88815c65
50d7599fd3a2b598a0d0be3969532f2906cb00dd
18856 F20101117_AABLUO sarykalin_s_Page_019.QC.jpg
d8b75df1aa869db8b04ec3095035df1d
931cdb8d4f7460f0998bb5cae3fa67f8e9f9476c
6738 F20101117_AABMYF sarykalin_s_Page_091thm.jpg
7261920d9bb85077222b9a619e364183
1d482b3393819fb5d4af68e2c900f730926f28d7
6024 F20101117_AABMXR sarykalin_s_Page_084thm.jpg
92d74390850ec350fcebf7b4b7f31a0b
1055b4b5fcbdd83ffc0816c14c124daa7f0d52f4
1051973 F20101117_AABMAJ sarykalin_s_Page_008.jp2
4fe4b6509800595f98530ee12578c96f
3db212701e3b1d2409ef75d251cfefccb04f40be
4822 F20101117_AABLVD sarykalin_s_Page_080thm.jpg
ca6c7c07415ca5aa5cef7e675bb94647
9760967a65ca6ba1eed93377b2b6683886b1041b
F20101117_AABLUP sarykalin_s_Page_057.tif
0358eb866f47e2af52bb0245b3c974f3
3881f8e31242866825e58b54e95631500ca1692a
20077 F20101117_AABMYG sarykalin_s_Page_092.QC.jpg
f3253cba60524d41bd2da4ac98209aa4
1724f12612afa9d7b3923c127d4dabaa5acb28f5
21854 F20101117_AABMXS sarykalin_s_Page_085.QC.jpg
bbc9bb98561738730b675c7c13a69ae0
64c292ff675e96466dbbea10c6f79cabc8f2c5c7
98930 F20101117_AABMAK sarykalin_s_Page_009.jp2
ec3550d09b9995544f47fed6b5b6de13
2e0b76a9e9c933c7b615745e44a0d8d8022020e3
F20101117_AABLVE sarykalin_s_Page_103.tif
80ad412eeef2782465fae416693e6724
e66bf84d6b90b2951f1207c798f9b865d077a760
2115 F20101117_AABLUQ sarykalin_s_Page_016.txt
031db5988a48b38d0b98a90e49495dd5
27a89c8950ae59e88a84ff1174b6e7de2e525f29
5565 F20101117_AABMYH sarykalin_s_Page_092thm.jpg
7b61b33acda0189a722d2c617057361e
fdafe813e0388d6ff2f6a4e9d8b8acafb332e006
5812 F20101117_AABMXT sarykalin_s_Page_085thm.jpg
266b1e44d75d8ae6a148e1b96d2ac913
9bb8fd001045f3f8d23be6d8a7ef2f0b2adeb0b7
29244 F20101117_AABMAL sarykalin_s_Page_010.jp2
a137dd5bc9814ea41cc2f75e9907f70a
ba34ff0e1a64b99a9f60b9d85ad0d5ce423b955f
F20101117_AABLVF sarykalin_s_Page_060.tif
2ea47a683459de026d355d813cae2a25
e7724d2205b5da2831d0a37c5bc308e9791adcc8
5640 F20101117_AABLUR sarykalin_s_Page_109thm.jpg
d47fff12e2389b002ef4fc7bd87cdd3f
480866a50575c81d648be0032d6750d7d06e488f
11936 F20101117_AABMYI sarykalin_s_Page_094.QC.jpg
3613ddf0ea5df529569eb015c42a5410
cfb72ae49f5d26a68e6ba8c81d36dea2c5175842
13380 F20101117_AABMXU sarykalin_s_Page_086.QC.jpg
12d3112c8591d4c3f93b1157d261ed8e
918e49bf0083bf4abd04b0e85767851f42fad04a
118070 F20101117_AABMAM sarykalin_s_Page_011.jp2
823ad6c1911fc53ba2881ca86dd1342a
a583f2c68104c4c557109a5a7b23597d477ac14b
F20101117_AABLVG sarykalin_s_Page_058.tif
333277c9fbbbbd8babbc0409b81b42c6
286748e07642b40fe6bfa2af051dd4a6be52ee0a
82551 F20101117_AABLUS sarykalin_s_Page_056.jp2
97e41606addc7b60ac8733e6f61110e1
bca4dddfc0b1b9ffd3f1d15a5f00507e5399ef87
1051924 F20101117_AABMBA sarykalin_s_Page_026.jp2
3f9d7fbc106f468ccd33494f3a39fa0b
2236a8a37b90b28818dd395b2d6723d10d2c4b6f
3865 F20101117_AABMYJ sarykalin_s_Page_094thm.jpg
72c5863933ef151ac741d79ea73ef330
e5faf7f97141b98a0002b37e65fc5281178ab7b2
4127 F20101117_AABMXV sarykalin_s_Page_086thm.jpg
4197220a4c1bf78edf9f52b183926b98
df4f6a778b0fb6666f12e2155571d2316e5c2835
99965 F20101117_AABMAN sarykalin_s_Page_012.jp2
fa3695f2a793702466d77f1efbc4c15b
479e71bf420d38ca2c38df90f4ff846db2335724
2151 F20101117_AABLVH sarykalin_s_Page_122.txt
026a98b3fff2e3d709fb41db545ccb75
b4bb12f679d659fecc15f463d06f9976b020ddee
877211 F20101117_AABLUT sarykalin_s_Page_090.jp2
31815e8b555be78061ce26da0a48687a
aeedda50a24a5d46d89bfcf7527a593a61a1c028
571240 F20101117_AABMBB sarykalin_s_Page_027.jp2
3e8b203e569772cf897797b3aca55aad
3b2d7fb46088690efd68b21fe589ed20cf778542
17439 F20101117_AABMYK sarykalin_s_Page_095.QC.jpg
1c30f5127ea0ff56cc2c10420b429ddf
4014e29caeb68ee56e818f722a7e961658106925
103824 F20101117_AABMAO sarykalin_s_Page_013.jp2
61f968d9cf828a6f4a00809cb95eb150
6ab0517597d5ec6d868e7207fccfcffbaed00c72
52394 F20101117_AABLVI sarykalin_s_Page_032.pro
ab3330016b77f74c90bb87244ff3a956
3f7eadf9e45f826c1f0c1378f9dd26b62e1100b1
79391 F20101117_AABMBC sarykalin_s_Page_029.jp2
de1f7837c8493ebd02650d26952282d1
121ad31c08ae3a5e2b4db520c02858963788d356
5225 F20101117_AABMYL sarykalin_s_Page_095thm.jpg
5af47e6d1be37ef4f8ffbe6050f94544
f1561274f4663070e6e524153e1c14fb970d845c
17684 F20101117_AABMXW sarykalin_s_Page_087.QC.jpg
a115dae0050d38ddb72fa2dd8ac81a77
c7a4f4c4b2c47d05e5b8722feb3b6186b7cd1bc0
118298 F20101117_AABMAP sarykalin_s_Page_014.jp2
67a6d37ed9365df20fe8c17b17769fc9
2e9716c3b9976042ea9065f1edfeae79e5e6c9ee
33970 F20101117_AABLVJ sarykalin_s_Page_028.pro
62d4a3d12c60b53d08c0f576a352549a
75d80e60b870f5f16b2833d04f1d852d9a7d555c
1672 F20101117_AABLUU sarykalin_s_Page_112.txt
4ab4a7d9aa410ee6871d719014a81fd4
94f0b50b1614d459cb2fec4a151d52369369fe9d
32 F20101117_AABMBD sarykalin_s_Page_030.jp2
ad0b8510bf5e14cb93c3c3de4c8cc4fb
7db7045a3a8b829f5804bcbfd71a8b7937a65ed5
23199 F20101117_AABMZA sarykalin_s_Page_103.QC.jpg
5ee089f4ad192f5baf41c17482a0792b
bdd10648995ace3f8a3c9fbd83d346044083c442
24563 F20101117_AABMYM sarykalin_s_Page_096.QC.jpg
ad7bbee7f4bf234a82c8bf2429e6c87e
319daa6f7ffa042a8f7dfb3b6749bdc8f458e480
4886 F20101117_AABMXX sarykalin_s_Page_087thm.jpg
86c566fe31b82932c8a174d4764304bf
920ed46eb7f40d38e46d4e58567ce1286031599f
81859 F20101117_AABMAQ sarykalin_s_Page_015.jp2
9d3a16af3964d7deec5ba21aaf32e111
8185c1634fbefbc706e1bedb1ede955422ed288d
191082 F20101117_AABLVK UFE0021770_00001.xml
b14cf008ff69e75232cab93680bb2da7
8532c0a2dea19af74906718726b4f85a19b8461d
2150 F20101117_AABLUV sarykalin_s_Page_032.txt
4b142e196d246f2ce7fc5666551461a1
eef28ab5145de9bce7047a79c199af558d2f295d
405809 F20101117_AABMBE sarykalin_s_Page_031.jp2
43700c362dec8ac3f2b7c223476e5177
05c34dd71de318341f1551f31de12f689fd1a57a
6275 F20101117_AABMZB sarykalin_s_Page_103thm.jpg
e3586e4ed34efa9c2c1ab4a97d47e583
9ea3dbbf8ef8bfae7ceadcb51ff55d1b8b49e0ff
6323 F20101117_AABMYN sarykalin_s_Page_096thm.jpg
dd7290f7be73fc9e160ad18096f10a8d
0b44bdf721dace9ea15dcfabbb98a9c844f65607
11751 F20101117_AABMXY sarykalin_s_Page_088.QC.jpg
5bdfa4f3b1a30e9ae0314f9e4f58cdf9
3a215384ddbf0f69b2a0ae2c6cd2f89cd2c5082c
997019 F20101117_AABMAR sarykalin_s_Page_016.jp2
fd0eb1f700733329f3c94992d90c0264
e3fdf839c0dd96319e8fa122d9171ab5f3a0a50e
46723 F20101117_AABLUW sarykalin_s_Page_020.jpg
94fddab573d3c101378d81b84738da83
20dce8ba0501ff1c997bd40c9b05963c7f11103e
113218 F20101117_AABMBF sarykalin_s_Page_032.jp2
e1b798308dc2268e19e8523959d8f6fc
bf0fd29e4fb7b95e770fa8ed31a48f2c60335cef
3913 F20101117_AABMZC sarykalin_s_Page_104thm.jpg
fb6d04922108fa6b1823699a850ee34f
867a416ee9170f0b4d1950567d1146d2cd0bc64f
21822 F20101117_AABMYO sarykalin_s_Page_097.QC.jpg
57449d76df419ad509733bbcf854d7e7
ae23837c83581bf29116bfab58ab2325fb9b7aa2
3291 F20101117_AABMXZ sarykalin_s_Page_088thm.jpg
a98ec854f3a8db0c539dfc7f18b974fe
5691223c1466f32254dc8ba89b177822cf3ef8b7
53542 F20101117_AABLWA sarykalin_s_Page_015.jpg
598a0313d1d979c126146565a37a5df6
7a6f10bd53c9ecaacc838694c0739563a4bc1f15
762248 F20101117_AABMAS sarykalin_s_Page_018.jp2
0b8a13e3eb703198df34ccf24b2e6b26
24c4d2f972eb8fa194363e10eb9eae49d7684e54
52679 F20101117_AABLUX sarykalin_s_Page_124.pro
b7b94bb4abad3670e9f88ea36eff78c8
c5d7dfe6f736fd31139f49d3545e283e9f76c5d9
123739 F20101117_AABMBG sarykalin_s_Page_034.jp2
bdb23e399e98159c207283a2e88b27a4
e4707c4e4ec65d731ed2c75c114b1e2062129897
15113 F20101117_AABMZD sarykalin_s_Page_105.QC.jpg
3b2c4aa3d5ba6b25796f8f60c601226a
8b42c6698b4c024909b3c0875df767947d068e04
5694 F20101117_AABMYP sarykalin_s_Page_097thm.jpg
e2e7ef5898ba489790750df4efe15201
2a7daa8b045d2f3046d64c1d917ed197089ed44c
69546 F20101117_AABLWB sarykalin_s_Page_016.jpg
0258fd2899e55db60e31beb404469cf8
384dac636aaaad24c2a9a90ce01bea85332d80a3
886317 F20101117_AABMAT sarykalin_s_Page_019.jp2
26ece570e4a61d8e3fbe10d230328f24
7912e6e0df9bd0b6dfe4bbc5d631616d01dbb099
22064 F20101117_AABLVN sarykalin_s_Page_001.jpg
e9b6fbe8f61dfdd2aa7319d75964cf7f
0e4ff361f16f00d00673a76482e929cfd295e574
47669 F20101117_AABLUY sarykalin_s_Page_067.jpg
f10e79f539c51e4a38a73c3256f2c320
bfe2828bee7cb8657999f7983734fb27e958d26e
104284 F20101117_AABMBH sarykalin_s_Page_036.jp2
1cf84daee09ac1dbcc4998256ed41738
907c10f544059a301187ea13932888a63285e796
4683 F20101117_AABMZE sarykalin_s_Page_105thm.jpg
6198ef55e3854e56c4d42dc7f60ac56b
67938eeec75bcc5061386616720ae2f235b92e14
21763 F20101117_AABMYQ sarykalin_s_Page_098.QC.jpg
0b9ef7a17eec4c252e2dd8274c95d5a2
cf5fb46b3a24c12ef02bfb97e3efad0fe20ba796
55227 F20101117_AABLWC sarykalin_s_Page_017.jpg
d083cc2ee95696e9cec24f9908e795b3
457e5e42ba9d348a60fee56988a1337030ae7cec
661187 F20101117_AABMAU sarykalin_s_Page_020.jp2
dc6d7065d2899fbd2dd699d919ef28b0
cbdd2e13f33be311cfff5fb2e7c48542074b86c3
9963 F20101117_AABLVO sarykalin_s_Page_002.jpg
2c22fa0d99d54bd61d3d7072539690b7
894e1b8bbfc84771cbd825841dfe2929ad23b3c8
5081 F20101117_AABLUZ sarykalin_s_Page_112thm.jpg
03bee894c797239b142bbd0a7f00ded6
90ac8bbdae3cadaa49e5dd423a1b09642107b258
908787 F20101117_AABMBI sarykalin_s_Page_037.jp2
fe2e62e86a8fff8185647e8d9d63e25e
eb829d52a574c52b0c7cef46aec85d50c5decf34
23881 F20101117_AABMZF sarykalin_s_Page_106.QC.jpg
aca30dea831055392a12de37c06aaeae
5ac282b97ba94b72c5410c7a9f37d34eda698797
5814 F20101117_AABMYR sarykalin_s_Page_098thm.jpg
b560ac61869576fb0c3a10bf947c66b1
fe7929a5105e4770cd0fc31575597224946e6739
55791 F20101117_AABLWD sarykalin_s_Page_018.jpg
2fd1c8d5123c95bb1e89dec443786e46
7d36d6c6f655daab4321e3c3a6da8aae2e89ff4c
787686 F20101117_AABMAV sarykalin_s_Page_021.jp2
1f9afd43d1fed00c4e028d9b7ecec87b
dc977d89b87389d950ee9032abacb25df9436c27
9338 F20101117_AABLVP sarykalin_s_Page_003.jpg
bfe08ec02c8892412864b74744b926a5
b9cc049d8e71c0c517b1f3bc096f80ba2a9af881
91599 F20101117_AABMBJ sarykalin_s_Page_038.jp2
83141e876689530628d2cc10b75bd2c2
35d24f8a828726da4eec7e513d725b64d500cead
6175 F20101117_AABMZG sarykalin_s_Page_106thm.jpg
3acc7589bfc45cab0d28d128b5d7a119
546e742c88a5f51af479829440e24101cb6756a6
23059 F20101117_AABMYS sarykalin_s_Page_099.QC.jpg
429c6f18c88b6a0970035a31c8c93c45
d12e354e908a460e66cabf98a4ff7a58cbdf3238
61122 F20101117_AABLWE sarykalin_s_Page_019.jpg
76a1ffb8df64c9331f5fd80ef840a473
fade7e0166f38794299f82c573274a06bbb1f083
685139 F20101117_AABMAW sarykalin_s_Page_022.jp2
cef74d06450fe3b01a5f9c477e044cc1
9d943d35eeddb9828cff198ef632cca2e17850eb
26158 F20101117_AABLVQ sarykalin_s_Page_004.jpg
544c18dabb6427554af356d8f665e55f
cce1954540c41340870fc38d2d445a90e111c6c4
86412 F20101117_AABMBK sarykalin_s_Page_039.jp2
8882785d3646b8e538cf7181de8ff4eb
4b88b3b45f024999fe2ce1ac70f4fe19a010fe86
25296 F20101117_AABMZH sarykalin_s_Page_107.QC.jpg
4c38b330475cfd17358d5626df99cedb
d0ff97ebdebd54ed9a9ace4e2db5f41d8c0b8f21
6241 F20101117_AABMYT sarykalin_s_Page_099thm.jpg
b5bcb17d2b53cfa11bd2466487b28b3a
efa4a08b987fbd272fa0b0c138942ee188e3c407
55492 F20101117_AABLWF sarykalin_s_Page_021.jpg
274ba20cb651459bb941a7d1d24b677a
c6cbb7d48b22ee3becd577148e02581fe41fcc4d
1051955 F20101117_AABMAX sarykalin_s_Page_023.jp2
9ec4bb74f5ade11152571caf85312faa
d4716fdf1df1b847f2f9fb604285f037b4e446de
79986 F20101117_AABLVR sarykalin_s_Page_005.jpg
3be8f23674144b0052d355abbc91c8c8
1b62b542b53ac702a267604a9a9c7c9429221fdd
976293 F20101117_AABMCA sarykalin_s_Page_055.jp2
872b520181ede1d7daa974ddc160ba68
9e31b78eb4f30a0ce584f92092b234e82e60c347
899226 F20101117_AABMBL sarykalin_s_Page_040.jp2
25d800f94550781d7da81eeadc48daff
6c8890640bc4188bf97d42db2070b062aa945354
6467 F20101117_AABMZI sarykalin_s_Page_107thm.jpg
aa366c4cceac03c60b3664f37e0d6124
ca9de3b5b74f3a911ce6cbfb9c37eb7a54d97de2
18652 F20101117_AABMYU sarykalin_s_Page_100.QC.jpg
a969334ab2a85758076188f5dbae85c6
092b95af9ba0fff6a0ecb12e7fb93b0b80b597a6
51654 F20101117_AABLWG sarykalin_s_Page_022.jpg
372a5882e22cbdf76a6c5d4888d27c56
e6f9edb22fa83d7c6b892f26972f17e0bff91c97
651164 F20101117_AABMAY sarykalin_s_Page_024.jp2
09afa9bf7f2a37f4c8df0e62073e31f6
c40f2a2a36a59dbd62879dbfc63f5ba37f590558
102318 F20101117_AABLVS sarykalin_s_Page_006.jpg
dbaf039ddc069aa2ec39fda0b7676bdb
2cd10d5fc1fc72481f2a167a6354ab57ee64c350
951004 F20101117_AABMBM sarykalin_s_Page_041.jp2
459e327b267360284eec1370a9d83753
038c3841883db5975995e16c3caf7a5e10d79277
24600 F20101117_AABMZJ sarykalin_s_Page_108.QC.jpg
43b56da4b8a2050d823121900f6ae306
3ddbb17103c3d1fa61ddc0af3fa078115f79e671
5317 F20101117_AABMYV sarykalin_s_Page_100thm.jpg
7cba674fad45c076a01dc4d16da97fb2
9c4218e7d20683cf3e9735bc2efc39198b35ec8e
73775 F20101117_AABLWH sarykalin_s_Page_023.jpg
d0ab4bec130d72fe43c9aaae6827dc27
9ce26518ba2696fb57b02a742f5fbc732c91ef98
875261 F20101117_AABMAZ sarykalin_s_Page_025.jp2
c91a94b0318c94063714e6092b6076c9
f07548e21ed762c98b86b9cab589472606105a3c
58383 F20101117_AABLVT sarykalin_s_Page_008.jpg
c5fed65aa394108b57d9d2d27f31eb95
8e9214c880218becbd04ba67a5775dfbc8fef767
1051986 F20101117_AABMCB sarykalin_s_Page_057.jp2
89e495542a6b680f29b1c8a8dc357ca8
955af1ccc0feee5647b88688d4e36a0eff6b580e
971491 F20101117_AABMBN sarykalin_s_Page_042.jp2
2aabfd0511f15a50d457eab0a0eb7863
30687769f867168be1a146db40aa634cb4e00dc2
6342 F20101117_AABMZK sarykalin_s_Page_108thm.jpg
3b0c68a966bdbf76d1d908af051dac5d
bdbdbf9bcb0eac6b210c8bbbdb011df23ad3e4ec
19476 F20101117_AABMYW sarykalin_s_Page_101.QC.jpg
4e9d7349c2443462ccfe4f93c8ce4a52
ed04d9eb932b9a3239c23aff90e2b01d9dc6aa28
47385 F20101117_AABLWI sarykalin_s_Page_024.jpg
af1f5c8737febd531ef698ba4b7fe7d8
0171cc44f611f92a399ba90143bb134962501538
61446 F20101117_AABLVU sarykalin_s_Page_009.jpg
a8077a6d6c8158693925797a579e58cd
46aa9cc3be3b296b635237258cb53b5d1d7c8a20
88078 F20101117_AABMCC sarykalin_s_Page_058.jp2
a5294da8a0b5b0c92f601a15526dd394
789dac13a7341dc107f70a60ec12dc971941da27
63884 F20101117_AABMBO sarykalin_s_Page_043.jp2
6e0558c2e5923d9c1ea59b35af788b31
ceaf02451991d72fc441897e607625205912a03d
22068 F20101117_AABMZL sarykalin_s_Page_109.QC.jpg
e5f7387e571431d9dc4a9b62dc279dcc
35ae794700bb8b16c3284c07be70a30c37afd829
59414 F20101117_AABLWJ sarykalin_s_Page_025.jpg
b728550e8e1d5e52e5c889822a3e11dd
405c1a9702a9ee98870d323aadbc403a3a29a9af
F20101117_AABMCD sarykalin_s_Page_059.jp2
21d70d6771d75268a34dc8ae9d0d5f47
232c430d4df7b7dbbf07535f52b6a5b05eda988d
F20101117_AABMBP sarykalin_s_Page_044.jp2
f06cd4fa56fa2dadf54682b709952ff2
674c9d1e11c1235bde3162af822c2ec2ff91c941
22473 F20101117_AABMZM sarykalin_s_Page_110.QC.jpg
e087b121ab748cad2dff47cf0e99eaa5
1b5eeee3fe9202047aa30d3b1552e2735929b034
5209 F20101117_AABMYX sarykalin_s_Page_101thm.jpg
6a69a67924e9734cfb771e144e14329e
b29f8b9d049447a0f678bc23ab457fd042f79259
22185 F20101117_AABLVV sarykalin_s_Page_010.jpg
0bae8013ad0f258aef986ef762ad59c6
213b44b104dc4cce775b45ad9e758fc0a72d8544
1051984 F20101117_AABMCE sarykalin_s_Page_060.jp2
fa7d47696a12b440e3959c4fe47c30a6
4e5ef4c5c582ae370b4dedcb343f1950698fde4a
1051974 F20101117_AABMBQ sarykalin_s_Page_045.jp2
d1728e766ac4dc5c44188ccfce00e6fd
10b15cc26d64f0c6ac5f4db18761d6a53d783561
80719 F20101117_AABLWK sarykalin_s_Page_026.jpg
3fda73f33ee9152ee110ec9244b453ac
e69f36f87549a6482d8614db878ae852f8839879
18891 F20101117_AABMZN sarykalin_s_Page_111.QC.jpg
194df7e6d2f3cf2bbf4fd4a819fc1bc2
2e4f44f16dae5ec9190140863adbeb9e8979d489
18562 F20101117_AABMYY sarykalin_s_Page_102.QC.jpg
1d0d0a4bf5927abf5aacf8a13f487034
50d13a32f4d910954d6a2da8570ed74376c00e43
74704 F20101117_AABLVW sarykalin_s_Page_011.jpg
ed14b3f175f85561ed702cc4074941fe
a5a05ef773d92441267f3e9880deb877359cf061
1051952 F20101117_AABMCF sarykalin_s_Page_061.jp2
f4380ec78e9a01c94c4a22e8a2d1e446
8bc9a643a95508760893538712fbaf933950e188
654393 F20101117_AABMBR sarykalin_s_Page_046.jp2
ae34b46fc441f97f43cf0b5e322656c3
c187f0734d7876ab15d49ba297b0d0506daaffee
43571 F20101117_AABLWL sarykalin_s_Page_027.jpg
a41ef6ff0664fa65e87053719620653e
20dbc96196a9beaa77407b9d189596ad9090732c
5161 F20101117_AABMZO sarykalin_s_Page_111thm.jpg
b840ed70b5aacc32ad78367c66a37df9
c7fe1bbadd536d8938a2ef43d21577f275f507a2
5034 F20101117_AABMYZ sarykalin_s_Page_102thm.jpg
5c2071367ab0c118ce6231e17f9a30f3
6bfc41dcbc63887cb886859f4f1f6bce57e87fb8
63893 F20101117_AABLVX sarykalin_s_Page_012.jpg
953cbcb41d7bd13472bae58a370f50ef
de7fffa8421690330b5c7d4e88423ef5ac458afa
108456 F20101117_AABMCG sarykalin_s_Page_062.jp2
77b9d3ca05e74482ae7eb47f72b6d0f8
1a89c78e954c2e67087916d1fe1c7b1672ff30ff
75575 F20101117_AABLXA sarykalin_s_Page_044.jpg
0b81c27f4031f0978262a672185a278b
681c66750f19b2abac4527ae007db2976253e445
784993 F20101117_AABMBS sarykalin_s_Page_047.jp2
37dc81cf495d8ccb333c7d1bff575f49
f9e6b299f11df19becd867f5f5e486d49b90538f
47632 F20101117_AABLWM sarykalin_s_Page_028.jpg
98d5e8cf1ab934ba19e7aff5c153975a
0e8dd53e571df7febc5a8778ed0be78c74b15dba
17386 F20101117_AABMZP sarykalin_s_Page_112.QC.jpg
18fd820d9f82a42d91a70c7a1626a4a8
52cd16b53b4eab4f4532d5ae12ad03645b0bf8ea
67616 F20101117_AABLVY sarykalin_s_Page_013.jpg
a6d72efde1d9bce9b4fbc3119ef6b028
727f0849c924c65d2f9caae44ed2331aee5b6255
77233 F20101117_AABMCH sarykalin_s_Page_063.jp2
72837cd0c65568fecc119d32a9ab4fed
33c916c677e72797316b0a95fb51241fbb0b3771
88595 F20101117_AABLXB sarykalin_s_Page_045.jpg
2a0e96f191b8baa75927e74b8c224e44
c49735850abf69a9d7b67aa0e6066cc230a59a0b
1002551 F20101117_AABMBT sarykalin_s_Page_048.jp2
5936983ed55ed731a57582295b3235c4
0120afa6ac847e5ee8ee2f21af202e3c568b154b
95379 F20101117_AABLWN sarykalin_s_Page_030.jpg
8000ebaa61580335ca09a123c3ad74a7
a5984a40a5fc2b8fc7ddb64c1b13b1fbdfe22ba9
19587 F20101117_AABMZQ sarykalin_s_Page_113.QC.jpg
e0ad93078e58d89bfd5fdd4eba750497
dcc901b0b8f2a3ae11e9b293464a5eb6f5e3d752
75444 F20101117_AABLVZ sarykalin_s_Page_014.jpg
c1a900141ae3c3effe6c1d359483a7d5
2b63170d7eb2d003ca836b4503fa3f752f7c3eb7
884809 F20101117_AABMCI sarykalin_s_Page_064.jp2
4e83feec4dc7a3cacb6a01ed7c60dca7
832aa3bc6799ce19b28ff3f40484146b4c4d3e58
49196 F20101117_AABLXC sarykalin_s_Page_046.jpg
d4c7231c65878ae691a3cae8f80a45d3
0551e60b717fe4c4f25c4b981cb52ba0bc0e9c75
110244 F20101117_AABMBU sarykalin_s_Page_049.jp2
48ec4cb99e5ba0c7442100dfca60b6cc
a500ef0d946ec3d9d0d09dd96dafbfad75541c4a
24571 F20101117_AABLWO sarykalin_s_Page_031.jpg
8e1a69293c9c866228ca0cbd66dbeeb7
ac1982c75ea34e67099c54e83a05601d2b9e8ade
5378 F20101117_AABMZR sarykalin_s_Page_113thm.jpg
d745f1e76f6924e5304003e83e7a0869
593d2fb66281504ba81bc62854ec857d693b137a
773615 F20101117_AABMCJ sarykalin_s_Page_066.jp2
db4effbdfe9cd3b02ac1cc115f662ea9
891da3e64e5828c9de19174508e66549873f5f0a
58489 F20101117_AABLXD sarykalin_s_Page_047.jpg
a62e759c28386d8cca3d61ceec282419
5fe8ee8c8ba4aab4c3d2f738644123b65712cd4e
90759 F20101117_AABMBV sarykalin_s_Page_050.jp2
eec9fa7c6f2f148803c930a785dfe6c9
25ddfa0c7d54dd54cd29ac3fe567ce062da85c75
72349 F20101117_AABLWP sarykalin_s_Page_032.jpg
f83497a7d7d306de8b1dc7ca11bb5c85
03e4b65b41248ea5d78052235295b6ffaef77d55
23292 F20101117_AABMZS sarykalin_s_Page_114.QC.jpg
46dac921ac75bfebda655fc8bd353204
475c5a7d9e19f842859ebe8f7cb7d379fe2e00ef
694726 F20101117_AABMCK sarykalin_s_Page_067.jp2
bcf3ce8ca83cae824de29910780d4033
8f03ea9f75aaae318cd6de7add47b3565d30c1b3
72545 F20101117_AABLXE sarykalin_s_Page_049.jpg
536996f1a15639c6fa29e4fe3dc29e14
333fbbffe54818fdf434debae3cca0cb2862e363
89891 F20101117_AABMBW sarykalin_s_Page_051.jp2
b3aefdf66176c55d8073ba9b24f31ee7
229a82141a4bf1b44e34d18e0397d93932abe146
78194 F20101117_AABLWQ sarykalin_s_Page_033.jpg
0a8992f2b2772c6befc38b32da30e86e
5ad2d2cef0486c423ad54cf41dfac300ef0827eb
6080 F20101117_AABMZT sarykalin_s_Page_114thm.jpg
edc85f8b7d1719465c1cac8d7e113aa7
a36490ea3e6d40508d0becb3df56db633e41fa82
932785 F20101117_AABMDA sarykalin_s_Page_083.jp2
3150a76ae5082460a4d49bb6f6217656
dbfb7b6f282287cfe6d1035b2468de438d896c42
87197 F20101117_AABMCL sarykalin_s_Page_068.jp2
55fb6017ec3d92f15f703cb710c923cd
532bd2206cf6f3e98258075e4882b85611f90320
59687 F20101117_AABLXF sarykalin_s_Page_050.jpg
3f03b25fc8cb42edc503788d1e67c6fb
d343fd58e01270ff4b03a7a78ecf5ec18eb50ea2
87469 F20101117_AABMBX sarykalin_s_Page_052.jp2
7a81d34ddd0176fa681af9b49866775c
1bd7b18270e911e1db1d316ed9b3e42250ef9230
77616 F20101117_AABLWR sarykalin_s_Page_034.jpg
cb16b64d6bd6f39155d3407a52692ff9
59bb3dcf0ec2331e5994211eaaacc84a2be32f30
23999 F20101117_AABMZU sarykalin_s_Page_115.QC.jpg
99ed9b556734f688c757d373acf543a9
af7898a9f80b1fce7f4384f65d5d721f3ea51bbb
954805 F20101117_AABMDB sarykalin_s_Page_084.jp2
45fd3d684843ddf57cf1bd23feb98aba
1496c811d42751b012d4f8e4d3dd9760036846a6
62126 F20101117_AABMCM sarykalin_s_Page_069.jp2
12d837a90555ba7b49ea46e8017ab637
b33460598073885eacd048cdff24842f2553c5ab
60311 F20101117_AABLXG sarykalin_s_Page_051.jpg
ef464fec34b32d101edef0dee032eae8
c45aff23e6f7f296d5322864097e779714b9d263
74465 F20101117_AABMBY sarykalin_s_Page_053.jp2
c7d9cbb877f1d0b0c94e70b35287b02c
a413789111e95be3e92bb98e9b4716266db74cc3
80897 F20101117_AABLWS sarykalin_s_Page_035.jpg
aa3180aedb3f819d88bd0aa7388a183f
ba1a3d691bbe677b3cb132a7d885c8d9dbc3e384
6033 F20101117_AABMZV sarykalin_s_Page_115thm.jpg
41392255554e143edb5fe0ccf24a70b0
cb05d235fd8b81229dab2489f249f7f0740fee98
100685 F20101117_AABMCN sarykalin_s_Page_070.jp2
99c773ee33a9e11edb147b642f5655f3
8a3e45b3d8dd6c2401c5b65afc9e5e1c05e12b00
58964 F20101117_AABLXH sarykalin_s_Page_052.jpg
156c9f76bb5f2e10fb0883a68a58a3ea
094dfa822804fe650abe653755f686909ac3ac8c
71694 F20101117_AABMBZ sarykalin_s_Page_054.jp2
1da373781a1b90ba35121c7670b29714
82bd9033c2337957710d6c3b58a82a841f0d7550
66480 F20101117_AABLWT sarykalin_s_Page_036.jpg
5b25ea30d1d07c69638381b81f208c90
ae2bd19f3123c9f08f12f495a64f06d1c6c733fb
23437 F20101117_AABMZW sarykalin_s_Page_116.QC.jpg
487e97e61f69c5d3cc2e0a5217f4c48e
9451fb51694312d94a2b4e3e26867eb1c2c0957b
974420 F20101117_AABMDC sarykalin_s_Page_085.jp2
3b0198f0a4aeea3252965f00a5dd40b2
44ccced1d3e23b787a597813e5451e6dc3e35bd0
101584 F20101117_AABMCO sarykalin_s_Page_071.jp2
604b5f9b311b0508d8e1c3b9ebee1302
4b45d44d7389c736e1a1be61f6b9d8939840e8a9
48191 F20101117_AABLXI sarykalin_s_Page_053.jpg
47e3c0b7d95d464f7187124575b83f46
f7ffda0d90a2495e8426301b478aae5861e7bd38
64050 F20101117_AABLWU sarykalin_s_Page_037.jpg
e3cc2bd500b4eeba65e85b9ffec69ee3
763ec3ef123f1ab592c038d5e0e24bf3f9e72814
5978 F20101117_AABMZX sarykalin_s_Page_116thm.jpg
b95ad0433a2560b8e82e7c0293542227
fbce959e5187af75043f7feb76871f4dc1e10b16
570755 F20101117_AABMDD sarykalin_s_Page_086.jp2
f18b97d2e3fe8b2fc9db98201326615d
5946e5a6bb8d50c7e9861bd4087ab75fb23b5344
1051968 F20101117_AABMCP sarykalin_s_Page_072.jp2
5911f7fb238ddbeadb769110e0686443
a8f17a6545604988af247cf64c09d0b50647cb8f
48331 F20101117_AABLXJ sarykalin_s_Page_054.jpg
bf18fe5071deb89427dda0d2055c2a13
fd14cc5754171c6b10ecbb68a895a24d2a3aafb9
59082 F20101117_AABLWV sarykalin_s_Page_038.jpg
58cf57ea78256254c898248cd615f65c
7cb2a69808c511ab05c1956fff0e37e1dac67ad1
754800 F20101117_AABMDE sarykalin_s_Page_087.jp2
91e9ab6d5e32d76b7528157cae106c39
8e87f39d0f1b54ed8209dd5347166820cc8d635b
121096 F20101117_AABMCQ sarykalin_s_Page_073.jp2
e49f4d0bf4fdda70c30be23aece76f28
e3f3c9fad34ae63de765e298478a37a231a3efc4
68020 F20101117_AABLXK sarykalin_s_Page_055.jpg
b262ca02c37605ac5dff6d3038d1747a
c3fa32b2486bda6a8a7b0c4841cf551d5ca1e6e0
18961 F20101117_AABMZY sarykalin_s_Page_117.QC.jpg
9e041a96e903e93a9c2ecf41f1cc0636
180539f7a6a2cc3eaa19f74eaa55ca21c26d1735
51716 F20101117_AABMDF sarykalin_s_Page_088.jp2
da585a6bf268275c83b195423b9327af
08c0dcd731cfeb78e397d853e1aab326cb7a7ce2
122492 F20101117_AABMCR sarykalin_s_Page_074.jp2
edeb8aff723f5aa482d71bc8eaac4d14
ef96ff3315f3eb75f82ab85f9d2f01fa7cb1a720
54777 F20101117_AABLXL sarykalin_s_Page_056.jpg
d3470082284a5596f1242ab606189537
096cbaeb081d5685a8c024f4a392bfb7f347c079
57224 F20101117_AABLWW sarykalin_s_Page_039.jpg
6c604ac6c37254e97cdfd3c0650a1437
68d532cd3d521c38ac6239129c9cccb58ac7fdc6
5527 F20101117_AABMZZ sarykalin_s_Page_117thm.jpg
9facfa9adcaae282ca8f94ffa2c08505
c1175697ccaccf3c426c55da23f4f2e45455046a
106272 F20101117_AABMDG sarykalin_s_Page_089.jp2
fc755a5a8f8b9daa99a9b5429333d998
10f22f76be899003e0d2d4ff8f0439252ec4d860
78186 F20101117_AABLYA sarykalin_s_Page_072.jpg
4a2388afa3b31b12d28ad190c63462b8
e0d9a6ccf108b27395e6e6f8231823f72b0ec742
71332 F20101117_AABMCS sarykalin_s_Page_075.jp2
4e1dc30545cf3e059c6ab5ccf9c35b24
bd71996dc0b82aeb12105700f1474242c983312f
73114 F20101117_AABLXM sarykalin_s_Page_057.jpg
9cff718f85756b285b02cbdb947dc925
b7f593e9e3868a1f7265e44f6995d1602510d0a7
64852 F20101117_AABLWX sarykalin_s_Page_040.jpg
902fa818aef40ce691fe0051719a57cb
168616100b0b498245c1850d9b345d0d7623cd80
77655 F20101117_AABLYB sarykalin_s_Page_073.jpg
dc426cbab0307f7131ad379c9668c57e
5c184c2b3050dac2c5e47833c4a5ecbee7d94ef3
673715 F20101117_AABMCT sarykalin_s_Page_076.jp2
22b3de892e73ac41f3fcf05335967cf6
183af988a45d5295801afd2009268868c2213139
59172 F20101117_AABLXN sarykalin_s_Page_058.jpg
28dc89f4bae867c7f31a998141bbc6be
08e1d3dc161212c41e335354417012f43d6556a8
68219 F20101117_AABLWY sarykalin_s_Page_042.jpg
4204a0c7434713109e51c3c171510179
725e895657dfc4ba130669f04cf03792dfe25a8b
1051954 F20101117_AABMDH sarykalin_s_Page_091.jp2
df88953a7815a07d1a7e69871353133a
1117f87b6b75860af094aa4562637f083800422a
77795 F20101117_AABLYC sarykalin_s_Page_074.jpg
4fd761369464d95c2bf5b00c0a37ddb4
836855c8b0a3b4c0915b9bb66caf49174334ea75
52562 F20101117_AABMCU sarykalin_s_Page_077.jp2
40eb1e52ac74e234a7f095ee844c5bbc
170190af556cff35a3e5548a26184969105a303c
87709 F20101117_AABLXO sarykalin_s_Page_059.jpg
9a30cf1eb9c8ef46bf3c9d96033d07be
7ed24da723bfb998b97c3bbb695307ddd814ff48
44386 F20101117_AABLWZ sarykalin_s_Page_043.jpg
5c3b87e8d9142d1821c9c45787a75a45
3d446b2d1d9e03d339caee5ba18dd1a999b6ccca
881214 F20101117_AABMDI sarykalin_s_Page_092.jp2
e7d1cd1d06c45a3df219ea71ffe9b398
2c2a9291998f1f0e03e568c4691499ec30c0eb01
47108 F20101117_AABLYD sarykalin_s_Page_075.jpg
7a184776b0fc09f0197bef9163c8155c
f1eed5ff1cbcf38d42717ab5bf26dc7036f8c3b8
97564 F20101117_AABMCV sarykalin_s_Page_078.jp2
63400e67e0401425206b92a1ce43c5a5
154d0902ca89893ae2e11aafc08e7054e6dfac43
89794 F20101117_AABLXP sarykalin_s_Page_060.jpg
15db1c270272cc45428a25fec83f62d9
ab6311620a9636d0f0fd6fe4bc59862274b7264b
648698 F20101117_AABMDJ sarykalin_s_Page_093.jp2
792f6317be4184ec536002c28fbd7672
91df5753bdf01b925ef8a9db30b8e504587aca60
51375 F20101117_AABLYE sarykalin_s_Page_076.jpg
eb2f2b852bf4f7e25316fe7180041706
36843bccec43163197a70aafd9dcb14397d36cb2
75620 F20101117_AABMCW sarykalin_s_Page_079.jp2
613c5ca5a3d93f2b8bfac81aebeba341
b888f1d1d237f678bba48634848ad1082352e550
84598 F20101117_AABLXQ sarykalin_s_Page_061.jpg
e35e1ed4c709fac409f14ef55ae71139
b69e5a7309a08831727c87e8858689997d54e5f1
474603 F20101117_AABMDK sarykalin_s_Page_094.jp2
d20475e6df8848864af2fb38514b5ba1
4ee3aa760abd2ea6b519d1a0d6a15941b245e1f5
35773 F20101117_AABLYF sarykalin_s_Page_077.jpg
4b1376bd7decf53e7e589961f6b56008
d6991f7d5a04de197ffc38140c15027178f59f95
787115 F20101117_AABMCX sarykalin_s_Page_080.jp2
bb108327f0be7186396ad47b41ac46fa
26417f32140e4a5175e7e639918f149976b3d069
69033 F20101117_AABLXR sarykalin_s_Page_062.jpg
d3fb34f7b4b844fcbd136e8d8d38d042
5715ce2ea94fa5de07cdffac422e4c1d99135dce
1025280 F20101117_AABMEA sarykalin_s_Page_110.jp2
1e2d198c3c0f43d8de53587859cfc806
427a227adb7b8ad0fffd37b6b3fe25359707a154
817977 F20101117_AABMDL sarykalin_s_Page_095.jp2
70eba99b7d3c71c10e229884c5b92375
b8fdb5aef6b094a2ba31f39a6eee7ca75ef7d890
60909 F20101117_AABLYG sarykalin_s_Page_078.jpg
2d76e2729faac0360e065fc97d6e5db3
309446d2d972b572bf63022c2f7b87ed85a94318
86586 F20101117_AABMCY sarykalin_s_Page_081.jp2
74ffb1d6d066cc1caa503c0232a1f88b
375262ffa4ed68830976df4c4737d64a7a7eb428
50635 F20101117_AABLXS sarykalin_s_Page_063.jpg
737f378f5a9f13c38f1b7185eabd83b2
2aa124e1481f75a24b426868d093266c88803c0e
842276 F20101117_AABMEB sarykalin_s_Page_111.jp2
20526bdc60c6c4ded510cf389c23f14e
c633da036956a00dba565786ad9409230d9b32e8
1051961 F20101117_AABMDM sarykalin_s_Page_096.jp2
dba3503afcfe06707161a3b4121cad64
78cd763f1d5730cfc3b44508d17bf58572a8023e
46779 F20101117_AABLYH sarykalin_s_Page_079.jpg
852cdf168cd969c2c4badddd667d3ea3
e8807692757de3394579c8472bff4ac8efa0b463
98102 F20101117_AABMCZ sarykalin_s_Page_082.jp2
93a896d993e2b5a95757cce75a675960
72c767a8784758a298e3df4d3261f2a2850495a9
65675 F20101117_AABLXT sarykalin_s_Page_064.jpg
a916d2a0a8f86399cd080f08dbbb1971
01a025e24513355e99c25572a0ea5d9a6118ba02
780229 F20101117_AABMEC sarykalin_s_Page_112.jp2
0f8b73f08446cf86c59e4e2f9aebb62b
ff11dfb7e7b1bad06b0eaf728a85fbed904b1d7f
955227 F20101117_AABMDN sarykalin_s_Page_097.jp2
aef45491e545e14f7ff1c1c9ad0b005a
d5d47abb3f2bf7eb2cd7c492acf7596e766daea8
58801 F20101117_AABLYI sarykalin_s_Page_080.jpg
01cc3b33109615424160147081440fe8
a320d013028fd09ff7e254824d8622ff1e157d83
62970 F20101117_AABLXU sarykalin_s_Page_065.jpg
20c1aee80a41b25213a50dee03642315
8f4a13db0548683165e691af346c57049e04816b
105231 F20101117_AABMDO sarykalin_s_Page_098.jp2
b908ddf9e109ddfa798b331baa872230
a1d942530e2b164819a549268bc2433b8191d4e6
57258 F20101117_AABLYJ sarykalin_s_Page_081.jpg
aea1560e48b71177b61916f3a09e6ec5
e95d55e8e362e748732e4420c657191c47ec2c15
50311 F20101117_AABLXV sarykalin_s_Page_066.jpg
0af6532110038539728706672fb7595c
c4c4f3de1f39fb82b0cdd506d1b819a49e8eb93f
902571 F20101117_AABMED sarykalin_s_Page_113.jp2
89a6b68f6eef29afc9ca5133f3130bf0
09f58270cec9f4b8322259a5c065cce68e6d7754
1051942 F20101117_AABMDP sarykalin_s_Page_099.jp2
84607320965e167ce71349736e4b774c
dcca44ebf9d4352584ed57ccaf3d4233b5a41f2f
63927 F20101117_AABLYK sarykalin_s_Page_082.jpg
54dde3aa2917ba9e8eab92555b4bd915
675426ed95db386e7a2867481b2bd8d96109a217
57802 F20101117_AABLXW sarykalin_s_Page_068.jpg
66c155d6f904a0f66f65f68a80f6e3cc
ea5a595d10062d154b762d2c123cfce4f6f084a5
1051981 F20101117_AABMEE sarykalin_s_Page_114.jp2
c374b1fbb99ba87b50d2c0210596488a
27e410a49acbc01d221c64313aa194a0bfd3a52e
836676 F20101117_AABMDQ sarykalin_s_Page_100.jp2
693ea4e45515c29bb158e230384a0547
4b4c83d7e412e2f98fe0e7fb7eb50c166253f5aa
67968 F20101117_AABLYL sarykalin_s_Page_083.jpg
86e692ba432e3ddf988b73c67e4416bc
f579a76ab6a815b0ab0eccbc760911b448d6654f
114614 F20101117_AABMEF sarykalin_s_Page_116.jp2
f49b92945ad095ed79c8a5bd1e7e0d87
e24018a4928d7d5712b3b935e85735f96fc4c81c
863541 F20101117_AABMDR sarykalin_s_Page_101.jp2
adc56e072e7ee49798caeec94178dd4a
7b401270f3bda6ca8382eee9875a5a35aa02d0d0
73471 F20101117_AABLYM sarykalin_s_Page_084.jpg
e51963ad1b4228ef8c47d6e052fe8279
4ef4e424d2d0d5960ac885d0eef718f915c9ec17
44125 F20101117_AABLXX sarykalin_s_Page_069.jpg
7a9830b2e8e0bb3d2374df740ea6c27a
56c445006273aa27981f08f586d8659a4887f8d0
824682 F20101117_AABMEG sarykalin_s_Page_117.jp2
cb2860cc2371d9b6162d910907373e28
4337b589c3427359285f80f2715f04b4b6f273c0
67571 F20101117_AABLZA sarykalin_s_Page_098.jpg
a3a97dbc56432e03de5d69bbdc53720b
aea826ef2f693ddaa0bb13016f55f6d5cd3267b2
799670 F20101117_AABMDS sarykalin_s_Page_102.jp2
d75fed582b8d0b004c56598fd1e1168b
90205a727537d50e52f18ff4eacad016ec340093
68972 F20101117_AABLYN sarykalin_s_Page_085.jpg
f89ebbcf77d73399aa666cbebd49c0d5
514febac3538661bf1498cc5c5139af93e4bd0cd
67585 F20101117_AABLXY sarykalin_s_Page_070.jpg
637a3d4c20dcbd14714be32bd32bcba7
749ebeea8fc7953ea673d4992b475f7c2848d614
673260 F20101117_AABMEH sarykalin_s_Page_118.jp2
bccd885d43be90bf8a5e12040d769201
f8508a0ff0958ed8d4f5c26253fedfb8aef76d98
75177 F20101117_AABLZB sarykalin_s_Page_099.jpg
18a590937a84d14a1457a4557a061733
1719bd8fd9bdae454bfee5be413befca31d4081e
1039910 F20101117_AABMDT sarykalin_s_Page_103.jp2
d0844b6e83202725b87b7a6bdabd13c7
3c67dee3cdd891112b9639e47686bbdb45232a4a
40288 F20101117_AABLYO sarykalin_s_Page_086.jpg
f92ee58952572e6b86b92c8accc719de
b4a13600aab216c5bc5deb60cd942370f294a5d4
68488 F20101117_AABLXZ sarykalin_s_Page_071.jpg
030f21c0d147417b0631bb1e34d01e71
f0c4b383d87c6d9570109175f049ef3d917b292d
854012 F20101117_AABMEI sarykalin_s_Page_119.jp2
f3952c1ff599bda8bc863e9e95efce67
716e9eff42ba22d26a9206bea87e19f8f9080782
60281 F20101117_AABLZC sarykalin_s_Page_100.jpg
caea1db2e1be305ab3ef703ad56a03fb
b7836ae232ced4fe35974ec9f3ae78e8f72dd3ca
483387 F20101117_AABMDU sarykalin_s_Page_104.jp2
510415a04125f9ab95dec4f91846d415
fcc3569853e441a7443df6009ef3e5bf822890a0
55136 F20101117_AABLYP sarykalin_s_Page_087.jpg
e907ed0eb5a71f1abff115b18bdb7fbb
f4ca49cf7038f404465e6107fce68937c35e8926
695923 F20101117_AABMEJ sarykalin_s_Page_120.jp2
d6ee964f25616405f2570c22c56b3c4d
076270ebe2711d570c0fb2e9d5262961dad5e8f0
64606 F20101117_AABLZD sarykalin_s_Page_101.jpg
2746a6a5e3de6eaf4578d5371f6ce80e
d76c6c2e12a35a39de3c4d7e7ff93b5be9946a42
642169 F20101117_AABMDV sarykalin_s_Page_105.jp2
5e294f10d1e2b2fa74f08d4acc1e4a53
b6fa0f90a9e43e06d78011a71ea02eddf51e2ed0
37898 F20101117_AABLYQ sarykalin_s_Page_088.jpg
40779602e8403cfec82a590f38c812f6
908220d8c686e585b168d7082c581ec7c6fa552d
67798 F20101117_AABMEK sarykalin_s_Page_121.jp2
4f3ba93b911fb47354c617ddc630b1f9
a225d43438525f3156f97a2637c02ae8e0e456b5
59282 F20101117_AABLZE sarykalin_s_Page_102.jpg
e10ddbaf888f60f893dd8a06aa78e3d5
cd5b9daa2d8aaa83041b9ee6f21f66126fdf97ca
117596 F20101117_AABMDW sarykalin_s_Page_106.jp2
e747c67e8e7838a6968f84eb413c9f8b
69438316176f5aef5712b8a92118ae337b3b4801
68314 F20101117_AABLYR sarykalin_s_Page_089.jpg
c8162fecd628114b675dc0854c4a9041
95194eb90d0c58afbca3e7f48d22ab303a79b66e
F20101117_AABMFA sarykalin_s_Page_012.tif
b380039a479ddba41762896f45cd9f59
9f8a366c9078dad5bd277ed7dafd4d5d8afcfa36
122326 F20101117_AABMEL sarykalin_s_Page_122.jp2
1e9fec2a86a167a464e9d20901b37511
44c8cc6e912dfc2f09e9c9a4da3fc472340543ae
72062 F20101117_AABLZF sarykalin_s_Page_103.jpg
b76674bb259e2eaee910cc99691826e0
7cad81db5c2fae7cabdfc02a69f22764b82411dd
125304 F20101117_AABMDX sarykalin_s_Page_107.jp2
f4568d35cb8aebed9152cf81c6080a63
db2f277aa096aff0493daece9e37e75c5231c4c3
66950 F20101117_AABLYS sarykalin_s_Page_090.jpg
7fea90557b28419a5187ab874f7800b3
3e64ee6025883a42b78eac0dcf3f63ad3a5fdcef
F20101117_AABMFB sarykalin_s_Page_013.tif
5234bde4fdd4ed477565ac9c7d573941
3b7624e253f264d80dec5f7cd0677f575d5c7417
130529 F20101117_AABMEM sarykalin_s_Page_123.jp2
1b9527ba6d2652946b61353dbe769e25
d18ded462cd21a322bb5abe67589871277b71809
38741 F20101117_AABLZG sarykalin_s_Page_104.jpg
da9e597c2b88fc359b30433a779e8733
4446b49146910955253dfa545909bd88ef71d0cd
121733 F20101117_AABMDY sarykalin_s_Page_108.jp2
24f4b254a5d8e25a8e330e4718729ca9
bbb527595bad5c23db7802cff1dc3281a87321b2
86876 F20101117_AABLYT sarykalin_s_Page_091.jpg
c4766b4aa778abfa22731fb42d23fd7f
aa2b55ce6efe3889456831e96fec1731906d4ee3
F20101117_AABMFC sarykalin_s_Page_015.tif
835cf79338f1ee19364ab4dec86be216
118135d02680beb8b6fe2c25973a08d5deb90258
118191 F20101117_AABMEN sarykalin_s_Page_124.jp2
bdf76e5f44b73e9ced93b077068d47fc
f7ddbe8ce73f58d6e5f9936c1bca89aa81b4cd35
47112 F20101117_AABLZH sarykalin_s_Page_105.jpg
696479dd163ede2ba721bc2f4f213914
d63af00bffdea74dcd2a82f702528a9b2aed5e3c
107946 F20101117_AABMDZ sarykalin_s_Page_109.jp2
498c369993eed86e093f72cd55454fff
a5850eb7c0f0eea5cc225dc8b18d827cd446433b
63515 F20101117_AABLYU sarykalin_s_Page_092.jpg
d3614e1f4ab728f8918464f9786b1f83
c7751799bad5d3806bd911eda1bb0ff5daa4aeb3
F20101117_AABMFD sarykalin_s_Page_016.tif
511201baa4e8f23dc3ee65d733b80ec8
893a7da500e3cd40b74934bf2e0d90b14d14e345
127432 F20101117_AABMEO sarykalin_s_Page_125.jp2
93c8b01da90db8e3cee409e84c135b37
4eb49afb2fe8c019b493ffb16866982af49a1513
75446 F20101117_AABLZI sarykalin_s_Page_106.jpg
a687e0581298ac7efe28f319887bcebb
269f8254e76928fda9ab2d3f13cc7c0ab68d7e56
47693 F20101117_AABLYV sarykalin_s_Page_093.jpg
e8cb64fd63885f5c177695ed272ca0bd
470ac688b1ebb43ed3671afa7608610f76fb573a
32838 F20101117_AABMEP sarykalin_s_Page_126.jp2
9614b6121321e221fc5a47bf6697b9fc
5e5051041b07fb0e5c0d848adcccb7f1c4d7fd71
79097 F20101117_AABLZJ sarykalin_s_Page_107.jpg
deb9798c8089dc47cc892ec72ef937aa
c46125c4e4239b5063f96622a1cff5bf30d13132
38687 F20101117_AABLYW sarykalin_s_Page_094.jpg
56038aa77cf2a8f42203f0c59f10021e
531a1d59321c4613780bb559a92214059c072f95
F20101117_AABMFE sarykalin_s_Page_017.tif
79d054a04ddd9aa77f15647bd858a133
3469e09eae606892c642701d244c7adb83466e9f
F20101117_AABMEQ sarykalin_s_Page_002.tif
d6a842cd1fb3160a7cba7a8f3af9d7e4
0d4cd041c33508b09d1ea5cb5c251a312f78e94c
76650 F20101117_AABLZK sarykalin_s_Page_108.jpg
e8ac43d004b796e1f998bc9e85e6816d
472ae4f578a84e246e6e314ec2b98eec59c04192
56526 F20101117_AABLYX sarykalin_s_Page_095.jpg
38c0c9ff39e36c7c92bda0fd348d60a2
1aa3ce8bd557ac199904155478ae090bc39bbb95
F20101117_AABMFF sarykalin_s_Page_018.tif
1ea4c2720dd99908300b79ca61b222b9
019e92166544723ba4f9a86a66e2569d35b28ba9
F20101117_AABMER sarykalin_s_Page_003.tif
58c62ad0636714fd4f9b3de698317ac1
6265e6e5bef7d62024da70a6335fd506cf343917
67877 F20101117_AABLZL sarykalin_s_Page_109.jpg
697f4a38ca06c3fbeea8a252fad329cd
c647b9e25975be26452ae2afa928b750fd27b03d
F20101117_AABMFG sarykalin_s_Page_019.tif
a9d9a6c6331280798488ad0b3ba39f62
f3cc25057f504232298819dc753d9c75f1eb0c9e
F20101117_AABMES sarykalin_s_Page_004.tif
5fe7b95ea659e2bfca3ab63d5bff258d
8d4b373eab283c2638d451e1f980110992ff8461
69889 F20101117_AABLZM sarykalin_s_Page_110.jpg
e6a527b5a623892156b0adccfe8919a5
0897f761669571f78c4a26ecc7fa153c32587124
78813 F20101117_AABLYY sarykalin_s_Page_096.jpg
0b41f765727bc01e9a2d5fae5f34cc72
e1fc5df5e85c94816699fcb34da7a0e741acff24
F20101117_AABMFH sarykalin_s_Page_020.tif
99282541c11697108f2352d8c615cd42
d8e09348b76dbf3a487fc146d00eecd48a187298
F20101117_AABMET sarykalin_s_Page_005.tif
a7f81b18b2dae103dd65038f2863a74c
51652899153188f4f6d8fd1a9e18db0a4e76e1e9
63692 F20101117_AABLZN sarykalin_s_Page_111.jpg
bdab3993bb3f48f132a3267bc2df44ab
1f187a2091e17e58b050ef20e10b66a7be7f9d01
69740 F20101117_AABLYZ sarykalin_s_Page_097.jpg
38fe736738435a219c9f4cb20b51a7dd
017be27bd3f42eb8b297a3d0be042b050f6f24fa
F20101117_AABMFI sarykalin_s_Page_021.tif
8e6af13ab7645fbfe03dade5a239aebd
de60ac0ebb31d868592a1b97dc65b04e66328fad
F20101117_AABMEU sarykalin_s_Page_006.tif
50aecd45a365f4c476fe4d466deb41e0
6447710e89a44e7834cf19ec4cda89c9437604a0
56541 F20101117_AABLZO sarykalin_s_Page_112.jpg
9f0b73a2c3f6fc4745eb39d78d70136e
96c82e7e0c3a350ab1234aedcfb058ba8570f4ff
F20101117_AABMFJ sarykalin_s_Page_022.tif
23e0f4de67e9fae1464720701885d2f8
54be08422429b955f1d244ff5568a691d90048ce
F20101117_AABMEV sarykalin_s_Page_007.tif
ac6ed8b5aeecbde561a49523a7331b41
26def10007df8d647ee67d40393ffa3e0b9c99f6
63480 F20101117_AABLZP sarykalin_s_Page_113.jpg
abf608f718bf3b4f8dee74fcf5898c5c
f78720e7b3a8be7137c041b758bd73830b548abd
F20101117_AABMFK sarykalin_s_Page_023.tif
385e05cb3b53e34b22fba5e65865f658
3e7a0bbb68b8566200962a1ac41166e61b797754
F20101117_AABMEW sarykalin_s_Page_008.tif
4971a533f5cb5321756f78c5dd2932ab
9015fbbca0ec111f1c8e4c441a81fd7bb0be9f24
73909 F20101117_AABLZQ sarykalin_s_Page_114.jpg
5abe11e3eda68f0dcf79142b06c1af9e
f897ca2bf48cc5e19978ba8c65715febe552b403
F20101117_AABMFL sarykalin_s_Page_024.tif
94fcb5e58dfdfdd5752d40e7b118fb7b
59a4398fb41a212a4dc061e5fc91e1894982e4a8
F20101117_AABMEX sarykalin_s_Page_009.tif
7179adf3a103b86efc826df41bdd6a10
e1957872bddd32808e409caab173d47c6887d897
75205 F20101117_AABLZR sarykalin_s_Page_115.jpg
ebcc47008e0cbabbb9bdc95f06cfffe8
c2266941f024c6dfd0d8957491b6321177649828
F20101117_AABMGA sarykalin_s_Page_044.tif
4bc1b34751b7e0e23989eebfd53b9116
4dba64eb423c2d7393ca21ef7b9fe7de9e975b00
F20101117_AABMFM sarykalin_s_Page_025.tif
d8a6d4634e68162164d06ea8308cfa50
bf923c91ac949a8cd35cdec5045721681f938e6c
F20101117_AABMEY sarykalin_s_Page_010.tif
8116cbd9ec028e08b5202e0f969a521f
534c85ee426ded4a8c72f2d839451cccea6cbe90
72566 F20101117_AABLZS sarykalin_s_Page_116.jpg
81bed014060638db34d5325bf282bb27
b033402b8d791b95496c7d665341c06efbcb2033
F20101117_AABMGB sarykalin_s_Page_045.tif
41c3d97ac8577554cebb57bb8acc686a
3bbfb797a97d0660f1d2afae5cb1a7a8eb856961
F20101117_AABMFN sarykalin_s_Page_027.tif
0a4b00e6e16269a2d0f13ad9fb7b2811
affbfd4d870e2e18c0b2682408cd9fba9b6e0747
F20101117_AABMEZ sarykalin_s_Page_011.tif
4462ea1a115e5dd468999294f05c8739
7006d5e43586b8458abc37cc28573c9c561dc5fe
59851 F20101117_AABLZT sarykalin_s_Page_117.jpg
b6debd30380ddfd315872d8d988c0572
72e0dc5de91ecab22e00be6f7ba4022aaeaed1c7
F20101117_AABMGC sarykalin_s_Page_046.tif
7f39b0b3e4a2f7174355cd5dc28efb27
7639a1dbd98de4d5a9f9f99036ebce563819c7c9







OPTIMIZATION METHODS IN FINANCIAL ENGINEERING


By

SERGEY V. SARYKALIN



















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007
































2007 Sergey V. Sarykalin




































To my parents.









ACKNOWLEDGMENTS

I want to thank my advisor Prof. Stan Uryasev for his guidance support, and

enthusiasm. I learned a lot from his determination and experience.

I want to thank my committee members Prof. Jason Karseski, Prof. Farid AitSahlia,

and Prof. R. Tyrrell Rockafellar for their concern and inspiration.

I want to thank my collaborators Vlad Bugera and Valeriy Ryabchekno, who were

alv--,i- great pleasure to work with.

I would like to express my deepest appreciation to my family and friends for their

constant support.









TABLE OF CONTENTS
page

ACKNOW LEDGMENTS ................................. 4

LIST OF TABLES ....................... ............. 7

LIST OF FIGURES .................................... 8

ABSTRACT . . . . . . . . . . 9

CHAPTER

1 INTRODUCTION ...................... .......... 11

2 TRACKING VOLUME WEIGHTED AVERAGE PRICE ........... 13

2.1 Introduction . .. . . . . . . .. 13
2.2 Background and Preliminary Remarks ................. 15
2.3 General Description of Regression Model .................. 18
2.3.1 Mean-Absolute Error ................... .... 18
2.3.2 CVaR-objective ................... ..... 19
2.3.3 M ixed Objective .................... ........ 20
2.4 Experiments and Analysis .................. ........ .. 21
2.4.1 M odel Design .................. ........... .. 21
2.4.2 Nearest Sample .................. ...... 23
2.4.3 Data Set ................. . . .... 23
2.4.4 Evaluation of Model Performance ................. 23
2.5 Experiments and Results .................. ......... .. 25
2.6 Conclusions .................. ................ .. 27

3 PRICING EUROPEAN OPTIONS BY NUMERICAL REPLICATION ..... 32

3.1 Introduction .................. ................ .. 32
3.2 Framework and Notations .................. ........ .. 37
3.2.1 Portfolio Dynamics and Squared Error ............... .. 37
3.2.2 Hedging Strategy .................. ......... .. 38
3.3 Algorithm for Pricing Options .................. ..... .. 41
3.3.1 Optimization Problem .................. ..... .. 41
3.3.2 Financial Interpretation of the Objective . . ..... 44
3.3.3 Constraints .................. ............ .. 45
3.3.4 Transaction Costs ...... . . .......... 45
3.4 Justification Of Constraints On Option Values And Stock Positions .. 46
3.4.1 Constraints for Put Options ................... . .46
3.4.2 Justification of Constraints on Option Values . . ... 47
3.4.3 Justification of Constraints on Stock Position . . 55
3.5 Case Study .................. ................ .. 58









3.5.1 Pricing European options on the stock following the geometric brownian
motion ...... ....... .... ....... 59
3.5.2 Pricing European options on S&P 500 Index . . ..... 59
3.5.3 Discussion of Results .................. ..... .. 60
3.6 Conclusions and Future Research ................ .... .. 63

4 METHODS OF REDUCING MAXIMIZATION OF OMEGA FUNCTION TO
LINEAR PROGRAMMING .................. .......... .. 73

4.1 Introduction .................. ................ .. 73
4.2 Omega Optimization .................. ........... .. 75
4.2.1 Definition of Omega Function .................. .. 75
4.2.2 General Problem .................. ......... .. 77
4.2.3 Two Reduction Theorems .... . . .... 78
4.3 Proofs Of Reduction Theorems For Omega Optimization Problem ..... 81
4.4 Applications of Reduction Theorems to Problems with Linear Constraints 84
4.5 Example: Resource Allocation Problem .............. .. .. 85
4.6 Conclusions .................. ................ .. 88

5 CALIBRATION OF GENERAL DEVIATION MEASURES FROM MARKET
DATA ............. ...... ........ .. .... . 89

5.1 Introduction .................. . . .... 89
5.1.1 Definitions and Notations .................. .... .. 89
5.1.2 General Portfolio Theory ....... . . ... 90
5.1.3 Generalized CAPM relations and Pricing Equilibrium . . 91
5.2 Intuition Behind Generalized CAPM Relations . . ...... 92
5.2.1 Two W-,v to Account For Risk ............... .. 92
5.2.2 Pricing Forms of Generalized CAPM Relations . . .... 93
5.3 Stochastic Discount Factors in General Portfolio Theory . ... 98
5.3.1 Basic Facts from Asset Pricing Theory. . . . ... 98
5.3.2 Derivation of Discount Factor for Generalized CAPM Relations 102
5.3.3 Geometry of Discount Factors for Generalized CAPM Relations .. 103
5.3.4 Strict Positivity of Discount Factors Corresponding to Deviation
Measures .... . . ..... .............. 103
5.4 Calibration of Deviation Measures Using Market Data . ... 106
5.4.1 Identification of Risk Preferences of Market Participants ....... 106
5.4.2 Notations ............. . . . .... 109
5.4.3 Implementation I of Calibration Methods . . .... 109
5.4.4 Implementation II of Calibration Methods . . ..... 110
5.4.5 Discussion of Implementation Methods . . ..... 115
5.5 Coherence of Mixed CVaR-Deviation ................ .... .. 116
5.6 Conclusions ............... .............. .. 121

REFERENCES. .... ....................... .... 122

BIOGRAPHICAL SKETCH ................... . ... 126









LIST OF TABLES


Table page

2-1 Performance of tracking models: stock vs. stock+index, full history regression .28

2-2 Performance of tracking models: stock vs. stock+index, best sample regression 28

2-3 Performance of tracking models: mixed objective, changing size of history and
best sam ple .................. ................... .. 29

2-4 Performance of tracking models: CVaR deviation, changing size of history and
best sample . ............... ..... .... 29

2-5 Performance of tracking models: mixed objective ... . . 29

3-1 Prices of options on the stock following the geometric Brownian motion: calculated
versus Black-Scholes prices. .................. .. ...... 68

3-2 S&P 500 options data set. .................. .. ........ 69

3-3 Pricing options on S&P 500 index: 100 paths .................. 70

3-4 Pricing options on S&P 500 index: 20 paths ................ 71

3-5 Summary of cashflow distributions for obtained hedging strategies presented on
Figures 3.6, 3.6, 3.6, and 3.6. .................. ..... 72

3-6 Calculation times of the pricing algorithm. .................. .... 72

3-7 Numerical values of inflexion points of the stock position as a function of the
stock price for some options. .................. .. ...... 72

4-1 Optimal allocation .................. . . .. 88









LIST OF FIGURES


Figure page

2-1 Percentages of remaining volume vs. percentages of total volume . ... 30

2-2 MAD, CVaR, and mixed deviations ............... ..... 30

2-3 Daily volume distributions ............... ......... .. 31

3-1 Implied volatility vs. strike: Call options on S&P 500 index priced using 100
sample paths .................. .................. .. 64

3-2 Implied volatility vs. strike: Put options on S&P 500 index priced using 100
sample paths .................. .................. .. 64

3-3 Implied volatility vs. strike: Call options in Black-Scholes setting priced using
200 sample paths .................. .............. .. .. 65

3-4 Implied volatility vs. strike: Put options in Black-Scholes setting priced using
200 sample paths .................. .............. .. .. 65

3-5 Black-Scholes call option: distribution of the total external financing on sample
paths .. .. . . . . . .... . . 66

3-6 Black-Scholes call option: distribution of discounted inflows/outflows at re-balancing
points . . . . . . ........ .. . 66

3-7 SPX call option: distribution of the total external financing on sample paths 67

3-8 SPX call option: distribution of discounted inflows/outflows at re-balancing points 67









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

OPTIMIZATION METHODS IN FINANCIAL ENGINEERING

By

Sergey V. Sarykalin

December 2007

C'!I wiC: Stanislav Uryasev
Major: Industrial and Systems Engineering

Our study developed novel approaches to solving and analyzing challenging problems

of financial engineering including options pricing, market foi I i.- and portfolio

optimization. We also make connections of the portfolio theory with general deviation

measures to classical portfolio and asset pricing theories.

We consider a problem faced by traders whose performance is evaluated using the

VWAP benchmark. Efficient trading market orders include predicting future volume

distributions. Several forecasting algorithms based on CVaR-regression were developed for

this purpose.

Next, we consider assumption-free algorithm for pricing European Options in

incomplete markets. A non-self-financing option replication strategy was modelled on

a discrete grid in the space of time and the stock price. The algorithm was populated by

historical sample paths adjusted to current volatility. Hedging error over the lifetime of

the option was minimized subject to constraints on the hedging strategy. The output of

the algorithm consists of the option price and the hedging strategy defined by the grid

variables.

Another considered problem was optimization of the Omega function. Hedge funds

often use the Omega function to rank portfolios. We show that maximizing Omega

function of a portfolio under positively homogeneous constraints can be reduced to linear

programming.









Finally, we look at the portfolio theory with general deviation measures from the

perspective of the classical asset pricing theory. We derive pricing form of generalized

CAPM relations and stochastic discount factors corresponding to deviation measures. We

-,.:.:, -i methods for calibrating deviation measures using market data and discuss the

possibility of restoring risk preferences from market data in the framework of the general

portfolio theory.









CHAPTER 1
INTRODUCTION

Fast development of financial industry makes high demands of risk management

techniques. Success of financial institutions operating in modern markets is largely affected

by the ability to deal with multiple sources of uncertainty, formalize risk preferences,

and develop appropriate optimization models. Recently, the synthesis of engineering

intuition and mathematics led to the development of advanced risk management tools.

The theory of risk and deviation measures has been created, with its applications to

regression, portfolio optimization, and asset pri i_:. which encouraged the use of novel risk

management methods in academia and industry and stimulated a lot or research in the

area of modelling and formalizing risk preferences. Our study makes a connection between

financial applications of the theory of general deviation measures and classical asset

pricing theory. We also develop novel approaches to solving and analyzing challenging

problems of financial engineering including options pricing, market forecasting, and

portfolio optimization.

C'! lpter 2 considers a broker who is supposed to trade a specified number of shares

over certain time interval (market order). Performance of the broker is evaluated by

Volume Weighted Average Price (VWAP), which requires trading the order according

to the market volume distribution during the trading period. A common approach to

this task is to trade the order following the average historical volume distribution. We

introduce a dynamic trading algorithm based on forecasting market volume distribution

using techniques of generalized linear regression.

C'! lpter 3 presents an algorithm for pricing European Options in incomplete markets.

The developed algorithm (a) is free from assumptions on the stock process; (b) achieves

0.5' .- :;' pricing error for European in- and at-the-money options on S&P500 Index; (c)

closely matches the market volatility smile; (d) is able to price options using 20-50 sample

paths. We use replication idea to find option price, however we allow the hedging strategy









to be non-self-financing and minimize cumulative hedging error over all sample paths.

The constraints on the hedging strategy, incorporated into the optimization problem,

reflect assumption-free properties of the option price, positions in the stock and in the

risk-free bond. The algorithm synthesizes these properties with the stock price information

contained in the historical sample paths to find the price of option from the point of view

of a trader.

C'! lpter 4 proves two reduction theorems for the Omega function maximization

problem. Omega function is a common criterion for ranking portfolios. It is equal to the

ratio of expected overperformance of a portfolio with respect to a benchmark (hurdle rate)

to expected underperformance of a portfolio with respect to the same benchmark. The

Omega function is a non-linear function of a portfolio return; however, it is positively

homogeneous with respect to instrument exposures in a portfolio. This property allows

transformation of the Omega maximization problem with positively homogeneous

constraints into a linear programming problem in the case when the Omega function

is greater than one at optimality.

C'! lpter 5 looks at the portfolio theory with general deviation measures from the

perspective of the classical asset pricing theory. In particular, we analyze the generalized

CAPM relations, which come out as a necessary and sufficient conditions for optimality in

the general portfolio theory. We derive pricing forms of the generalized CAPM relations

and show how the stochastic discount factor emerges in the generalized portfolio theory.

We develop methods of calibrating deviation measures from market data and discuss

applicability of these methods to estimation of risk preferences of market participants.









CHAPTER 2
TRACKING VOLUME WEIGHTED AVERAGE PRICE



2.1 Introduction

Volume-weighed average price (VWAP) is one of the commonly used trade evaluation

benchmarks in the stock market. VWAP of a set of transactions is the sum of prices

of these transactions weighted by their volumes. For example, in order to calculate the

daily VWAP of the market, one should sum up the amounts of money traded for each

transaction and divide it by the total volume of stocks traded during the di-. A trader's

performance can then be evaluated by comparing the VWAP of a trader's transactions

with the market VWAP. Selling a block of stocks is well performed if the VWAP of

selling transactions is higher then the VWAP of the market and vice versa.

There are several types of benchmarks similar to VWAP. VWAP, as it is defined

above, is reasonable for evaluation of relatively small orders of liquid stocks. VWAP

excluding own transactions is appropriate when the total volume of transactions

constitutes a significant portion of the market's daily volume. For highly volatile stocks,

value-weighted average price is also used, where prices of transactions are weighted by

dollar values of this transactions. VWAP benchmarks are widespread mostly outside

USA, for example, in Japan.

The purpose of the VWAP trading is to obtain the volume-weighted price of

transactions as close to the market VWAP as possible. An investor may act differently

when seeking for VWAP execution of his order. He can make a contract with a broker

who guarantees selling or buying orders at the daily WVAP. Since the broker assumes

all the risk of failing to achieve the average price better than VWAP and is usually risk

averse, commissions are quite large.



This chapter is based on joint work with Vladimir Bugera and Stan Uryasev.









An order may be sent to electronic systems where it is executed at the daily VWAP

price (VWAP crosses). These orders are matched electronically before the beginning of a

trading dv and executed during or after trading hours. VWAP crosses normally have low

transaction costs; however, the price of execution is not known in advance and there may

exist the possibility that the order will not be executed.

An investor with direct access to the market may trade his order directly. But since

VWAP evaluation motivates to distribute the order over the trading period and trade

by small portions, this alternative is not preferable due to intensity of trading and the

presence of transaction costs.

The most recent approach to VWAP trading is participating in VWAP automated

trading, where a trading period is broken up into small intervals and the order is

distributed as closely as possible to the market's daily volume distribution, that is

traded with the minimal market impact. This strategy provides a good approximation

to market's VWAP, although it generally fails to reach the benchmark. More intelligent

systems perform careful projections of the market volume distribution and expected price

movements and use this information in trading. A more detailed survey of VWAP trading

can be found in Madgavan (2002).

Although VWAP-benchmark has gained popularity, very few studies concerning

VWAP strategies are available. Several studies, Bertsimas and Lo (1998), Konishi and

Makimoto (2001) have been done about block trading where optimal splitting of the

order in order to optimize the expected execution cost is considered. In the setup of block

trading, only prices are uncertain, whereas the purpose of VWAP trading is to achieve a

close match of the market VWAP, which implies dealing with stochastic volumes as well.

Konishi (2002) develops a static VWAP trading strategy that minimizes the expected

execution error with respect to the market realization of VWAP. A static strategy is

determined for the whole trading period and does not change as new information arrives.









It is especially suitable for trading low liquid stocks, due to statistical errors in historical

data for such stocks that make them unsuitable for forecasting.

In this chapter we develop dynamic VWAP strategies. We consider liquid stocks

and small orders, that make negligible impact on prices and volumes of the market. The

forecast of volume distribution is the target; the strategy consists in trading the order

proportionally to projected market daily volume distribution. We split a trading di- into

small intervals and estimate the market volume consecutively for each interval using linear

regression techniques.

2.2 Background and Preliminary Remarks

Consider the case when only one stock is available for trading. If at time r a

transaction of trading v units of the stock at a price p we denote this transaction by

{r, v,p}. Let Q = {{rTk, Vk,Pk}, k = 1,.., K} be a set of all transactions in the market
during a div. Then the VWAP of the stock is


VWAP = kPkVk (2-1)
Ek Vk

If a trading d i- is split into N equal intervals {(tl,-t] n = 1,.., N}, tT (n/N)T ,

where T is the length of the di -, then the corresponding expression for the daily VWAP

is given by

VWAP = N (2-2)

where

V. Ilk (23)
k: TckE(tnl,tn]
is the volume traded during time period (t,-, t,],


P, J (Ck:kE(t, 1,..,t,] PkVk)/VT, if VT > 0
(24)
[ 0, if V= 0

can be thought of as an average market price during the nth interval.









Consider an order to sell X units of stock during a trading day. We assume that

an execution of this order does not affect the prices and the volume distribution of the

market, which is reasonable for relatively small volumes X. We describe a trading strategy

by the sequence
N
{xn, n=l ,..,N}, Yxn= 1,
n=l
where x, is the proportion of the order amount X traded during the 1th time interval.

This strategy in terms of amount of stock would be


{xX n= ,..,N}.

Definition in terms of proportions is more appropriate because the value of VWAP

depends on proportions i/ lN 1 V, rather than on volumes Vi:

N .
VWAP= P U Y, N V (25)
n=l j=1 V
Values of x, are assumed to be nonnegative (i.e. the trader is not allowed to buy

stocks).

We construct the dynamic trading strategy by forecasting the volumes of stock

traded in the market during each interval of a trading d4i-. We assume that during a small

interval (about 5 min) we can perform transactions at the average market price during this

interval. Then, from (2-5) it follows that a possible way to meet the market VWAP is

to trade the order proportionally to the market volume during each interval, yielding the

same daily distribution of the traded volume as the market's one. For each interval of a

d4i- we make a forecast of the market volume that will be traded during this interval and

then trade according to this forecast. At the end of the di- we obtain the forecast of the

full daily volume distribution; the order is traded according to this distribution.

The way of dynamic computing of the distribution should be discussed first. Direct

estimations of proportions of the market volume vl, ..., VN does not guarantee that the









obtained proportions will sum up to one, since our procedure of finding each proportion

, does not take into account the previously found proportions vj, j < i. To avoid this

problem, we construct the distribution using the fractions of the remaining volume, that

has not yet been traded at current time, rather than of the total daily volume. To make

it more rigorous, suppose that (VI, V2, ..., VN) is the distribution of volume (in number of

stocks) during a div. In terms of fractions of the daily volume this distribution can be

represented as

(v1, v2, ...,VUN), '

An alternative representation is


(W W2, ..., N), tk', = '
ij=k Vy

where 1,,, is a fraction of the remaining volume after the (k 1)t interval, that is traded

during the kth interval. Figure 2-1 demonstrates the two representations of the volume

distribution. Note, that WN is al- i- equal to 1. There is a one-to-one correspondence

between representations (vi, ..., Nv) and (w1, ..., wN); the transitions between them are

given by formulas


wi "t, k= 2,..., N (2-6)
1 ii 1

and
k-1
Vi1 =W1, vk (1 t), k 2,...,N. (2-7)
i=1

The last equations follow from the fact that


(1 (1- 0-i'_ ) ..." (1- *-t _. ) t= ... .m = 1,...,i- 1.
Vi-m + --- + VN

Thus, for each interval i we make a forecast of the fraction i, of the remaining

volume. The fraction i, corresponds to the amount of the stock Vit' = r ,,, to be









traded during 1th interval, where Vir'e is the number of shares left to trade at the end of

the (i- 1)t interval. At the end of the d-., E I V =d X.

2.3 General Description of Regression Model

In the algorithm is described in detail in the next section we use the linear regression

to make a forecast of the market volume distribution. For every interval i the fraction

,' is represented as a linear combination of several informative values obtained from the

preceding time intervals. In this section we discuss some general questions regarding the

types of deviation functions we use for the regression.

Consider the general regression setting where a random variable Y is approximated by

a linear combination

Y ~ ciX1 + ... + cX, + d (2-8)

of indicator variables Xi, ..., X,. In our study the variables are modelled by a set of

scenarios

{(Y"; X, ...,X) S =l,...,S} (2-9)

For a scenario s the approximation error is


e Y" ciX' -...- cX d. (2-10)


We consider our regression model as an optimization problem of minimizing the

.. i-.regated approximation error. Below we describe penalty functions we use as the

objective.

2.3.1 Mean-Absolute Error

In the first regression model, the minimized objective is the mean-absolute error of

the approximation (2-8)

DMAD(C) = EBec. (2-11)









For the case of scenarios (2-9), the optimization problem is

is
min DMAD S Ys cX cX d (2-12)


2.3.2 CVaR-objective

The objective we used in the second regression model will be referred to as CVaR-

objective. Mean-absolute deviation equally penalizes all outcomes of the approximation

error (2-10), however our intention penalize the largest (by the absolute value) outcomes

of the error. To give a more formal definition of the CVaR-objective and show the

relevance of using it in regression problems, wee need to refer to the newly developed

theory of deviation measures and generalized linear regression, see Rockafellar et al.

(2002b).

CVaR-objective consists of two CVaR-deviations (Rockafellar et al. (2005a)) and

penalizes the a-highest and the a-lowest outcomes of the estimation error (2-10) for

a specified confidence level a (a is usually expressed in percentages). We will use a

combination of CVaR-deviations as an objective:


DCVaR(c) = CVaRf(c) + CVaR(-e) = (2-13)

= CVaR(c) + CVaR(-Q).


This expression is the difference between the average of a highest outcomes of random

variable X and the average of a lowest outcomes of X.

DCvaR(c) does not depend on the free term d in (2-8) and the minimization (2-13)

determines the optimal values of variables c, ..., c, only. The optimal value of the term d

can be found from different considerations; we use the condition that the estimator (2-8)

is non-biased.









Thus, the regression problem takes the form:


min,d CVaR [Y Y] + CVaR [F Y]
s.t. E [Y] E[Y] (2-14)

Y = ciXi + d.

Since
a 1
CVaR(1_) [-X] = CVaR, [X] -- E[X], (2-15)

optimization program (2-14) becomes:

minc,d aCVaRa [Y Y] + (1 a)CVaRi [Y Y]
s.t. E [Y] E[Y] (2-16)

Y = E ciXi + d.

The term E[Y Y] is not included into the objective function since E[Y Y] = 0 due to
the first constraint.
For the case of scenarios (2-9) the optimization problem (2-16) can be reduced to the
following linear programming problem.

min aX+ + (1 a)X-
s.t. Ef [Y' I ciX8 + d] 1YS

Xa Ca + c_S Zs 1 Za
XI-a > I-a + zS_ 1Z Eas (2-17)
> Y" (E c~x2 + d)
z a> vs (YE cXr + d) a1
Variables: c, d ER for i =1, ..., n; Xa, Xi-a E R; Z ,Z, z > 0 for s = 1,...,S.

2.3.3 Mixed Objective

Generally -I' '1. '- one can construct different penalizing functions using combinations
the mean-absolute error function and CVaR-objectives with different confidence levels a.
Denote the objective in (2-14) by DvjaR, then the problem with the mixed objective is









stated as follows


I
min /DMAD + YA DVa3
i= (2-18)

subject to constraints in (2-17),

where / E [0, 1], i 1,..., I, 3 + Ei / 1.

In our experiments, we used convex combinations of two CVaR-objectives, one with

the confidence level 50'.

min 3. D R + (1 ) D
(2-19)
subject to constraints in (2-17),

and of the mean-absolute error function and the CVaR-deviation:

min 3. DMAD + (1 /) Dvan
(2-20)
subject to constraints in (2-17) without the first one,

where the balance coefficient 3 E [0, 1]. For comparison, different types of deviations are

presented on Figure 2-2.

2.4 Experiments and Analysis

2.4.1 Model Design

Suppose that historical records for the last S di- are available, where each d4iv is

split into N equal intervals. The purpose of our study is to estimate relative volumes for

each interval of a d v. Suppose we want to forecast the fraction of the remaining volume

,,', that will be traded in the market during the kth interval. In order to forecast it',

we use the information about volumes and prices of the stock represented by variables

p k-1),s' ,, k-),s' 1, where p,, ..., p are variables taken from the ith interval and

L is the number of the preceding intervals.

We consider the following regression model
L
"", ~ + + P ) (2-1)
= 1









If from the beginning of the di- up to the current time the number of intervals is less than

L then missing intervals are picked from the previous di-. In order to approximate the kth

(k < L) interval of the d4i- parameters from intervals 1 through k 1 of the current d4i

and intervals N (L k) through N of the previous d4i are used in linear combination

(2-21).

Values of the corresponding parameters Pk-),s and fractions of the remaining volume

wf, s = 1,...,, S, i = 1, ..., L, j 1, ..., P, are collected from the preceding S d' of the
history. Thus, we have the set of scenarios


{ t, Pk- ),s (kk, -P ),s IL1) s 1,..., S}. (2-22)

Denote the linear combination
L
-1 k-1),s +-+ Pk-)s) (2-23)
l 1

as wf, the collection of 7 _k as '.

In our study we consider the following optimization problems:

Pl: MAD

min Ewk kw (2-24)

P2: CVaR
mint CVaR (wk- wk) + CVaR (wC w)
(2-25)
s.t. E[wk] E[,k]
P3: MAD+CVaR

mmn SEw k + (1 3) (CVaR(k k) + CVaR(k w k)) (2-26)

P4: Mixed CVaR

mint 3 (CVaRso(0k wk) + CVaRso%(WC- wk)) +

+(1 3) (CVaR (wk k) + CVaR (wk wk)) (2-27)
s.t. E[wk] =E[w],









where 3 e [0, 1]. Each of these problems can be reduced to linear programming ones.

By solving these problems, the optimal value of is obtained. The forecast of ,'- is

then made by the expression (2-21).

2.4.2 Nearest Sample

It is reasonable to choose for regression the "nearest" scenarios in the sense of

similarity of historical dv- to the current day. Since for each div we are interested in the

values of variables p(k-1), ",k-1),, I 1,..., L, we define the "dI-i ii'- between the

current dv and the scenario s in the following way:


D= P (p P -1)8,)2. (2-28)
i=l 1l=1

After calculating distances to all S scenarios, we choose Sbest closest scenarios corresponding

to lowest values of Di in (2-28). By doing so, we eliminate "outliers" with unusual,

with respect to the current d4i-, behavior of the market which favors the accuracy of

forecasting.

2.4.3 Data Set

The model was verified with the historical prices of IBM stock for the period April

1997 August 2002. Each d- is split into 78 5-minute intervals (daily trading hours are

9:30 AM 4:00 PM). For some experiments besides prices and volumes of the IBM stock

we also used prices and volumes of index SPY.

2.4.4 Evaluation of Model Performance

We evaluated the performance of the model by applying it to the historical data set

and forecasting the volume distribution of the IBM stock for the period of 100 (Feb. 2002

- Aug. 2002). In order to make the forecast for one d i-, a set of scenarios from the last

S admissible d4 -, was used. The d4iy is "a lihi'-- l!i if this d4i- and the previous d(i- are

full trading di -, starting and ending in usual hours, and there are no trading interruptions

during these di ,- We compared the forecasted distributions with the actual ones and

found the estimation error by averaging estimation errors for each interval over all output









d -,v Suppose that the algorithm was used to forecast volume distributions for K d iv- If

the historical volumes are

{(vk, ...,v \) k= 1,...,K}; (2-29)

the forecasted volumes are

{(Q ,...,v) k= ,...,K}, (2-30)

then the estimation error is
SK N
MAD = |v kj|. (2-31)
kin 1
k=l1 n=l1

We also calculated another error


SD =i( ( 4)2. (2 32)
SD (n n
k=l n=l

As a benchmark measuring the relative accuracy of the model "average daily

volumes" (ADV) strategy was used. This very simple strategy provides a good approximation

to VWAP. Suppose a set of historical volumes of the market:


{ (V1 ..,V), = ,..,S}. (2-33)

Denote
S N
V" = v, Vtotal V. (2-34)
s=l n=l
Then the average volume distribution is

V"
(V1, ,VN) V (2-35)
Vtotal

ADV strategy is trading according to this distribution.

An example of average volume distribution versus the actual volume evolution is

presented in Figure 2-3. It can be seen that daily volume exhibits the "U-shape" and that

the average distribution provides a good approximation to the daily volume evolution.









In the case of the data set described above, we calculated the average volume

distribution over S admissible d v-. The estimation error of the ADV strategy was

calculated using (2-32). The relative gain in accuracy of the regression algorithm was

judged by the value of
MADADV MAD
GMAD MADADV 1011' (2 36)

Relative gain in standard deviation is

SDADV SD
GSD SDAD 1("1' (237)


2.5 Experiments and Results

In our experiments, we varied the type of the objective, coefficients in the objective,

the "1. i,l !i, of the regression L, the number of admissible historical d- v- S and the

number of (nearest) scenarios Sbest used in the regression.

With respect to the parameters (2-21) we took from each interval, the experiments

were divided into two groups.

In the first group, the experiments were based on using only prices and volumes of the

stock as useful information. Namely, from each interval we used the following information:


In V and In P-los (2-38)
Open

where V is market volume during the interval, Popen and Pciose are open and close prices of

the interval. R = Pcose/Popen is, therefore, the return during the interval. Logarithms were

used to take into account the possibility that the ratios of returns and volumes, aside from

returns and volumes themselves, contain some information about the future volume. A

linear combination of logarithms of parameters can be represented as a linear combination

of the parameters and their ratios.









In the second group, we added volumes and returns of INDEX SPY to the set of

parameters taken from intervals. From each interval P = 4 parameters were used:


In V, In P ls, In VSPY and In close (2-39)
P psPYY
open open

The idea of using index information comes from the fact that evolutions of index and

stock are correlated and that the ratios of returns and prices of stock and index may also

contain useful information.

Tables 2-1, 2-2 show the results for the mean-absolute deviation used as an objective

and different values of L, S and Sbest. These tables show that including INDEX data does

not improve the accuracy of prediction. Also, as one can notice, there is a balance between

the number of terms term = L P in the linear combination (2-21) and the number of

scenarios (Sbest) used in the regression model. As Nterm increases, the model becomes

more flexible and more scenarios are needed to achieve the same level of accuracy. For

example, the best two models that use stock data (P = 2), have values of Sbest and Nterm

equal 450 and 4, 200 and 2, respectively. Also, when the index data is used, the number

of parameters P doubles, and the number of scenarios in the best models increases to

700 800 for the same regression length L.

In the case of CVaR-objective and mixed objective (Tables 2-3, 2-4), different values

of L, S and Sbest yielded a similar order of superiority as in the case of the mean-absolute

deviation.

Two more facts can be seen from the results. First, that the most successful models

use information only from the last one or two intervals, which means that the information

about the future volume is concentrated in the past few minutes. Second, the idea of

choosing the closest scenarios from the preceding history does work, especially when

a small portion of non-similar d ,v- (50 or 100 out of 500 or 800 potential scenarios) is

excluded. This agrees with the observation that most of the d ,v- are "reg- 11i enough to

be used for the estimation of the future.









In Table 2-5 we changed the form of the mixed objective, that is, differed 3 and a in

(2-17). We found that the best models have all weight put on the CVaR objective and for

a fixed balance 3 the models with small values of a are superior.

The most accurate model turned out to be the one with CVaR- objective having the

relative gain !' .

2.6 Conclusions

In this study we designed several VWAP trading strategies based on dynamic

forecasting of market volume distribution. We made estimations of market volume during

small time intervals as a linear combination of market prices and volumes and their ratios.

We found that prices and volumes do not contain much information about the future

volume. Linear regression techniques proved to be quite efficient and easily implementable

for forecasting the volumes, although the considered sets of indicator parameters do not

justify the use of regression instead of the simple average strategy.









Table 2-1. Performance of tracking models: stock vs. stock+index, full history regression
S Sbest L MAD, SD, GMAD, GSD,
STOCK
500 500 2 34.0 41.1 3.5 3.2
500 500 1 34.1 41.4 3.4 2.5
500 500 3 34.2 41.3 3.0 2.8
800 800 2 34.3 42.1 2.7 0.9
800 800 1 34.4 42.5 2.6 -0.1
800 800 3 34.4 42.2 2.4 0.5
STOCK+INDEX
500 500 1 34.1 41.0 3.2 3.5
500 500 2 34.2 41.0 3.1 3.5
800 800 1 34.2 41.2 3.1 3.1
800 800 2 34.2 40.8 3.1 3.9
800 800 3 34.3 40.7 2.7 4.0
500 500 3 34.4 41.2 2.4 3.0


Table 2-2. Performance of tracking models: stock vs. stock+index, best sample regression
S Sbest L MAD, SD, GMAD, GSD,
STOCK
500 450 2 34.0 40.8 3.7 4.0
500 200 1 34.0 39.4 3.6 7.1
500 450 1 34.0 41.0 3.6 3.3
500 400 2 34.0 40.5 3.6 4.6
500 400 1 34.0 41.0 3.5 3.3
800 500 2 34.2 41.0 3.1 3.4
500 450 3 34.2 41.1 3.0 3.1
800 700 2 34.3 42.1 2.9 0.9
STOCK+INDEX
500 450 1 34.1 40.6 3.3 4.4
500 450 2 34.1 40.1 3.2 5.4
800 750 1 34.2 40.8 3.2 3.8
800 750 2 34.2 40.8 3.2 4.0
800 700 1 34.2 40.9 3.1 3.7
800 700 2 34.2 40.5 3.1 4.6
500 400 1 34.3 41.0 3.0 3.5
500 400 2 34.3 41.1 2.8 3.3











Table 2-3. Performance of tracking models: mixed objective, changing size of history and
best sample
S Sbest L a, /3, MAD,. SD, GMAD, GSD,'
STOCK
500 450 2 30 50 34.0 40.8 3.7 4.0
500 200 1 30 50 34.0 39.4 3.6 7.2
500 450 1 30 50 34.0 41.0 3.6 3.4
500 400 2 30 50 34.0 40.5 3.6 4.6
500 400 1 30 50 34.0 41.0 3.5 3.3
500 500 2 30 50 34.0 41.1 3.5 3.2
500 500 1 30 50 34.1 41.4 3.4 2.5
800 500 2 30 50 34.2 41.0 3.0 3.4


Table 2-4. Performance of tracking models: CVaR deviation, changing size of history and
best sample


S Sbest L MAD, .
STOCK
500 400 2 30 100 33.9
500 200 1 30 100 33.9
500 200 2 30 100 33.9
500 450 2 30 100 33.9
500 480 2 30 100 33.9
500 400 3 30 100 34.0
500 400 1 30 100 34.0
500 450 1 30 100 34.0


SD, GMAD, GSD,


40.7
39.7
39.4
40.8
40.8
40.4
41.1
41.0


Table 2-5. Performance of tracking models: mixed objective
S Sbest L a, /3, MAD, SD, GMAD, GSD,
STOCK


500 450 2 20
500 450 2 30
500 450 2 10
500 450 2 5
500 450 2 10
500 450 2 20
500 450 2 5
500 450 2 30


100 33.9
100 33.9
100 33.9
30 33.9
30 33.9
30 34.0
100 34.0
30 34.0


40.7
39.6
39.8
40.6
40.7
40.7
41.6
41.7





























12 00%


10.00oo
-- Day 1 -7r Day 2

8.00% .
-o-Day 3 -*-Average

2 6.00%


4.00%


2.00%


0.00%
1 3 5 7 9 11 13 151719 21 23 25 27 29 31 3 35 37 39
Time

Figure 2-3. Daily volume distributions









CHAPTER 3
PRICING EUROPEAN OPTIONS BY NUMERICAL REPLICATION



3.1 Introduction

Options pricing is a central topic in financial literature. A reader can find an excellent

overview of option pricing methods in Broadie and Detemple (2004). The algorithm

for pricing European options in discrete time presented in this paper has common

features with other existing approaches. We approximate an option value by a portfolio

consisting of the underlying stock and a risk-free bond. The stock price is modelled by

a set of sample-paths generated by a Monte-Carlo or historical bootstrap simulation.

We consider a non-self-financing portfolio dynamics and minimize the sum of squared

additions/subtractions of money to/from the hedging portfolio at every re-balancing

point, averaged over a set of sample paths. This error minimization problem is reduced

to quadratic programming. We also include constraints on the portfolio hedging strategy

to the quadratic optimization problem. The constraints dramatically improve numerical

efficiency of the algorithm.

Below, we refer to option pricing methods directly related to our algorithm. Although

this paper considers European options, some related papers consider American options.

Replication of the option price by a portfolio of simpler assets, usually of the

underlying stock and a risk-free bond, can incorporate various market frictions, such

as transaction costs and trading restrictions. For incomplete markets, replication-based

models are reduced to linear, quadratic, or stochastic programming problems, see, for

instance, Bouchaud and Potters (2000), Bertsimas et al. (2001), Dembo and Rosen (1999),

Coleman et al. (2004), Naik and Uppal (1994), Dennis (2001), Dempster and Thompson



This chapter is based on the paper Ryabchenko, V., Sarykalin, S., and Uryasev,
S. (2004) Pricing European Options by Numerical Replication: Quadratic Programming
with Constraints. Asia-P i..: I. Financial Markets, 11(3), 301-333.









(2001), Edirisinghe et al. (1993), Fedotov and Mikhailov (2001), King (2002), and Wu and

Sen (2000).

Analytical approaches to minimization of quadratic risk are used to calculate an

option price in an incomplete market, see Duffie and Richardson (1991), F6llmer and

Schied (2002), F6llmer and Schweizer (1989), Schweizer (1991, 1995, 2001).

Another group of methods, which are based on a significantly different principle,

incorporates known properties of the shape of the option price into the statistical analysis

of market data. Ait-Sahalia and Duarte (2003) incorporate monotonic and convex

properties of European option price with respect to the strike price into a polynomial

regression of option prices. In our algorithm we limit the set of feasible hedging strategies,

imposing constraints on the hedging portfolio value and the stock position. The properties

of the option price and the stock position and bounds on the option price has been studied

both theoretically and empirically by Merton (1973), Perrakis and Ryan (1984), Ritchken

(1985), Bertsimas and Popescu (1999), Gotoh and Konno (2002), and Levi (85). In

this paper, we model stock and bond positions on a two-dimensional grid and impose

constraints on the grid variables. These constraints follow under some general assumptions

from non-arbitrage considerations. Some of these constraints are taken from Merton

(1973).

Monte-Carlo methods for pricing options are pioneered by Boyle (1977). They

are widely used in options pricing: Joy et al. (1996), Broadie and Glasserman (2004),

Longstaff and Schwartz (2001), Carriere (1996), Tsitsiklis and Van Roy (2001). For a

survey of literature in this area see Boyle (1997) and Glasserman (2004). Regression-based

approaches in the framework of Monte-Carlo simulation were considered for pricing

American options by Carriere (1996), Longstaff and Schwartz (2001), Tsitsiklis and Van

Roy (1999, 2001). Broadie and Glasserman (2004) proposed stochastic mesh method which

combined modelling on a discrete mesh with Monte-Carlo simulation. Glasserman (2004),

showed that regression-based approaches are special cases of the stochastic mesh method.









The algorithm uses the hedging portfolio to approximate the price of the option. We

aimed at making the hedging strategy close to real-life trading. The actual trading occurs

at discrete times and is not self-financing at re-balancing points. The shortage of money

should be covered at any discrete point. Large shortages are undesirable at any time

moment, even if self-financing is present.

The pricing algorithm described in this paper combines the features of the above

approaches in the following way. We construct a hedging portfolio consisting of the

underlying stock and a risk-free bond and use its value as an approximation to the

option price. We aimed at making the hedging strategy close to real-life trading. The

actual trading occurs at discrete times and is not self-financing at re-balancing points.

The shortage of money should be covered at any discrete point. Large shortages

are undesirable at any time moment, even if self-financing is present. We consider

non-self-financing hedging strategies. External financing of the portfolio or withdrawal

is allowed at any re-balancing point. We use a set of sample paths to model the underlying

stock behavior. The position in the stock and the amount of money invested in the bond

(hedging variables) are modelled on nodes of a discrete grid in time and the stock price.

Two matrices defining stock and bond positions on grid nodes completely determine the

hedging portfolio on any price path of the underlying stock. Also, they determine amounts

of money added to/taken from the portfolio at re-balancing points. The sum of squares

of such additions/subtractions of money on a path is referred to as the squared error on a

path.

The pricing problem is reduced to quadratic minimization with constraints. The

objective is the averaged quadratic error over all sample paths; the free variables are stock

and bond positions defined in every node of the grid. The constraints, limiting the feasible

set of hedging strategies, restrict the portfolio values estimating the option price and stock

positions. We required that the average of total external financing over all paths equals

to zero. This makes the strategy "self-financing on avi I, We incorporated monotonic,









convex, and some other properties of option prices following from the definition of an

option, a non-arbitrage assumption, and some other general assumptions about the

market. We do not make assumptions about the stock process which makes the algorithm

distribution-free. Monotonicity and convexity constraints on the stock position are

imposed. Such constraints reduce transaction costs, which are not accounted for directly in

the model. We aim to prohibit sharp changes in stock and bond positions in response to

small changes in the stock price or in time to maturity.

We performed two numerical tests of the algorithm. First, we priced options on the

stock following the geometric Brownian motion. Stock price is modelled by Monte-Carlo

sample-paths. Calculated option prices are compared with the known prices given by the

Black-Scholes formula. Second, we priced options on S&P 500 Index and compared the

results with actual market prices. Both numerical tests demonstrated reasonable accuracy

of the pricing algorithm with a relatively small number of sample-paths (considered cases

include 100 and 20 sample-paths). We calculated option prices both in dollars and in the

implied volatility format. The implied volatility matches reasonably well the constant

volatility for options in the Black-Scholes setting. The implied volatility for S&P 500 index

options (priced with 100 sample-paths) tracks the actual market volatility smile.

The advantage of using the squared error as an objective can be seen from the

practical perspective. Although we allow some external financing of the portfolio along the

path, the minimization of the squared error ensures that large shortages of money will not

occur at any point of time if the obtained hedging strategy is practically implemented.

Another advantage of using the squared error is that the algorithm produces a

hedging strategy such that the sum of money added to/taken from the hedging portfolio

on any path is close to zero. Also, the obtained hedging strategy requires zero average

external financing over all paths. This justifies considering the initial value of the hedging

portfolio as a price of an option. We use the notion of "a price of an option in the

practical setting which is the price a trader agrees to buy/sell the option. In the example









of pricing options on the stock following the geometric Brownian motion the algorithm

finds hedging strategy which delivers requested option 1p .iments at expiration with high

precision on many considered sample paths. Therefore, we claim that the initial value of

the portfolio can be considered as an estimate of the market price.

We assume an incomplete market in this paper. We use the portfolio of two

instruments the underlying stock and a bond to approximate the option price and

consider many sample paths to model the stock price process. As a consequence, the

value of the hedging portfolio may not be equal to the option 1p ioff at expiration on

some sample paths. Also, the algorithm is distribution-free, which makes it applicable to

a wide range of underlying stock processes. Therefore, the algorithm can be used in the

framework of an incomplete market.

Usefulness of our algorithm should be viewed from the perspective of practical options

pricing. Commonly used methods of options pricing are time-continuous models assuming

specific type of the underlying stock process. If the process is known, these methods

provide accurate pricing. If the stock process cannot be clearly identified, the choice of

the stock process and calibration of the process to fit market data may entail significant

modelling error. Our algorithm is superior in this case. It is distribution-free and is based

on realistic assumptions, such as discrete trading and non-self-financing hedging strategy.

Another advantage of our algorithm is low back-testing errors. Time-continuous

models do not account for errors of implementation on historical paths. The objective in

our algorithm is to minimize the back-testing errors on historical paths. Therefore, the

algorithm has a very attractive back-testing performance. This feature is not shared by

any of time-continuous models.









3.2 Framework and Notations

3.2.1 Portfolio Dynamics and Squared Error

Consider a European option with time to maturity T and strike price X. We suppose

that trading occurs at discrete times tj, j = 1, ..., N, such that


0 = to < tl < ... < tN = T, tj+l tj = const, j = 0,1,..., N 1.


We denote the position in the stock at time tj by uj, the amount of money invested in the

bond by vj, the risk-free rate by r, and the stock price at time tj by Sj.

The price of the option at time tj is approximated by the price cj of a hedging

portfolio consisting of the underlying stock and a risk-free bond. The hedging portfolio is

rebalanced at times tj, j 1, ..., N- 1. Suppose that at the time tj_- the hedging portfolio

consists of uj-1 shares of the stock and _1 dollars invested in the bondI The value of

the portfolio right before the time tj is uj-~Sj + (1 + r)vj-1. At time tj the positions in

the stock and in the bond are changed to uj and vj, respectively, and the portfolio value

changes to ujSj + vj. We consider a non-self-financing portfolio dynamics by allowing the

difference

aj = ujSj + vj (uj-Sj + (1 + r)vj-1) (3-1)

to be non-zero. The value aj is the excess/shortfall of the money in the hedging portfolio

during the interval [tjl, tj]. In other words, aj is the amount of money added to (if

aj > 0) or subtracted from (if aj < 0) the portfolio during the interval [tj-1, tj]. Thus, the

inflow/outflow of money to/from the hedging portfolio is allowed.



1 Below, the number of shares of the stock and the amount of money invested in the
bond are referred to as positions in the stock and in the bond.









We require that the value of the hedging portfolio at expiration be equal to the option

p, .-off h(SN), UNSN + VN = h(SN), where


Smax{0, S X} for call options;
h(S)
max{0, X S} for put options.

The non-self-financing portfolio dynamics is given by


Uj+lSj+l + Vj+l = UjSj+l + (1 + r)vj + aj, j = 0,..., N 1, (3-2)

where the portfolio value at time tj is cj = jSj + vj, j = 0,..., N.

The degree to which a portfolio dynamics differs from a self-financing one is an

important characteristic, essential to our approach. In this paper, we define a squared

error on a path,
N
A (ae -r)2, (3 3)
j=1
to measure the degree of "non-self-f"n i~1 i'i The reasons for choosing this particular

measure will be described later on.

3.2.2 Hedging Strategy

We assume that the composition of the hedging portfolio depends on time and the

stock price. We define a hedging strategy as a function determining the composition of

a hedging portfolio for any given time and the stock price at that time. If the hedging

strategy is defined, the corresponding portfolio management decisions for the stock price

path So, S', ..., SN are given by the sequence (uo, ',,), (Ul, vl), ..., (UN, UN).

A hedging strategy is modelled on a discrete grid with a set of approximation rules.

Consider a grid consisting of nodes {(j, k); j = 0,..., N, k = 1,..., K} in the time and the

stock price. The index j denotes time and corresponds to time tj; the index k denotes the

stock price and corresponds to the stock price Sk (we use the tilde sign for stock values

on the grid to distinguish them from stock values on sample-paths). Stock prices 5k,









k = 1,..., K on the grid are equally distanced in the logarithmic scale, i.e.

S, < S2 < ... < SK, ln(Sk+) n(Sk) = cost.

Thus, the node (j, k) of the grid corresponds to time tj and the stock price Sk. To every
node (j, k) we assigned two variables Uf and V, representing the composition of the

hedging portfolio at time tj with the stock price Sk. The pair of matrices

U1 U/ ... UN Vo1 V,1 ... V1
Su2 ...U2 /2 V2 /2
[Uf ] U02 1 [Vk] 2 1 (3-4)

U0K K ... K VK K ... vrK

are referred to as a hedging str il, 1; These matrices define portfolio management decisions
on the discrete set of the grid nodes. In order to set those decisions on any path, not

necessarily going through grid points, approximation rules are defined.

We model the stock price dynamics by a set of sample paths


(S, Sp',... 1,...,P} (3-5)

where So is the initial price. Let variables and j define the composition of the hedging
portfolio on path p at time tj, where p = 1,..., P, j = 0,..., N. These variables are
approximated by the grid variables Uf and Vk as follows. Suppose that {So, Sf, ..., S } is

a realization of the stock price, where SJ denotes the price of the stock at time tj on path

p, j = 0,..., N, p = 1,..., P. Let u and vp denote the amounts of the stock and the bond,
respectively, held in the hedging portfolio at time tj on path p. Variables up and vo are
linearly approximated by the grid variables Uf and V0 as follows


uP = apf (j')+ + (1 a~P) ( 'p), ~f PVk(p)+l + (1 a P)Vk(jp), (3 6)










In SP In Sk(j,p)
where a = and k(j,p) is such that Sk(j,p) < S < Sk(j,p)+i.
S n Sk(j,p)+l In Sk(j,p)
According to (3-1), we define the excess/shortage of money in the hedging portfolio

on path p at time tj by


a uJ+,S1, + v, (ujSj + (1 + r)vJ).


The squared error p on path p equals

N
p (ae rj)2. (3 7)
j=1

We define the average squared error S on the set of paths (3-5) as an average of squared

errors Sp over all sample paths (3-5)

P N
S ^ i>ri, (3-8)
p=1 j=1

The matrices [Utf] and [Vjk] and the approximation rule (3-6) specify the composition

of the hedging portfolio as a function of time and the stock price. For any given stock

price path one can find the corresponding portfolio management decisions {(uj, vj)j =

0,..., N 1}, the value of the portfolio cj = Sjuj + vj at any time tj, j =0,..., N, and the

associated squared error.

The value of an option in question is assumed to be equal to the initial value of the

hedging portfolio. First columns of matrices [Ui] and [Vj], namely the variables Uk and

VOk, k 1,..., K, determine the initial value of the portfolio. If one of the initial grid

nodes, for example node (0, k), corresponds to the stock price So, then the price of the

option is given by UkSo + VOk. If the initial point (t = 0, S = So) of the stock process falls

between the initial grid nodes (0, k), k = 1,..., K, then approximation formula (3-6) with

j = 0 and So = So is used to find the initial composition (uo, ,,) of the portfolio. Then,

the price of the option is found as uoSo + vo.









3.3 Algorithm for Pricing Options

This section presents an algorithm for pricing European options in incomplete

markets. Subsection 3.1, presents the formulation of the algorithm; subsections 3.2 3.4

discuss the choice of the objective and the constraints of the optimization problem.

3.3.1 Optimization Problem

The price of the option is found by solving the following minimization problem.

N P
mmin S- EE ({ S + v I (1 + r)v I}e-r")2 (3-9)
j=1 p=l

subject to
tN P
EE {upsp + -u Is (1 + r)v }e -
j=1 p=1

UNk+ + V = h(), k 1,...,K,

approximation rules (3-6),

constraints (3-10)-(3-18) (defined below) for call options,

or constraints (3-19)-(3-27) (defined below) for put options,

free variables: U k, j = 0,..., N, k= 1,..., K.

The objective function in (3-9) is the average squared error on the set of paths (3-5). The

first constraint requires that the average value of total external financing over all paths

equals to zero. The second constraint equates the value of the portfolio and the option

p ioff at expiration. Free variables in this problem are the grid variables Uf and Vk; the

path variables up and vo in the objective are expressed in terms of the grid variables using

approximation (3-6). The total number of free variables in the problem is determined

by the size of the grid and is independent of the number of sample-paths. After solving

the optimization problem, the option value at time tj for the stock price Sj is defined

by ujSj + vj, where uj and vj are found from matrices [Ut] and [VJ], respectively, using
C~~IVIU III II*UI~U C3I I*II









approximation rules (3-6). The price of the option is the initial value of the hedging

portfolio, calculated as uoSo + ,,

The following constraints (3-10)-(3-18) for call options or (3-19)-(3-27) for put

options impose restrictions on the shape of the option value function and on the position

in the stock. These restrictions reduce the feasible set of hedging strategies. Subsection 3.3

discusses the benefits of inclusion of these constraints in the optimization problem.

Below, we consider the constraints for European call options. The constraints for

put options are given in the next section, together with proofs of the constraints. Most

of the constraints are justified in a quite general setting. We assume non-arbitrage and

make 5 additional assumptions. Proofs of two constraints on the stock position (horizontal

monotinicity and convexity) in the general setting will be addressed in subsequent papers.

In this paper we validate these inequalities in the Black-Scholes case.

The notation CO stands for the option value in the node (j, k) of the grid,


Ckf UjS1] + V1k

The strike price of the option is denoted by X, time to expiration by T, one period

risk-free rate by r.

Constraints on Call Option Value

Immediate exercise constraints. The value of an option is no less than the value of

its immediate exercise2 at the discounted strike price,

C > t Xe -T(T- (3-10)



2 European options do not have the feature of immediate exercise. However, the right
part of constraint (3-10) coincides with the immediate exercise value of an American
option having the current stock price S and the strike price Xei-(T-).









* Option price sensitivity constraints.


j 0,...,N k 1,...,K 1.

This constraints bound sensitivity of an option price to changes in the stock price.

* Monotonicity constraints.

0. Vertical monotonicity. For any fixed time, the price of an option is an increasing

function of the stock price.

ck+1 > "
j C + > Cf, j = 0,...,N; k= 1,...,K 1. (3-12)

0. Horizontal monotonicity. The price of an option is a decreasing function of

time.

C +1 j 0, j ..., N- 1; k= 1,...,K. (3-13)

* Convexity constraints. The option value is a convex function of the stock price.

Cf+l < c+Ick + (1 !- 3+lf)Ck+2

where +1 is such that S+1 = f+1j + (1 +1)S2, (3 14)

j 0, ...,N; k 1,..., K 2.

Constraints on Stock Position for Call Options

Let us define k, such that Sk < X < SC+l.

* Stock position bounds. The stock position value lies between 0 and 1

S< U < 1, j =0,...,N, k 1,...,K. (3-15)

* Vertical monotonicity. The position in the stock is an increasing function of the

stock price,

Gk+l > j, j =0,...,N; k =1,...,K 1. (3-16)









Horizontal monotonicity. Above the strike price the position in the stock is an

increasing function of time; below the strike price it is a decreasing function of time,

Uk < U if k > k; U > U if k < k. (3-17)


Convexity constraints. The position in the stock is a concave function in the stock

price above the strike and a convex function in the stock price below the strike,

(1- +l)uk+2 + +l U < U+1, if k > k,

(1 -1)jk-2 u1k k- if k < k, (3-18)

where /3 is such that S= 3S-1 (1 /t, I (k + 1), k 1).


3.3.2 Financial Interpretation of the Objective

There are two reasons for considering the average squared error: financial interpretation

and accounting for transaction costs. The financial interpretation is discussed here, while

the accounting for transaction costs is considered in subsection (3.3.4).

The expected hedging error is an estimate of "non-self-financity" of the hedging

strategy. The pricing algorithm seeks a strategy as close as possible to a self-financing

one, satisfying the imposed constraints. On the other hand, from a trader's viewpoint, the

shortage of money at any portfolio re-balancing point causes the risk associated with the

hedging strategy. The average squared error can be viewed as an estimator of this risk on

the set of paths considered in the problem.

There are other v--v to measure the risk associated with a hedging strategy. For

example, Bertsimas et al. (2001) considers a self-financing dynamics of a hedging

portfolio and minimizes the squared replication error at expiration. In the context of

our framework, different estimators of risk can be used as objective functions in the

optimization problem (3-9) and, therefore, produce different results. However, considering

other objectives is beyond the scope of this paper.









3.3.3 Constraints

We use the value of the hedging portfolio to approximate the value of the option.

Therefore, the value of the portfolio is supposed to have the same properties as the value

of the option. We incorporated these properties into the model using constraints in the

optimization problem. The constraints (3-10)-(3-14) for call options and (3-19)-(3-23) for

put options follow under quite general assumptions from the non-arbitrage considerations.

The type of the underlying stock price process pl .1' no role in the approach: the set

of sample paths (3-5) specifies the behavior of the underlying stock. For this reason,

the approach is distribution-free and can be applied to pricing any European option

independently of the properties of the underlying stock price process. Also, as shown in

section 5 presenting numerical results, the inclusion of constraints to problem (3-9) makes

the algorithm quite robust to the size of input data.

The grid structure is convenient for imposing the constraints, since they can be stated

as linear inequalities on the grid variables Uf and Vk. An important property of the

algorithm is that the number of the variables in problem (3-9) is determined by the size of

the grid and is independent of the number of sample paths.

3.3.4 Transaction Costs

The explicit consideration of transaction costs is beyond the scope of this paper. We

postpone this issue to following papers. However, we implicitly account for transaction

costs by requiring the hedging strategy to be "smooth", i.e., by prohibiting significant

rebalancing of the portfolio during short periods of time or in response to small changes in

the stock price. For call options, we impose the set of constraints (3-16)-(3-18) requiring

monotonicity and concavity of the stock position with respect to the stock price and

monotonicity of the stock position with respect to time (constraints (3-25)-(3-27) for put

options are presented in the next section). The goal is to limit the variability of the stock

position with respect to time and stock price, which would lead to smaller transaction

costs of implementing a hedging strategy. The minimization of the average squared error is









another source of improving "smoothness" of a hedging strategy with respect to time. The

average squared error penalizes all shortages/excesses ap of money along the paths, which

tends to flatten the values a over time. This also improves the "smoothness" of the stock

positions with respect to time.

3.4 Justification Of Constraints On Option Values And Stock Positions

3.4.1 Constraints for Put Options

This subsection presents constraints in optimization problem (3-9) for pricing

European put options.

Constraints on value of Put options.

Immediate (::; i, -. constraints.


P > Xe- -'T sk (3-19)

Option price sensitivity constraints.

pk (3-20)
j 0,...,N 1, k = 1,...,K 1.

Monotonicity constraints.

0. Vertical monotonicity.


P j 1 j 0,...,N; k 1,..., K (3-21)

0. Horizontal monotonicity.


Pl < Pk + X(e-r(T-tj+l) e-(T-tj)), = 0,...,N 1; k =1,...,K. (3-22)

Convexity constraints.

p+l < Q k+lpk + (1 Di klpk+2

where 3+l is such that S,+1 Q+1, + (1 +1)S +2 (3-23)

j = 0,...,N; k = ,...,K -2.









Constraints on stock position for put options

In the following constraints, k is such that S < X < Sk+

Stock Position Bounds

0 < < 1, j 0,...,N; k 1,...,K. (3 24)

Vertical monotonicity

+i >U k j 0,...,N; k 1,..., K- 1. (3-25)

Horizontal monotonicity

UT < Uk, if k > uk > U if k < k (3-26)


Convexity constraints

(1 3+1)U+2 + 3+1k < Uk+1, if k > k,

(1 /-1)uk-2 +2 -1 > U-1, if k < ,
(3-27)
where /3 is such that S. 0.-1 + (1 t- 3/ 1

S(k + 1), (k 1).


3.4.2 Justification of Constraints on Option Values

This subsection proves inequalities on put and call option values under certain

assumptions. Properties of option values under various assumptions were thoroughly

studied in financial literature. In optimization problem (3-9) we used the following

constraints holding for options in quite a general case. We assume non-arbitrage and make

technical assumptions 1-5 (used by Merton (1973) for deriving properties of call and put

option values. Some of the considered properties of option values are proved by Merton

(1973). Other inequalities are proved by the authors.

The rest of the section is organized as follows. First, we formulate and prove

inequalities (3-10)-(3-14) for call options. Some of the considered properties of option









values are not included in the constraints of the optimization problem (3-9), they are

used in proofs of some of constraints (3-10)-(3-14). In particular, weak and strong scaling

properties and two inequalities preceding proofs of option price sensitivity constraints and

convexity constraints are not included in the set of constraints.

Second, we consider inequalities (3-19)-(3-23) for put options. We provide proofs of

vertical and horizontal option price monotonicity; proofs of other inequalities are similar to

those for call options.

We use the following notations. C(St, T, X) and P(St, T, X) denote prices of call and

put options, respectively, with strike X, expiration T, when the stock price at time t is St.

When appropriate, we use shorter notations Ct and Pt to refer to these options.

Similar to Merton (1973), we make the following assumptions to derive inequalities

(3-10)-(3-14) and (3-19)-(3-23).

Assumption 1. Current and future interest rates are positive.

Assumption 2. No dividends are paid to a stock over the life of the option.

Assumption 3. Time homogeneity assumption.

Assumption 4. The distributions of the returns per dollar invested in a stock for any

period of time is independent of the level of the stock price.

Assumption 5. If the returns per dollar on stocks i and j are identically distributed,

then the following condition hold. If Si = Sj, T, = Tj, X, = Xj; then Claimi(Si, T, Xi)

Claimj(Sj, Tj, Xj), where Claimi and Claimj are options (either call or put) on

stocks i and j respectively.

Below are the proofs of inequalities (3-10)-(3-14).

1. "Immediate (::; i, constraints. Merton (1973)


C > [St X e-rT-t +.









E Put-Call parity, Ct Pt + X e-r(T-t) = St, and non-negativity of a put option

price (Pt > 0) imply Ct > St Xe-'T-t). This inequality combined with Ct > 0 gives

Ct > Max(0, St X e-r-t)) = [St X e-r(T-t)]+. U

2. Scaling property.

a) Weak scaling property. Merton (1973)

For any k > 0 consider two stock price processes S(t) and k S(t). For these processes,

the following inequality is valid C(k St, T, k X) = k C(St, T, X), where St is the value of

the process S(t) at time t.

O At expiration T, the price of the first stock is ST, the value of the second stock is

k ST. By definition, the values of call options written on the first stock (with strike X)

and on the second stock (with strike k X) are C(St, T, X) = Max[O, ST X] and C(k

St, T, kX) = Max[O,k.STr-kX], respectively. From Max[O, k.ST-k.X] = k.Max[O, ST-

X] and non-arbitrage considerations, it follows that C(k St, T, k X) = k C(St, T, X). U

b) Strong scaling property. Merton (1973)

Under assumptions 4 and 5, the call option price C(S, T, X) is homogeneous of degree

one in the stock price per share and exercise price. In other words, if C(S, T, X) and

C(k S, T, k X) are option prices on stocks with initial prices S and k S and strikes X

and k X, respectively, then C(k S, T, k X)= k C(S, T, X).

O Consider two stocks with initial prices S1 and S2; define k = S2/S1. Let zi(t) be

the return per dollar for stock i, i = 1, 2. Consider two call options, A and B, on stock 2.

Option A is written on 1/k shares of stock 2 and has strike price Xi; option B is written

on one share of stock 2 and has strike X2 = k X1. Prices C2(S1, T, Xi) and C2(S2, T, X2)

of these options are related as C2(S2T, ,TX2, k X) k C2(, T, X),

according to the weak scaling property.

Now consider an option C with the strike Xi written on one share of the stock 1.

Denote its price by C',(SI, T, X). Options A and C have equal initial prices S1 = -S2,

time to expiration T, and X1. Moreover, the distribution of returns per dollar zi(t) for









stocks i = 1,2 are the same. Hence, from assumption 5, C1(SI, T, XI) C2(SI, T, XI),

and, therefore, C2(S2, T, X2) k C(SI, T, X1), which concludes the proof. U

3. Option price sensitivity constraints.

a) First, we derive an inequality taken from Merton (1973). In part b) we apply it to

obtain the sensitivity constraint on the call option price.

For any X1, X2 such that 0 < X1 < X2, the following inequality holds


C(St,T, X1) < C(St,T,X2) + (X2 XI) e-rTr-t)

O Consider two portfolios. Portfolio A contains one call option with strike X2

and (X2 X1) e-r'(T-t) dollars invested in bonds. Portfolio B consists of one call

option with strike X1. Both options are written on the stock following the process St.

At expiration, the value of portfolio A is max{0, ST X2} + X2 X1, the value of

portfolio B is max{0, ST X1}. The value of portfolio A is alv--v- greater than the value

of portfolio B at expiration. This statement with non-arbitrage considerations implies that

C(St, T, X) + (X2 XI) er.(T-t) > C(St, T, X1). U

b) Consider two options with strike X and initial prices S2 and S1, S2 > S1. Denote

7 = S2/S1. The following inequality takes place,


C(S, T,X) < yC(SI, T,X) + X(7 )e-r(T-t).

1 SI
D Let a Using inequality presented in a), we write C(SI, T, aX) <
7 S2
C(S1, T, X) + (X aX)e-r(T-t). Applying scaling property to the left-hand side of this
S2 S1
inequality yields C(Si, T, aX) C(S T, X) C(S2 a, T, aX) = aC(S2, T, X).
Therefore,

aC(S, T,X) < C(SI, T,X)+ X( a)e-r'r-t)

Dividing by a and substituting 1/a = 7 we get C(S2, T, X) < 7C(SI, T, X) + X(7 -

l)e-r(T-t).









4. Vertical option price monotonicity.

For two options with strike X and initial prices S1 and S2, S2 > SI, there holds


C(S1, T, X) < C(S, T, X) .
S2

O For any strike X1 < X, from non-arbitrage assumptions we have C(SI, T, X) <

C(S1, T, X1). Applying scaling property to the right-hand side gives
X S1
C(SIJT,X) < C(SI X T,X). By setting X2 2 < X, we get C(S, T,X) <

SC(S2, T, X).
5. Horizontal option price monotonicity.

Let C(t, S, T, X) denote the price of a European call option with initial time t, initial

price at time t equal to S, time to maturity T, and strike X. Under the assumptions 1, 2

and 3 for any t, u, t < u, the following inequality holds,


C(t, S,T,X) > C(u, S, T, X).

O Similar to C(t, S, T, X), define A(t, S, T, X) to be the value of American call option

with parameters t, S, T, and X meaning the same as in C(t, S, T, X). Time homogeneity

assumption 2 implies that two options with different initial times, but equal initial and

strike prices and times to maturity should have equal prices: A(t, S, T, X) = A(u, S, T +

u t, X). On the other hand, non-arbitrage considerations imply A(u, S, T + u t, X) >

A(u, S, T, X). Combining the two inequalities yields A(t, S, T, X) > A(u, S, T, X). Since

the value of an American call option is equal to the value of the European call option

under assumption 1, the above inequality also holds for European options: C(t, S, T, X) >

C(u, S, T, X). U

6. Convexity. Merton (1973).

a) C is a convex function of its exercise price: for any X1 > 0, X2 > 0 and A E [0, 1]


C(S,T, A X, + (1 A) X2) < A. C(S,T,X) + (1 A) C(S, T, X2).









E Consider two portfolios. Portfolio A consists of A options with strike X1 and (1- A)

options with strike X2; portfolio B consists of one option with strike A X1 + (1 A) X2.

Convexity of function max{0, x} implies that the value of portfolio A at expiration in no

less than the value of portfolio B at expiration. Amax{0, ST X1} + (1 A) max{0, ST -

X2} > max{0, ST (A X1 + (1 A) X2)}. Hence, from non-arbitrage assumptions,

portfolio A costs no less than portfolio B: A C(S, T, X1) + (1 A) C(S, T, X2) >

C(S,T, A X, + (1 A) X2). -

b) Under the assumption 4, option price C(S, T, X) is a convex function of the stock

price: for any S1 > 0, S2 > 0 and A E [0, 1] there holds,


C(A S, + (1 A) S, T,X) < A C(S,, T,X) +(1 A) C(S2, T, X).

O Denote S3 = AS1 + (1 A)S2. ('!I...-. X1, X2 and a such that Xi = X/S1,

X2 = X/S2, a S1/S3 e [0, 1], and denote X3 = aX1 + (1 a)X2.

Consider an inequality C(1, T, X) < a C(1,T, X) + (1 a) C(1,T, X2) following

from convexity of option price with respect to the strike price (proved in a) ). Since

AS,
aS3 = AS1, ( -1 )S3 = S3 3 AS, = (1 A)S2, (3-28)


multiplying both sides of the previous inequality by S3 gives S3 C(1, T, X3) < A Si

C(1, T, Xi) + (1 A) S2 C(1, T, X2). Further, using the weak scaling property, we get

C(S3, T, S3 X3) < A C(SI, T, S Xi) + (1 A) C(S2, T,S2 X2). Using definitions of Xi

and X2 and expanding S3X3 as

a( 1 a
S3(aX + (1 a)X2) = S3X + -S2


(SA11 S3 -S1 1\ (A 1-
ASl S3 S- AS A 3- A
= S 3 SI S3 S 2 ) 3 S>3 S>3 )









we arrive at C(S3,T,X) < .- C(Si,T,X) + (1- A) C(S2,T,X), as needed. U

Constraints on European put option values are presented below. We state them in

the same order as the constraints for call options. Proofs are given for vertical option

price monotonicity constraints; other inequalities can be proved using put-call parity and

considerations similar to those in the proofs of corresponding inequalities for call options.

1. Iii' 'i il.. (::, i- constraints.



Pt > [X e-rT-) St]+

2. Scaling property.

a) Weak scaling property.

For any k > 0, consider two stock price processes S(t) and k S(t). For these processes

the following inequality holds: P1(k St, T,k X) k P2(St, T, X), where P1 and P2 are

options on the first and the second stocks respectively.

b) Strong scaling property.

Under the assumptions 4 and 5, put option value P(S, T, X) is homogeneous of

degree one in the stock price and the strike price, i.e., for any k > 0, P(k S, T, k X)

k P(S,T,X).

3. Option price sensitivity constraints.

a) For any X1, X2, 0 < X1 < X2, the following inequality is valid,


P(St,T, X2) < P(St,T, X) + (X2 Xi) eT-t)


b) For initial stock prices S1 and S2, S1 < S2


P(S1,T,X) < 7P(S2,T,X) + X(1- )e-(T-t


where 7 = S1/S2.

4. Vertical option price monotonicity.









a) For any a E [0, 1] the following inequality is valid:


P(S,X -a)

D Consider portfolio A consisting of one option with strike a X, and portfolio

B consisting of a options with strike X. We need to show that portfolio B ah--iv

outperforms portfolio A. This follows from non-arbitrage consideration since at expiration

the value of portfolio B is greater or equal to the value of portfolio A: [X a ST]+ <

a [X ST]+, 0 < a < 1. 0

b) For any S1, S2, S1 < S2, there holds P(S2,T,X) < P(S, T,X).

D Consider an inequality P(SI, aX) < aP(SI, X), 0 < a < 1, proved above. Set

a = S1/S2 c [0, 1]. Applying the weak scaling property, we get

1
P(S1 -a,, T, aX) < aP(SI, T, X),
a
1
P(S -,T,X) < P(S, T, X),
a
P(S2,T,X) < P(S, T,X).


5. Horizontal option price monotonicity.

Under assumptions 1, 2, and 3, for any initial times t and u, t < u, the following

inequality is valid:


P(t, S, T, X) > P(u, S, T, X) + X (e-r.(T-) -r(T-u))


where P(r, S, T, X) is the price of a European put option with initial price 7, initial price

at time 7 equal to S, time to maturity T, and strike X.

6. Convexity.

a) P(S, T, X) is a convex function of its exercise price X

b) Under assumption 4, P(S, T, X) is a convex function of the stock price.









3.4.3 Justification of Constraints on Stock Position

This subsection proves/validates inequalities (3-15)-(3-18) and (3-24)-(3-27) on

the stock position. Stock position bounds and vertical monotonicity are proven in the

general case (i.e. under assumptions 1-5 and the non-arbitrage assumption); horizontal

monotonicity and convexity are justified under the assumption that the stock process

follows the geometric Brownian motion.

The notation C(S, T, X) (P(S, T, X)) stands for the price of a call (put) option

with the initial price S, time to expiration T, and the strike price X. The corresponding

position in the stock (for both call and put options) is denoted by U(S, T, X).

First, we present the proofs of inequalities (3-15)-(3-18) for call options.

1. Vertical monotonicity (Call options).

U(S, t, X) is an increasing function of S.

O This property immediately follows from convexity of the call option price with

respect to the stock price, (property 6(b) for call options). U

2. Stock position bounds (Call options).


0< (S, T, X) < 1

D Since the option price C(S, t, X) is an increasing function of the stock price S, it follows

that U(S, t, X) = C(S, t,X) > 0.

Now we need to prove that U(S, t, X) < 1. We will assume that there exists such

S* that C'(S*) > a for some a > 1 and will show that this assumption contradicts the

ineqiality3 C(S,t,X) < S.



3 This inequality can be proven by considering a portfolio consisting of one stock and
one shorted call option on this stock. At expiration, the portfolio value is ST-max{0, ST-
X} > 0 for any ST and X > 0. Non-arbitrage assumption implies that S > C(S, t, X).









Since U(S, t, X) increases with S, for any S > S* we have U(S, t, X) > a,

j U(s, t, X)ds > j2 ads, C(S, t,X) C(S*, t,X) > aS aS*, C(S,T,x) >
C(S*,t, X) aS* + aS, C(S,t, X) > S + (a 1)S + C(S*,t, X) aS*.

Let f(s) = (a )s + C(S*) aS*. Since (a 1) > 0, there exists such S, > S* that

f(SI) > 0. This implies C(S1, t, X) > S1 which contradicts inequality C(S, t, X) < S. 0

The previous inequalities were justified in a quite general setting of assumptions 1-5

and a non-arbitrage assumption. We did not manage to prove the following two groups

of inequalities (horizontal monotonicity and convexity) in this general setting. The proofs

will be provided in further papers. However, here we present proofs of these inequalities in

the Black-Scholes setting.

3) Horizontal monotoe :. .1: (Call options)

U(S, t, X) is an increasing function of t when S > X,

U(S, t, X) is a decreasing function of t when S < X.

O We will validate these inequalities by analyzing the Black-Scholes formula and

calculating the areas of horizontal monotonicities for the options used in the case study.

The Black-Scholes formula for the price of a call option is


C(S,T,X) S N(d) XeTN(-d2),

where S is the stock price, T is time to maturity, r is a risk-free rate, a is the volatility,

1 fy z2
N(y) =e 2 dZ, (3-29)

and di and d2 are given by expressions

1 SerT 1
dl In X + a ,
an X( )+aT

T1 Se, 1
d2 (tr I -J T.
aT XJ









Taking partial derivatives of C(S, T, X) with respect to S and t, we obtain


C'(S,T, X)= U(S,T,X)= (S,T,X)= N(d,),


exp (T(r+ )+ ))} (-T(2r + 2) + 21n (4))
C,(S, T,X) U,(S, T, X) 3

The sign of UJ(S, T, X) is determined by the sign of the expression F(S) -T(2r + c2) +

2 In () F(S) > 0 (implying U (S,T,X) > 0) when S > L and F(S) < 0 (implying

Uti(S,T,X) < 0) when S < L, where L X eT(r+2/2)

For the values of r = 1' a = 31 T = 49 d ,v L differs from X less than

2.5'. For all options considered in the case study the value of implied volatility did not

exceed 31 and the corresponding value of L differs from the stike price less than 2.5'.

Taking into account resolution of the grid, we consider the approximation of L by X in the

horizonal monotonicity constraints to be reasonable. U

4) Convexity (Call options).

U(S, t, X) is a concave function of S when S > X,

U(S, t, X) is a convex function of S when S < X.

O We used MATHEMATICA to find the second derivative of the Black-Scholes

option price with respect to the stock price (Us (S, t, X)). The expression of the second

derivative is quite involved and we do not present it here. It can be seen that US(S, t, X)

as a function of S has an inflexion point. Above this point U(S, t, X) is concave with

respect to S and below this point U(S, t, X) is convex with respect to S. We calculated

inflexion points for some options and presented the results in the Table (3-7).

The Error( .) column contains errors of approximating inflexion points by strike

prices. These errors do not exceed :'. for a broad range of parameters. We conclude that

inflexion points can be approximated by strike prices for options considered in the case

study. U

Next, we justify the constraints (3-24)-(3-27) for put options.









1. Vertical monotonicity (Put options).

U(S, t, X) is an increasing function of S.

D This property immediately follows from convexity of the put option price with

respect to the stock price (property 6(b) for put options). U

2. Stock position bounds (Put options).



-1< U(S,T,X) < 0

D Taking derivative of the put-call parity C(S, T, X) P(S, T, X) + X e-T = S with

respect to the stock price S yields C'(S, T,X) P,(S,T,X) 1. This equality together

with 0 < C'(S, T, X) < 1 implies -1 < P'(S, T, X) < 0, which concludes the proof. U

3) Horizontal monotonicity (Put options).

U(S, t, X) is an increasing function of t when S > X,

U(S, t, X) is a decreasing function of t when S < X.

D Taking the derivatives with respect to S and T of the put-call parity yields

C',tS,T, X) P,(S,T,X). Therefore, the horizontal monotonic properties of U(S,T,X)

for put options are the same as the ones for call options. U

4) Convexity (Put options).

U(S, t, X) is a concave function of S when S > X,

U(S, t, X) is a convex function of S when S < X.

D Put-call parity implies that C"s(S, T, X) = P s(S, T, X). Therefore, the convexity

of put options is the same as the convexity of call options. U

3.5 Case Study

This section present the results of two numerical tests of the algorithm. First, we

price European options on the stock following the geometric Brownian motion and

compare the results with prices obtained with the Black-Scholes formula. Second, we price

European options on S&P 500 index (ticker SPX) and compare the results with actual

market prices.









Tables 3-1, 3-3, and 3-4 report "relaul', values of strikes and option prices, i.e.

strikes and prices divided by the initial stock price. Prices of options are also given in

the implied volatility format, i.e., for actual and calculated prices we found the volatility

implied by the Black-Scholes formula.

3.5.1 Pricing European options on the stock following the geometric brown-
ian motion

We used a Monte-Carlo simulation to create 200 sample paths of the stock process

following the geometric brownian motion with drift 10C' and volatility 21i' The initial

stock price is set to $ 62; time to maturity is 69 d4i-- Calculations are made for 10 values

of the strike price, varying from $ 54 to $ 71. The calculated results and Black-Scholes

prices for European call options are presented in Table 3-1.

Table 1 shows quite reasonable performance of the algorithm: the errors in the price

(Err( .), Table 3-1) are less than for most of calculated put and call options.

Also, it can be seen that the volatility is quite flat for both call and put options.

The error of implied volatility does not exceed for most call and put options

(Vol.Err( .), Table 3-1). The volatility error slightly increases for out-of-the-money

puts and in-the-money calls.

3.5.2 Pricing European options on S&P 500 Index

The set of options used to test the algorithm is given in Table 3-2. The actual market

price of an option is assumed to be the average of its bid and ask prices. The price of the

S&P 500 index was modelled by historical sample-paths. Non-overlapping paths of the

index were taken from the historical data set and normalized such that all paths have the

same initial price So. Then, the set of paths was i .i-- i,' d" to change the spread of paths

until the option with the closest to at-the-money strike is priced correctly. This set of

paths with the adjusted volatility was used to price options with the remaining strikes.

Table 3-3 di-pl'i--1 the results of pricing using 100 historical sample-paths. The pricing

error (see Err( .), Table 3-3) is around 1.0' for all call and put options and increases









for out-of-the-money options. Errors of implied volatility follow similar patterns: errors

are of the order of 1 for all options except for deep out-of-the-money options. For deep

in-the-money options the volatility error also slightly increases.

3.5.3 Discussion of Results

Calculation results validate the algorithm. A very attractive feature of the algorithm

is that it can be successfully applied to pricing options when a small number of sample-paths

is available. (Table 3-4 shows that in-the-money S&P 500 index options can be priced

quite accurately with 20 sample-paths.) At the same time, the method is flexible enough

to take advantage of specific features of historical sample-paths. When applied to S&P

500 index options, the algorithm was able to match the volatility smile reasonably well

(Figures 3.6, 3.6). At the same time, the implied volatility of options calculated in the

Black-Scholes setting is reasonably flat (Figures 3.6, 3.6). Therefore, one can conclude that

the information causing the volatility smile is contained in the historical sample-paths.

This observation is in accordance with the prior known fact that the non-normality of

asset price distribution is one of causes of the volatility smile.

Figures 3.6, 3.6, 3.6, and 3.6 present distributions of total external financing

(E1, a e-rj) on sample paths and distributions of discounted money inflows/outflows

(ape-rj) at re-balancing points for Black-Scholes and SPX call options. We summarize

statistical properties of these distributions in Table (3-5).

Figures 3.6, 3.6, 3.6, and 3.6 also show that the obtained prices satisfy the non-arbitrage

condition. With respect to pricing a single option, the non-arbitrage condition is

understood in the following sense. If the initial value of the hedging portfolio is considered

as a price of the option, then at expiration the corresponding hedging strategy should

outperform the option 'p 'off on some sample paths, and underperform the option p .'ioff

on some other sample paths. Otherwise, the free money can be obtained by shorting the

option and buying the hedging portfolio or vise versa. The algorithm produces the price

of the option satisfying the non-arbitrage condition in this sense. The value of external









financing on average is equal to zero over all paths. The construction of the squared error

implies that the hedging strategy delivers less money than the option i, off on some paths

and more money that the option 'p ,ioff on other paths. This ensures that the obtained

price satisfies the non-arbitrage condition.

The pricing problem is reduced to quadratic programming, which is quite efficient

from the computational standpoint. For the grid consisting of P rows (the stock price

axis) and N columns (the time axis), the number of variables in the problem (3-9) is 2PN

and the number of constraints is O(NK), regardless of the number of sample paths. Table

3-6 presents calculation times for different sizes of the grid with CPLEX 9.0 quadratic

programming solver on Pentium 4, 1.7GHz, 1GB RAM computer.

In order to compare our algorithm with existing pricing methods, we need to consider

options pricing from the practical perspective. Pricing of actually traded options includes

three steps.

Step 1: Choosing stock process and calibration. The market data is analyzed

and an appropriate stock process is selected to fit actually observed historical prices. The

stock process is calibrated with currently observed market parameters (such as implied

volatility) and historically observed parameters (such as historical volatility).

Step 2: Options pricing. The calibrated stock process is used to price options.

Analytical methods, Monte-Carlo simulation, and other methods are usually used for

pricing.

Step 3: Back-testing. The model performance is verified on historical data. The

hedging strategy, implied by the model, is implemented on historical paths.

Most commonly used approach for practical pricing of options is time continuous

methods with a specific underlying stock process (Black-Scholes model, stochastic

volatility model, jump-diffusion model, etc). We will refer to these methods as process-specific

methods. In order to judge the advantages of the proposed algorithm against the

process-specific methods, we should compare them step by step.









Comparison at step 1. C'! .... ; the model may entail modelling error. For

example, stocks are approximately follow the geometric Brownian motion. However, the

Black-Scholes prices of options would fail to reproduce the market volatility smile.

Our algorithm does not rely on some specific model and does not have errors related

to the choice of the specific process. Also, we have realistic assumptions, such as discrete

trading, non-self-financing hedging strategy, and possibility to introduce transaction costs

(this feature is not directly presented in the paper).

Calibration of process-specific methods usually require a small amount of market

data. Our algorithm competes well in this respect. We impose constraints reducing

feasible set of hedging strategies, which allows pricing with very small number of sample

paths.

Comparison at step 2. If the price process is identified correctly, the process-specific

methods may provide an accurate pricing. Our algorithm may not have any advantages in

such cases. However, the advantage of our algorithm may be significant if the price process

cannot be clearly identified and the use of the process-specific methods would contain a

significant modelling error.

Comparison at step 3. To perform back-testing, the hedging strategy, implied by a

pricing method, is implemented on historical price paths. The back-testing hedging error is

a measure of practical usefulness of the algorithm.

The major advantage of our algorithm is that the errors of back-testing in our

case can be much lower than the errors of process-specific methods. The reason being,

the minimization of the back-testing error on historical paths is the objective in our

algorithm. Minimization of the squared error on historical paths ensures that the need

of additional financing to practically hedge the option is the lowest possible. None of the

process-specific methods possess this property.









3.6 Conclusions and Future Research

We presented an approach to pricing European options in incomplete markets. The

pricing problem is reduced to minimization of the expected quadratic error subject to

constraints. To price an option we solve the quadratic programming problem and find a

hedging strategy minimizing the risk associated with it. The hedging strategy is modelled

by two matrices representing the stock and the bond positions in the portfolio depending

upon time and the stock price. The constraints on the option value impose the properties

of the option value following from general non-arbitrage considerations. The constraints on

the stock position incorporate requirements on "smoothness" of the hedging strategy. We

tested the approach with options on the stock following the geometric Brownian motion

and with actual market prices for S&P 500 index options.

This paper is the first in the series of papers devoted to implementation of the

developed algorithm to various types of options. Our target is pricing American--I ile and

exotic options and treatment actual market conditions such as transaction costs, slippage

of hedging positions, hedging options with multiple instruments and other issues. In this

paper we established basics of the method; the subsequent papers will concentrate on more

complex cases.






















0.05


0.10


strike (relative, shifted to start from zero)
-I--Actual Vol --- Calculated Vol


Figure 3-1.


35.00
30.00
25.00
20.00
15.00
10.00
5.00
nn-
000
0.00


Figure 3-2.


Implied volatility vs. strike: Call options on S&P 500 index priced using 100
sample paths. Based on prices in columns Calc. Vol(..) and Act. Vol(..) of
Table 3-3.
Calculated Vol(.) = implied volatility of calculated options prices (100
sample-paths), Actual Vol( .) = implied volatility of market options prices,
strike price is shifted left by the value of the lowest strike.


0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45


strike (relative, shifted to start from zero)
I---Actual Vol-A- Calculated Vol


Implied volatility vs. strike: Put options on S&P 500 index priced using 100
sample paths. Based on prices in columns Calc. Vol(..) and Act. Vol(..) of
Table 3-3.
Calculated Vol(.) = implied volatility of calculated options prices (100
sample-paths), Actual Vol( .) = implied volatility of market options prices,
strike price is shifted left by the value of the lowest strike.


20.00

S15.00

= 10.00

S 5.00
E


0.00 4-
0.00


0.15


0.20


--
b--=~Fii~~














C-- -- r-rr-r-r- -


I I I
0.100 0.150 0.200
strike (relative, shifted to stait from zero)
I-- Actual Vol -A- Calculated Vol


0.250


0.300


Figure 3-3.









nr nnl


3 .UU


S20.00

. 15.00
o
10.00

C 5.00


0 00


Implied volatility vs. strike: Call options in Black-Scholes setting priced using
200 sample paths. Based on prices in columns Calc. Vol(..) and B-S. Vol(..) of
Table 3-1.
Calculated Vol(.) = implied volatility of calculated options prices (200
sample-paths), Actual Vol( .) = flat volatility implied by Black-Scholes
formula, strike price is shifted left by the value of the lowest strike.


IP- 's- a


~1


0.100 0.150 0.200
strike i l-lir.l'.., sifted to tarlt from zero)
I--ActualVol -- Calculated Vol


Figure 3-4.


Implied volatility vs. strike: Put options in Black-Scholes setting priced using
200 sample paths. Based on prices in columns Calc. Vol(.. ) and B-S. Vol(.. ) of
Table 3-1.
Calculated Vol(.) = implied volatility of calculated options prices (200
sample-paths), Actual Vol( .) = flat volatility implied by Black-Scholes
formula, strike price is shifted left by the value of the lowest strike.


25.00
20.00
15.00
10.00
5.00


0.00 *-
0.000


0:050


0.000


0.050


0.250


0.300













30
> 25
C 20
0 15
0 10
S"- I I
LL 5

C'N C4N CO. COD Ct CO CO II CN C CN t CO. CD. CN CO. CDO (D
C CD CD CD CD CD CD CD 0
I I I I I I I I

Dollars

Figure 3-5. Black-Scholes call option: distribution of the total external financing on
sample paths.
Initial price=- '.," strike= -~. time to expiration 70, risk-free rate 10' ,
volatility 21 I'
Stock price is modelled with 200 Monte-Carlo sample paths.







3000
S2500
2000
3 1500
a 1000
u. 500




Dollars

Figure 3-6. Black-Scholes call option: distribution of discounted inflows/outflows at
re-balancing points.
Initial price=-'-., strike= -~. time to expiration 70, risk-free rate 10'.
volatility 21 I'
Stock price is modelled with 200 Monte-Carlo sample paths.













16
14
Q 12
C 10
= 1
L 4
S2
0


Dollars


Figure 3-7.


1200
1000
800
600
400
200
0

IN


SPX call option: distribution of the total external financing on sample paths.
Initial price=$1183.77, strike price=$1190 time to expiration 49 d-,,-
risk-free rate 2.;' .
Stock price is modelled with 100 sample paths.


N I


Figure 3-8.


>P'


Dollars


N" NF


SPX call option: distribution of discounted inflows/outflows at re-balancing
points.
Initial price=$1183.77, strike price=$1190 time to expiration 49 d -,
risk-free rate 2.;;'
Stock price is modelled with 100 sample paths.


,~~~Q,G~


4 P ^ 9 ^f
le

















Table 3-1. Prices of options on the stock following the geometric Brownian motion:
calculated versus Black-Scholes prices.


Strike Calc. B-S Err( .) Calc.Vol. ( .)
Call options
1.145 0.0037 0.0038 -3.78 19.63
1.113 0.0075 0.0074 1.35 19.91
1.081 0.0134 0.0133 0.65 19.87
1.048 0.0226 0.0227 -0.04 19.79
1.016 0.0364 0.0361 0.80 19.94
1.000 0.0446 0.0445 0.19 19.82
0.968 0.0651 0.0648 0.47 19.94
0.935 0.0891 0.0892 -0.08 19.59
0.903 0.1166 0.1168 -0.11 19.29
0.871 0.1464 0.1465 -0.07 18.71
Put options
1.145 0.1274 0.1276 -0.16 19.73
1.113 0.0995 0.0994 0.04 20.03
1.081 0.0738 0.0738 0.05 20.02
1.048 0.0514 0.0514 -0.10 19.97
1.016 0.0334 0.0332 0.71 20.14
1.000 0.0258 0.0258 0.15 20.02
0.968 0.0147 0.0144 1.82 20.19
0.935 0.0070 0.0071 -1.60 19.89
0.903 0.0029 0.0031 -5.77 19.71
0.871 0.0010 0.0011 -12.88 19.52


Initial price- i .2


B-S.Vol. (

20.00
20.00
20.00
20.00
20.00
20.00
20.00
20.00
20.00
20.00

20.00
20.00
20.00
20.00
20.00
20.00
20.00
20.00
20.00
20.00


time to expiration 69 div- risk-free rate-


.) Vol.Err( .)

-1.86
-0.46
-0.65
-1.04
-0.28
-0.92
-0.31
-2.07
-3.56
-6.44

-1.36
0.17
0.12
-0.16
0.68
0.11
0.93
-0.56
-1.45
-2.41
10' volatility 21 I' 200


sample paths generated by Monte-Carlo simulation.
Strike($) option strike price, Calc. obtained option price (relative), BS Black-Scholes
option price (relative), Err=(Found BS)/BS, Calc.Vol. obtained option price in
volatility form, BS.Vol.( ) Black-Scholes volatility,
Vol.Err(,) (Calc.Vol. BS. Vol.)/BS. Vol.

























Table 3-2. S&P 500 options data set.


Strike Bid Ask P
Call options


1500 N/A
1325 0.3
1300 0.45
1275 1.15
1250 3.7
1225 8.6
1210 13.2
1200 17.5
1190 22.1
1175 30.8
1150 48.0
1125 68.3
1100 90.2
500 682.1
Strike($) option


0.5
0.5
0.8
1.65
4.2
9.6
14.8
18.9
24.1
32.8
50.0
69.5
92.2
684.1


rice Rel.Pr Strike Bid


N/A
0.4
0.625
1.4
3.95
9.1
14.0
18.2
23.1
31.8
49.0
68.9
91.2
683.1


N/A
0.0003
0.0005
0.0012
0.0033
0.0077
0.0118
0.0154
0.0195
0.0269
0.0414
0.0582
0.0770
0.5771


strike price, Bid($) option


1500
1300
1275
1225
1210
1200
1190
1175
1150
1125
1100
1075
1050
1025
bid price,


Put
311.3
112.7
88.8
46.9
36.9
31.0
26.1
19.8
12.5
8.0
5.1
3.3
2.2
1.55
Ask($)


Ask Price Rel.Pr


options
313.3
114.7
90.8
48.9
38.9
33.0
28.1
21.4
14.0
9.0
5.9
4.1
3.0
2.05


312.3
113.7
89.8
47.9
37.9
32.0
27.1
20.6
13.25
8.5
5.5
3.7
2.6
1.8


0 'i. :
0.0960
0.0759
0.0405
0.0320
0.0270
0.0229
0.0174
0.0112
0.0072
0.0046
0.0031
0.0022
0.0015


option ask price,


Price($) option price (average of bid and ask prices), Rel.Pr relative option price


)














Table 3-3. Pricing options on S&P 500 index: 100 paths
Strike Cale. Actual Err( .) Calc.Vol.(' ) Act.Vol.( ) Vol.Err( .)
Call options
1.119 0.0002 0.0003 -40.00 13.17 14.14 -6.82
1.098 0.0005 0.0005 -5.28 12.80 12.92 -0.90
1.077 0.0013 0.0012 11.57 12.70 12.40 2.42
1.056 0.0035 0.0033 5.70 13.03 12.80 1.78
1.035 0.0079 0.0077 3.15 13.38 13.18 1.52
1.022 0.0117 0.0118 -0.75 13.43 13.49 -0.48
1.014 0.0156 0.0154 1.32 13.91 13.77 1.03
1.005 0.0195 0.0195 0.01 14.07 14.06 0.01
0.993 0.0269 0.0269 0.18 14.63 14.60 0.23
0.971 0.0416 0.0414 0.50 15.57 15.40 1.09
0.950 0.0589 0.0582 1.12 16.81 16.13 4.25
0.929 0.0775 0.0770 0.62 18.04 17.35 3.94
0.422 0.5789 0.5771 0.33 69.39 N/A N/A
Put options
1.267 0.2633 0 2'i : -0.20 22.50 29.02 -22.44
1.098 0.0956 0.0960 -0.47 13.88 15.14 -8.35
1.077 0.0756 0.0759 -0.36 13.71 14.18 -3.32
1.035 0.0406 0.0405 0.33 14.22 14.11 0.77
1.022 0.0319 0.0320 -0.25 14.29 14.35 -0.40
1.014 0.0274 0.0270 1.26 14.75 14.51 1.62
1.005 0.0229 0.0229 -0.01 14.89 14.90 -0.01
0.993 0.0176 0.0174 1.38 15.47 15.30 1.10
0.971 0.0111 0.0112 -0.52 16.43 16.47 -0.28
0.950 0.0070 0.0072 -1.95 17.58 17.72 -0.79
0.929 0.0045 0.0046 -3.42 18.84 19.05 -1.09
0.908 0.0028 0.0031 -10.00 20.02 20.57 -2.68
0.887 0.0015 0.0022 -32.27 20.46 22.24 -7.99
0.866 0.0011 0.0015 -26.00 22.46 23.78 -5.54
Initial price=$1183.77, time to expiration 49 div~ risk-free rate 2.;'. Stock price is
modelled with 100 sample paths. Grid dimensions: P = 15, N = 49.
Strike option strike price (relative), Calc. calculated option price (relative),
Actual actual option price (relative), Err (Calc. Actual)/Actual, Calc.Vol. calculated
option price in volatility form, Act.Vol.( )actual option price in volatility terms,
Vol.Err( -)=(Calc.Vol. Act.Vol.)/Act.Vol.














Table 3-4. Pricing options on S&P 500 index: 20 paths
Strike Cale. Actual Err( .) Calc.Vol. ( ) Act.Vol.( .) Vol.Err( .)
Call options
1.119 0.0005 0.0003 45.00 14.95 14.14 5.78
1.098 0.0010 0.0005 88.80 14.48 12.92 12.09
1.077 0.0020 0.0012 66.86 13.95 12.40 12.50
1.056 0.0047 0.0033 41.80 14.39 12.80 12.38
1.035 0.0092 0.0077 19.84 14.43 13.18 9.42
1.022 0.0132 0.0118 11.41 14.47 13.49 7.26
1.014 0.0160 0.0154 4.03 14.20 13.77 3.13
1.005 0.0195 0.0195 0.00 14.06 14.06 0.00
0.993 0.0264 0.0269 -1.66 14.28 14.60 -2.15
0.971 0.0393 0.0414 -5.01 13.67 15.40 -11.23
0.950 0.0548 0.0582 -5.76 12.01 16.13 -25.52
0.929 0.0737 0.0770 -4.35 8.39 17.35 -51.65
0.422 0.5790 0.5771 0.34 N/A N/A N/A
Put options
1.267 0.2633 0 2'i : -0.19 23.45 29.02 -19.16
1.098 0.0959 0.0960 -0.13 14.82 15.14 -2.11
1.077 0.0762 0.0759 0.40 14.67 14.18 3.45
1.035 0.0415 0.0405 2.49 14.92 14.11 5.72
1.022 0.0332 0.0320 3.69 15.20 14.35 5.93
1.014 0.0278 0.0270 2.74 15.03 14.51 3.54
1.005 0.0229 0.0229 0.01 14.90 14.90 0.01
0.993 0.0168 0.0174 -3.31 14.90 15.30 -2.63
0.971 0.0089 0.0112 -20.72 14.58 16.47 -11.48
0.950 0.0030 0.0072 -58.73 12.99 17.72 -26.73
0.929 0.0000 0.0046 -100.00 4.38 19.05 -77.00
0.908 0.0000 0.0031 -100.00 6.07 20.57 -70.50
0.887 0.0000 0.0022 -100.00 7.68 22.24 -65.48
0.866 0.0000 0.0015 -100.00 8.98 23.78 -62.21
Initial price=$1183.77, time to expiration 49 div~ risk-free rate 2.;'. Stock price is
modelled with 20 sample paths. Grid dimensions: P = 15, N = 49.
Strike option strike price (relative), Calc. calculated option price (relative),
Actual actual option price (relative), Err (Calc. Actual)/Actual, Calc.Vol. calculated
option price in volatility form, Act.Vol.( )actual option price in volatility terms,
Vol.Err( -)=(Calc.Vol. Act.Vol.)/Act.Vol.










Table 3-5.


Summary of cashflow distributions for obtained hedging strategies presented on
Figures 3.6, 3.6, 3.6, and 3.6.
Total financing Re-bal. cashflow Total financing Re-bal. cashflow
Black-Scholes Call SPX Call


mean 0.0 0.0 0.0 0.0
st.dev. 0.6274 0.0449 16.1549 1.2730
median 0.0770 -0.0008 0.2695 -0.0314
Total financing ($) = the sum of discounted inflows/outflows of money on a path; Re-bal.
cashflow ($) = discounted inflow/outflow of money on re-balancing points.
Black-Scholes Call: Initial price= -'-.2 strike= -,'.2 time to expiration 70, risk-free
rate 10 '., volatility =2''. Stock price is modelled with 200 Monte-Carlo sample paths.
SPX Call: Initial price=$1183.77, strike price=$1190, time to expiration 49 d-,v- risk-free
rate 2.;:'. Stock price is modelled with 100 sample paths.



Table 3-6. Calculation times of the pricing algorithm.
# of paths P N Building time(sec) CPLEX time(sec) Total time(sec)
20 20 49 0.8 8.2 9.0
100 25 49 1.6 12.6 14.2
200 25 70 5.5 31.7 37.2
Calculations are done using CPLEX 9.0 on Pentium 4, 1.7GHz, 1GB RAM.
# of paths = number of sample-paths, P = vertical size of the grid, N = horizontal size of
the grid, Building time = time of building the model (preprocessing time), CPLEX time =
time of solving optimization problem, Total time = total time of pricing one option.


Table 3-7. Numerical values of inflexion points of the stock position as a function of the
stock price for some options.
Expir.(d (- ) Strike($) Inflexion($) Error( .)
0 62 60.126 3.02
35 62 61.056 1.52
69 62 61.975 0.04
0 54 52.368 3.02
35 54 53.178 1.52
69 54 53.974 0.05
0 71 68.855 3.02
35 71 69.919 1.52
69 71 70.967 0.05
Expir.(d -,( ) = time to expiration, Strike($) = strike price of the option, Inflexion($)
inflexion point, Error( ) (Strike-Inflexion)/Strike.









CHAPTER 4
METHODS OF REDUCING MAXIMIZATION OF OMEGA FUNCTION TO LINEAR
PROGRAMMING

4.1 Introduction

The classical mean-variance portfolio theory is based on the assumption that the

returns are normally distributed. One of important characteristics of a portfolio is the

Sharpe ratio, the ratio of the excess return over the risk-free rate to the standard deviation

of a portfolio. Maximization of the Sharpe ratio in portfolio management allows to pick

a portfolio with the highest return or with the lowest risk. However, if the standard

deviation does not adequately represent risk, Sharpe-optimal portfolios can produce highly

non-optimal returns. Critique of the classical approach to the portfolio management is

based on the fact that the mean and the variance of a non-normal random variable does

not fully describe its distribution and, in particular, do not account for heavy tails of

distributions, which are of particular interest for investors. Introduction of higher-order

moments into portfolio analysis leads to more accurate solutions. One of the areas in

which the mean-variance framework fails is the hedge fund analysis. Properties of tails

of return distributions are the key characteristics of hedge funds. Portfolio measurement

should incorporate the information about higher-order moments of return distributions in

order to adequately represent hedge fund risk.

One of the alternatives to the mean-variance approach is the Omega function, recently

introduced in Shadwick and Keating (2002). Omega function 2r(rh) is the ratio of the

upper and the lower partial moments of an asset rate of return r against the benchmark

rate of return rh. The upper partial moment is the expected outperformance of an asset

over a benchmark; lower partial moment is the expected underperformance of an asset

with respect to the benchmark. The Omega function has several attractive features which

made it a popular tool in risk measurement. First, it takes the whole distribution into

account. A single value 2,(rPh) contains the impact of all moments of the distribution.

A collection of Qr(rh) for all possible rh fully describes the return distribution. Second,









Omega function has a simple and intuitive interpretation. For a fixed benchmark return

L, the number Qr(rh) is a ratio of the expected upside and the expected downside of an

asset with respect to the benchmark. It also contains the investor's risk preferences by

specifying the benchmark return. Third, given a benchmark rh, comparison of two assets

with returns rl and r2 is done by comparing their Omega values Q,,(rh) and Q,,(rh). The

asset with greater Omega is preferred to the asset with lower Omega.

The choice of the Omega-optimal portfolio with respect to a fixed benchmark with

linear constraints on portfolio weights leads to a non-linear optimization problem. Several

approaches to solving this problem has been proposed, among which are the global

optimization approach in Avouyi-Govi et al. (2004) and parametric approach employing

the family of Johnson distributions in Passow (2005). Mausser et al. (2006) proposes

reduction of the Omega maximization problem to linear problem using change of variables.

The -~i-.- --1. reduction is possible if the Omega function is greater than 1 at optimality,

several non-linear methods are s, i.-.- -1-. 1 otherwise.

This paper investigates reduction of the Omega-based portfolio optimization problem

with fixed benchmark to linear programming. We consider a more general problem than

Mausser et al. (2006) by allowing short positions in portfolio instruments and considering

constraints of the type h(x) < 0 with the positively homogeneous function h(.), instead of

linear constraints in Mausser et al. (2006). We prove that the Omega-maximizing problem

can be reduced to two different problems. The first problem has the expected gain as

an objective, and has a constraint on the low partial moment. Second problem has the

low partial moment as an objective and a constraint on the expected gain. If the Omega

function is greater than 1 at optimality, the Omega maximization problem can be reduced

linear programming problem. If the Omega function is lower than 1 at optimality, the

proposed reduction methods lead to the problem either of maximizing a convex function,

or with linear objective and a non-convex constraint.









To illustrate the use of the reduction theorems, we consider a resource allocation

problems frequently arising in the hedge fund management. We show how this problem

can be reduced to linear programming and illustrate this with a case study based on the

data set of a real hedge fund.

4.2 Omega Optimization

4.2.1 Definition of Omega Function

Let t = 1, ..., T denote time periods. Each time period produces one scenario of

asset returns. Consider a portfolio of N instruments, with instrument i having the rate of

return ri at time t. The benchmark rate of return is called the hurdle rate of return and is

denoted by rh. The difference
-t t
fit t i rh

is the excess rate of return of instrument i over the hurdle rate at time t.

Let xi be the exposure in instrument i in the portfolio; the corresponding weights are

i' /Zi 1Xi, i 1,..., N.

The loss function measuring underperformance of the portfolio with respect to the

hurdle rate at time t is defined by

N
L(t, x) =- (r Yi .
i= 1

The lower partial moment q(x) and the upper partial moment q](x) of the loss

function L(t, x) are defined as follows

1
S(x) T L(tx), where S+(x) {t L(t,x) > 0,t ,...,T};
tEs+ (x)

1
q](x) T L(t,x), where S_(x) {t I L(t,x) < 0,t 1,..., T}.
tES (x)









The expected gain with respect to the hurdle rate rh is


1 T
q(x) T -L(t,x).
t=1

Assumption (Al)

We make the assumption that there are no x / 0 such that L(t, x) = 0 for all

t =1, ..., T. In other words, we assume that there are N linear independent vectors among


t Fl, ** ], t it T.

Since the number of scenarios T is usually much 'i.r-. -r than the number of instruments N

in the portfolio, the assumption Al is almost ahv--, satisfied. This assumption prohibits

the case when both functions q(x) and rl(x) simultaneously equal to zero for some x / 0.

The Omega function is the ratio of the two partial moments

r(x)


which can be expressed as

rq(x) rq(x) rq(x) + rq(x) q(x) + rq(x) q(x)
(x q(x) q(x) q(x) i+ x)'

Note that both functions q(x) and qr(x) are positively homogeneous' This is trivial

for q(x) as it is linear with respect to x, and holds for qr(x) since linearity of the loss

function L(t, x) with respect to x implies




qr(Ax) i L(t, Ax)T i L(t,Ax)/T i) AL(t,x)/T AT (x).
tIs+ (AX) tES+ (x) tES+ (x)



1 A function f(x) is called positively homogeneous if f(Ax) = Af(x) for all A > 0.









4.2.2 General Problem

This paper deals with solving the following non-linear problem. We consider a fixed

hurdle rate rh and form a portfolio of N instruments subject to restrictions expressed by

K inequalities. The goal is to maximize the Omega function of the portfolio.


(Po)


q(w)
max (w) = 1 +
nl~w)


hk(w) > k= ,..., K,

E 1"' 1,


where functions hk(x) are positively homogeneous.

It is not necessary for variables in problem Po to

(x) is invariant to scaling its argument, since

q(Ax) Aq(x)
(Ax)= -+-- -+---
r (Ax) Arj(x)

for any feasible x and A > 0. Moreover, if constraints

x, they also hold for Ax, A > 0.

Consider the following alternative to Po.


(Ps) max Q(x) 1 +


hk(x) >


xi =1 R,
xi E R,


be weights. Note that the function


q(x)
(x) > O, ...,K hold for somex)


hk(x) > 0, k 1, ...,K hold for some


q(x)


'ITAX)


0, k 1,...,K,

> 0,

i = 1, ..., I.









In order to simplify notations, we denote feasible sets in problems Po and 7P by K"

and K, respectively,
I
K, = {x hk(x) > 0, X=i =1, xi e R, k= 1,...,K, i= 1,...,I},
i= 1
I
K {x hk(x) > 0, x > 0, x R, k ,...,K, i 1,...,I}.
i=

Problems Po and Po can be written by maxK. 2(x) and maxK 2(x), respectively. The

relationship between problems Po and 7P is stated in the following lemma.

Lemma 1. If the problem Po has a solution, then the problem 7P also has a solution,

and vice versa. Moreover, if x* is the solution to Po and w* is the solution to PQ, then

x* = Aw* for some A > 0, and Q(x*) = Q(w*).

Proof:

Let w* be optimal solution to Po. If (x) > Q(w*) for some x E K in 7P, then for

A= (Zi xi)-1 we have 2(i x) = 2(x) > 2(w*). Since x c Ku,, we have a contradiction
with w* being the optimal solution to Po. Therefore, the objective in 7P' is bounded from

above by Q(w*). Since w* E K, it is the optimal solution to Po: x* = w*. Any solution of

the form x* = Aw*, A > 0 will also be optimal.

Conversely, suppose x* is the optimal solution to 7P. Then the objective function in

Po are bounded from above by Q(x*). Take A* = ( x)- 1, then A*x* is feasible point

in Po, and Q(A*x*) = Q(x*), therefore w* = A*x* is the optimal solution to 0Po.

4.2.3 Two Reduction Theorems

In the following reduction theorems, we use the notion of reduction of one problem

to another, which is understood in the following sense. Problem P, can be reduced to

problem P2 (notation is P1 -- P2) if both problems have finite solutions or are unbounded.

Furthermore, xI and x* are optimal solutions to pi and P2, respectively, then x = Ax* for

some A > 0.

Equivalence of problems Pi and P2 is denoted by PI <= Pa2









In order to state the first theorem, we introduce the following sets


Dq+ {x | q(x) > 0}n K,

D
D>1 = {25 {x I(2x) > 1},


and define problems

(p,< 1) max q(x)
K n D, and

( 7,>1) max q(x).
KnD,il
Theorem 1. Suppose that the feasible region in problem Po is bounded. Then Po either

has a finite solution or is unbounded. If Dq+ K,, / 0, then problem Po can be reduced to

problem P<,_1. If Dq+ K, = 0, the problem Po can be reduced to problem P,2i1.

For the second theorem, we introduce sets Dq o = {x I q(x) = 0}, Dq>i {-x I q(x) >

1}, and Dq-1 = {x I q(x) > -1}, and define problems


(Pq> ) mmin r(x)
K Dq>l

and

(Pq>-1) max (r (x).
K n Dq> 1
Theorem 2. Suppose that the feasible region in problem Po is bounded. If Dq+ K,, / 0

and D,=o [" K / 0, then problem Po is unbounded and the objective function in problem

Vq>1 is equal to zero at ol,':,,',:ahl; If Dq+ n Kw / 0 and D,=o Kw = 0 then problem

Po can be reduced to Pq>1. If Dq+ 0 K, = 0, and Dq=o n K, / 0, then the objective

function in Po is equal to zero at 'ol/'.:,,il.:;' and the problem Pq>-1 is unbounded. If

Dq,+ Kw = 0 and Dq=o n Kw = 0, the problem Po can be reduced to problem Pq>-1.
Proofs of both theorems are given in the next section.









Theorems 1 and 2 require the knowledge if Dq+ K, = 0, i.e. if q(x) > 0 at least at

one feasible point x in problem Po. This information can be obtained by maximizing q(x)

over the feasible region in Po, i.e. by solving

max q(w)

s.t.

hk(w)> 0, k = ,...,K, (41)

i=1 = 1,
,,e R, i= 1,...,I.

If q(x) > 0 at optimality in (4-1), then problem Po can be reduced to P,<1 (or Pq>1),

otherwise it can be reduced to P,>1 (or Pq>_1). The alternative to solving (4-1), one

could solve
max q(x)

s.t.

hk(x) > k = ,...,K, (4-2)

E 1 xi > 0,

xi e R, i = 1,..., I,

where the variables are not restricted to be weights. If q(x*) > 0 at optimality in (4-2),

then q(A*x*) > 0 for A 1- /(ZE X*), where A*x* is a feasible point in Po. If q(x*) < 0

in (4-2), then q(x) < 0 for all feasible points in (4-1), since the feasible region in (4-2)

contains the feasible region in (4-1).

Another prescription to determine if Dq+ Q K = 0 is to solve P,<1 first. If

Dq+ K K / 0, then the reduction to P,<1 is correct, and q(x) > 0 ( or 2(x) > 1) at

optimality. If Dq+ Q K = 0, then problem P, < 1 has no solution or have the optimal

objective value equal to zero. To see this, note that if Dq o K K[ / 0, then there exists

a point x* such that q(x*) = 0 and rl(x) = 1, therefore the objective in P,<1 is equal

to zero at optimality. If D,=o Q K =K 0, then q(x) < 0 for any x E Dq_. However,









any point of the form Ax, x e Dq_, is feasible to P,_<1 for significantly low A, moreover,

q(Ax) = Aq(x) 0 as A 0, therefore problem P,1<1 never attains its maximum.
Alternatively, the problem Pq>1 can be attempted. If Dq+ Kw / 0, then the

solution to Pq>1 after normalizing gives the solution to Po. If Dq+ Kw = 0, the problem

Pq>1 is infeasible, due to the constraint q(x) > 1.
4.3 Proofs Of Reduction Theorems For Omega Optimization Problem

We use the following notations.

Dq+ = Ix q(x) > 0}n K, Dq_ {x I q(x) < 0} K,

Dqo {x | q(x) = 0}, Dq = {x | q(x) = 1},

Dq> = {x I q(x) > 1}, Dq>_ {x q(x) > -1},

and

D+ = {x y(x) > 0}, D,,o = {x y(x) = 0},

D,1 {x I y() 1}, D<_1 {x | 7y() < 1},

D,>I = {x I r(x) > 1}.

Theorem 1

Suppose that the feasible region in problem Po is bounded. Then Po either has a finite

solution or is unbounded. If Dq+ K, / 0, then problem Po can be reduced to problem

P,7<1. If Dq+ n K. = 0, the problem Po can be reduced to problem P,>1.
Proof: Consider the case when Dq+ K 7 0. If K. D,=o / 0, both problems

Po and 'P,<1 are unbounded. Indeed, there exists x E K, such that q(x) > 0 and

rl(x) = 0, therefore, (x) = +oo and the problem Po is unbounded. On the other hand,
Ax E K D,<1 for any A > 0 and q(Ax) = Aq(x) +oo as A +oo, therefore, the

problem Po<0 is also unbounded.
If K. D,=o = 0, feasible sets in both problems Po and ,1<1 are bounded and

closed, and objective function are continuous, therefore both problems have finite









solutions. By Lemma 1, the problem Po can be reduced to Po. The following sequence
of reductions of the problem PO leads to the problem 'P,<1.

(1/) (2) (') (T/
P = max (x) < max (x) -4 max Q (x) < max q(x) < max q(x)= P, K Dq+ D,+ D,= D,+ 0 D,+ D,+ 0n ,<

(1') Since (x) > 1 for any x e Dq+ and (x) < 1 otherwise. Therefore, the maximum in

PO will never be attained in the set Dq_.
(2') Let x* be solution to maxDq+ (x), x** be solution to maXDq nD0 1 (().
Then maxDq +n l (x) < Q(x*). Take A* 1/((x*), then r(A*x*) = and

2(A*x*) (x*), so x** = A*x*.
q(x) q(x)\
(3') maXDq 0nD,, 1 + maxD D, t = 1+ -) 1 + maXDq +nD, q(x).

(4') Suppose that x* is the solution to maxDq+ED ,< q(x) and ql(x*) < 1. Take
A* 1/t(x*) > 1. Then qr(A*x*) = 1, q(A*x*) = A*q(x*) > q(x*), which is a contradiction.

Therefore, rl(x) = 1 at optimality in problem P,<1, and the equivalence (4') is justified.

Now consider the case Dq+ n K = 0. Definitions of functions q(x) and rl(x) imply

that Dq_ D,=o = 0, so rl(x) > 0 for any x E K,. By the same argument as above, both
problems Po and P1>1 have finite solutions, and Po PO.

First, consider the case when Dq=o n K / 0. In this case, the optimal solution x* to

Po gives Q(x*) = 1, and q(x*) = 0, Tl(x*) > 0. Taking A* 1/l(x*), yields q(A*x*) = 0,
q(A*x*) = 1, so x**A*x* is the optimal solution to 'P,>1, and q(x**) = 0.
If Dq=o n K, = 0, then q(x) < 0 for all x E Dq_. The following sequence of
reductions leads to the problem P,>1.

(1") (.. 3//)
P = max Q(x) max Q(x) max q(x) max q(x) = P>1.
Dq- Dq- nD,=l Dq nOD,=1 Dq nD,>

(1") and (2") are proven similarly to (1') and (2'), so here we consider (3"). We need to

show that if x* is the optimal solution to P,>1, then l(x*) = 1. Indeed, suppose that









q(x*) > 1. Take A* 1/t(x*) < 1. Then q(A*x*)= 1 and q(A*x*) = A*q(x*) > q(x*), which

is a contradiction. The equivalence (3") is justified. U

Theorem 2

Suppose that the feasible region in problem Po is bounded. If Dq+ K 0 and

D, o n K, / 0, then problem Po is unbounded and the objective function in problem Pq>1

is equal to zero at o/l/.:a,,l.:1;, If Dq+ K, / 0 and D,=o K, = 02 then problem Po can

be reduced to Pq>1. If Dq+ n K, = 0, and Dq=o K,, / 0, then the objective function in

PO is equal to zero at ol'i:,,i'al.h/, and the problem Pq>-1 is unbounded. If Dq+ nK = 0

and Dq=o Kw = 0, the problem Po can be reduced to problem 'Pq>-1.

Proof:

If Dq+ K / 0 and D,=o 0 K, / 0, then unboundedness of Po is already shown in

Theorem 1. Consider problem Pq_>1. Take x E D ,o n K,, then q(x) > 0, l(x) = 0. Taking

A = 1/q(x), we have q(AX) = 1, q(AX) = 0. Since qr(x) > 0 for all x, the optimal objective

value in problem Pq>1 is zero.

If Dq+ Kw / 0 and D, o 0 K, = 0, then problem Po can be reduced to problem

PO as was shown in the proof of Theorem 1. The following sequence of reductions
transforms the problem Po into Pq>1.

( (2') (3') (T)
'PI = max -(x) ( max 2(x) +- max Q2(x) ( min r (x) min ((x) = Pq>1.
K Dq+ Dq D,=1 Dq+ Dq, 1 Dq+ n Dq>

(1') is already shown in Theorem 1.

(2') Let x* be solution to maxDq@ -(x), x** be solution to maxDq ~Dq=1 (().

Then maxDq +nD 1 (x) < -(x*). Take A* = 1/q(x*), then q(A*x*) = and

2(A*x*) 2(x*), so x** = *x*.

(I x) 1m 1
(3') maXDqn Dq 1 + maXDq Dq=1 -- 1++ D +
TO) )I TO)) minDqmnnqi TOY)









(4') Suppose that x* is the solution to maxDq, nDq> r(x) and q(x*) > 1. Take

A* = 1/q(x*) < 1. Then q(A*x*) = 1, q(A*x*) = A*q(x*) < q(x*), which is a contradiction.

Therefore, q(x) = 1 at optimality in problem Pq>1, and the equivalence (4') is justified.

Now consider the case Dq+ K, = 0 and D,=o K,, / 0. Take x E Do K,,.

From the assumption Al, it follows that rl(x) > 0. Since maxK.w (x) < 0 and -(x) = 0,

we conclude that the optimal objective value in problem Po is zero. As for problem Pq<1,

points of the form Ax for A > 0 are all feasible, and r\(Ax) +oo as A +oo, so

problem Pq~< is unbounded.

If Dq+ K, = 0 and D,=o 0 K, = 0, then the feasible region of problem Po

is closed and bounded, and the objective function is continuous, therefore, problem Po

has a solution. According to Lemma 1, Po can be reduced to PO. Consider the following

sequence of reductions.
(1/) (2 (3"//
P' = max ((x) max 2() max (x) max q(x) = Pq>-1.
Dq- Dq Dq 1 Dq Dq, 1 D, n Dq>-

(1") Let x* be solution to maXDq Q (x), x** be solution to maXD, n0Dq1 2(X).

Then maxDq0Dq_ (x) < -(x*). Take A* = -1/q(x*) > 0, then q(A*x*) = -1 and

2(A*x*) = (x*), so x** A**.
S q(x) 1 1
(//) max Dq 1 + maXDqnDq-1 -- 1I--Xq--
(") ma7() m aI 1 XD(x) ) maxD,q nDq- (Y)
(3") Suppose that x* is the solution to maXDq 0Dq> y(x) and q(x*) > -1. Take

A* = -1/q(x*) > 1. Then q(A*x*) -1, r(A*x*) = A*(x*) > q(x*), which is a

contradiction. Therefore, q(x) = -1 at optimality in problem Pq>_1, and the equivalence

(3") is justified. U

4.4 Applications of Reduction Theorems to Problems with Linear Constraints

The set of constraints

hk(x) > 0, k= 1,...,K (43)









with positively homogeneous functions hk(-) in problem Po is quite general. For example,

any set of linear inequalities on portfolio weights

N
Ax < b, Xi = 1
i= 1

can be written in the form (4-3) by taking

N
h(x) b Xi Ax
i=1


In this subsection, we discuss application of Theorems 1 and 2 to problems with linear

constraints. In the case when Dq+ n Kw / 0 (alternatively, ((x) > 1 at optimality), the

problem Po can be reduced to P,1<1 or Pq>1. In problem P
be reduced to linear programming. Recall that q7(x) = z [L(t, x)]+. Introduction of

additional variables zt, t = 1, ..., T, allows to enforce the constraint qr(x) < 1 by replacing it

with
T
> z < 1, zt >L(t,x), Zt >0, for t 1,...,T.
t=1
The problem PqI> can similarly be reduced to linear programming. The minimization

of the convex function qr(x) can be reduced to maximization of t1 Zt with additional

constraints Zt > L(t,x), zt > 0, t 1,...,T.

If (x) < 1 at optimality, the problem Po is reduced to Pi>1 or q_>-1. Both of these

problems cannot be reduced to linear programming due to the presence of the constraint

qr(x) > 1 in P,>1 or maximization of the convex objective qr(x) in Pgq-1.

4.5 Example: Resource Allocation Problem

As an example of applying Theorem 1, we solve the following problem arising in

hedge fund management. Consider N fund managers among which the resources should be

allocated. Let i, be the fraction of resources allocated to manager i, i = 1,..., N. Some

managers have similar strategies; there are M different strategies among all managers. Let

Jm be a set of managers pursuing strategy m, then j,,J- wj is the fraction of resources










allocated to strategy m, m = 1,..., M. The following optimization problem allocates

money to groups of managers with similar strategies and to individual managers within

each strategy.


q(w)
max Q (w)= 1 +

s.t.

S1 "' = 1 budget constraint,

bK < E:j wj < b, m 1, ...,n M constraints on allocation to strategies,

i < "' < Ui, where 1i > -oo, i = 1, ...,I box constraints for individual positions,

e R, i = 1,..., I.
(4-4)

The constraint zfi -, x 1 allows to rewrite the set of constraints


bl< xj < b, m= ...,M, (4-5)
J Jr,

li < Xi < Ui, i = 1, (4-6)


in the following form

I I
b x s< xj ,< b zx, m ..., M, (4-7)
i= 1 jEJ i 1
I I
Sj xi < Xi < uiy xi, i = 1, ..., 1. (4-8)
i 1 i 1

For any x satisfying (4-7)-(4-8), Ax for A > 0 will also satisfy (4-7)-(4-8). Therefore,

constraints (4-7)-(4-8) are special case of the constraints of type (4-3).

According to Theorem 1, the problem (4-4) can be reduced to the following problem.










max q(x)

s.t.

I(x) < 1 (or r(x) > 1)
(4-9)
i E i < Ei < i 1 i ,, 1, ..,


xi E R, i = 1,..., I.

where the loss constraint is r](x) < 1 if q(x) > 0 (Q(x) > 1) at least at one feasible point of

the problem (4-4), and r](x) > 1 otherwise.

We solve the above problem with the constraint T](x) < 1 by reducing it to the

following linear programming program. In the case study described below the solution

gives Q(x*) > 1 at optimality, implying that the reduction to LP is valid. The following

LP formulation uses explicit expressions of functions q(x) and Tq(x).

max -T I Yif (rTh rT)xi

s.t.

(410)1 <
Zt > Ei 1(h ')x, t 1,...,T, (4 10)

b'i E Xi < ECj. x < b x, m= 1, ..., M

1 Ei < x < ui E, xi, (where 1i > -oc), i = 1,...,I
zt > 0, t- 1, ..., T.

xi E R, i = 1,..., I.

We solve the allocation problem 4-10 for a portfolio consisting of 10 strategies. We

used historical daily rates of return for the funds from October 1, 2003, to March 17, 2006.

The daily hurdle rate is set to rh = 0.00045. The optimal solution x* to (4-10) gives

Q(x*) = 1.164 > 1, which indicates that the reduction of (4-4) to (4-10) is correct. The

solution to (4-4) is obtained by normalizing the solution x*. The optimal allocation is

given in the Table 4-1.









Table 4-1. Optimal allocation
Strategy Allocation ( .)
Manager 1 10.00
Manager 2 20.00
Manager 3 7.50
Manager 4 2.22
Manager 5 0.00
Manager 6 7.50
Manager 7 0.00
Manager 8 12.78
Manager 9 20.00
Manager 10 20.00


4.6 Conclusions

We considered a problem of maximizing the Omega function of a portfolio with a

fixed benchmark with positively homogeneous constraints. We proved that this problem

can be reduced either to maximizing the expected gain with constraint of the low partial

moment or to maximizing/minimizing the low partial moment with constraint on the

expected gain. We showed that in case when the Omega function is greater than 1 at

optimality, the proposed reductions lead to linear programming. We illustrate the use of

the proposed methodology with the resource allocation problem from the real hedge fund

practice.









CHAPTER 5
CALIBRATION OF GENERAL DEVIATION MEASURES FROM MARKET DATA

5.1 Introduction

General portfolio theory with general deviation measures, developed by Rockafellar

et al. (2005a, 2006), was shown to have similar results to the classical portfolio theory

Markowitz (1959). Replacement of the standard deviation in the classical portfolio

optimization problem by some general deviation measure leads to generalization of

concepts of masterfund, efficient froniter, and the CAPM formula. In particular, the

necessary and sufficient conditions of optimality in the portfolio problem with general

deviation measures were called CAPM-like relations in Rockafelar et al. (2006). In this

chapter, we refer to them as generalized CAPM relations; and refer to the underlying

theory as the generalized portfolio theory.

This paper makes a connection between the general portfolio theory and the classical

asset pricing theory by examination of generalized CAPM relations. In particular, we

derive discount factors, corresponding to the CAPM-like relations and consider pricing

forms of generalized CAPM relations. We propose a method of calibrating deviation

measures from market data and discuss v--iv of identifying risk preferences of investors in

the market within the framework of the general portfolio theory.

5.1.1 Definitions and Notations

Following Rockafellar et al. (2005b), we define random variables as elements of

2(Q) = 2(, M4, P), where Q is a space of future states w, M4 is a a-algebra on Q, and

P is a probability measure on (2, AM). The inner product between elements X and Y in

[2(Q) is

(X,Y) E[XY] = X(w)Y(w)dP(w).

In this paper we will use the notions of a deviation measure D, its associated risk envelope

Q, and a risk identifier Q(X) for a random variable X E 2(Q) with respect to D. The









reader is referred to Rockafellar and Uryasev (2002) and Rockafellar et al. (2005a, 2006)

for details.

5.1.2 General Portfolio Theory

The general portfolio theory (Rockafellar et al. (2005a)) is derived in the following

framework. The market consists of n risky assets with rates of return modelled by r.v.'s ri

for i = 1,..., n and a risk-free asset with the constant rate of return modelled by a constant

r.v. ro. Several modelling assumptions are made about these rates of returns.

Investors solve the following portfolio optimization problem.


min D(xoro + xiri + ... + xr) (5-1)

s.t. E(xoro + x1ir + ... + Xnr) > ro + A

xo + l + +... + = 1

xi E R, i = 0,..., n.


In the case of a finite and continuous deviation measure D, generalized CAPM

relations come out as necessary and sufficient conditions for optimality in the above

problem. It was shown in Rockafellar et al. (2005b) that problem (5-1) has three different

types of solution depending on the magnitude of the risk-free rate, corresponding to cases

of the master fund of positive type, the master fund of negative type, and the master

fund of threshold type. Master fund of positive type is the one most commonly observed

in the market, when return of the market portfolio is greater than the risk free rate, and

investors would take long positions in the master fund when forming their portfolios.

In this paper, we consider the case of master funds of positive type and the

corresponding CAPM-relations

cov(-rs,, Qj)
Eri ro = D(r) [E I ro], i = ..., n, (5-2)









where rM is the rate of return of the master fund, Q2 is the risk identifier for the master

fund rf corresponding to the deviation measure D.

5.1.3 Generalized CAPM relations and Pricing Equilibrium

Relationships (5-2) closely resemble the classical CAPM formula. However,

generalized CAPM relations cannot pl i, the same role in the general portfolio framework

as CAPM formula pl i in the classical theory, as discussed in Rockafellar et al. (2005b).

The group of investors using the deviation measure D is viewed only as a subgroup of

all the investors, generalized CAPM relations do not necessarily represent the market

equilibrium, as the classical CAPM formula does, and therefore cannot be readily used as

a tool for asset pricing. Another difficulty with using relations (5-2) for asset pricing is

that neither the master fund nor the asset beta for a fixed master fund can be uniquely

determined.

For the pricing using the generalized CAPM relations to make sense, we make the

following assumptions.

(Al) All investors in the considered economy use the same deviation measure D.

(A2) The master fund can be identified in the market (or some proxy for the master

fund exists). If the set of risk identifiers for the master fund is not a singleton, the choice

of a particular risk identifier from this set has negligible effect on asset prices obtained

though the generalized CAPM relations. Therefore, we can fix a particular risk identifier

for the purpose of asset pricing.

Assumption A2 makes sense because for most basic deviation measures members

of the risk identifier set QD(r') for a given master fund r' differ on a set of the form

{' = C}, where C is a constant. For deviation measures considered in Rockafellar et

al. (2006), the risk identifier set for standard deviation and semideviations is a singleton;

C = -VaRo(X) for CVaR-deviation with confidence level a; C = Ei for mean absolute

deviation and semideviations. Since asset prices in generalized CAPM (5-2) depend on the

risk identifier Q' though (-ri, Q'), assumption A2 -,.. that Prob{r = C} = 0.









This is true for continuous distributions of r{, and usually holds in practice when the

distribution of the master fund is modelled by scenarios.

Assumption A2 cannot be satisfied for worst-case deviation and semideviations, see

Rockafellar et al. (2006)

Under assumptions Al and A2, all quantities in generalized CAPM relations are fixed

and well-defined, and the relations represent pricing equilibrium. In further chapters we

will closely examine generalized CAPM relations under these assumptions.

5.2 Intuition Behind Generalized CAPM Relations

5.2.1 Two Ways to Account For Risk

Consider an asset with price 7 and uncertain future p lioff (. In a risk-neutral world,

the asset will be priced as follows.


= E[ (5-3)
1 + ro

where ro is a risk-free rate of return. The price of an asset is the discounted expected

value of its future p ivoff. The asset with random p i-off ( would have the same price as an

asset with 1p ]Ii = E[(] with probability 1 in the future.

If the risk is present, the price of an asset p liing ( with certainty in future would,

generally speaking, differ from the price of the asset having random p i-off (, such that

E[(] = The formula (5-3) needs to be corrected for risk. There are two --- -4 to do it.

The first way is to modify the discounted quantity:


7- = 0 (asset), (5-4)
l+ ro

where J(asset) is called the cer'ah',,l equivalent. It is a function of asset parameters and

is equal to the p i-off of a risk-free asset having the same price as the risky asset with

1p 'ioff (.









The second way is to modify the discounting coefficient:


7 = E[(], rra(asset) -1. (5-5)
1 + rra(asset)

where rra(asset) is the risk-adjusted rate of return.

Pricing forms of the classical CAPM (see, for example, Luenberger (1998)) are as

follows.

Certainty equivalent form of CAPM:

1 (E[ cov(,TrM)(ErM -ro))
t = E [[(] COV, r(E ro.- (5-6)
1+ro o /

RB-1:- dliusted form of CAPM:

S= E []. (5-7)
1 + ro + 3(ErM ro)

Here asset beta 3 = co ( rM) rM is the rate of return of the master fund, and r is the rate

or return of the asset (r =(( r)/Tr).

Relevant to further discussion, there is a measure of asset quality known as the

Shapre Ratio
E[r] ro
S 7 (r)

It is a risk-return characteristic, measuring the increase in the access return of an asset if

the asset volatility in increased by 1. The higher the Sharpe Ratio, the better the asset.

Classical CAPM implies that master fund has the highest Shapre Ratio in the economy.

5.2.2 Pricing Forms of Generalized CAPM Relations

We now derive pricing forms of the generalized CAPM relations. Substituting

ri = /Ti 1 into (5-2), we get











E( cov((/7~ Q1, Q)
(ro + 1) [E, ro]

Er ro
E( 7i#(ro + 1) cov(, Q,)

1 E, -ro D
F = r E + EcOV (Ci, rQf) (5-8)
1 + ro ((m

Pricing formula (5-8) the certainty equivalent pricing form of generalized CAPM

relations (5-2) (compare it to (5-4)), where the certainty equivalent

Er D ro
Q((,) E( + (r) cov(, QM) (5-9)

is the p ioff of a risk-free asset having the same price 7rj.

We could rearrange the formula (5-2) in a different way, namely

cov(-ri, Q) )
Er ro = D( [El, ro]


r ro r


EE( ErE ro
S( (ro + )) + r cov(- )
ii D(rM)

E(i
TTi = (5-10)

1 + ro + D(r-,) cov(- r, Q)

ErD ro
when D(rM) cov(-ri, Q) / 1 + r0.









Formula (5-10) is the risk-adjusted pricing form of generalized CAPM relations (5-2)

(compare with (5-10)), where the risk-adjusted rate of return is



TVM)
E, ro
rm(ri) r0 + D(r ) cov(-r4, Q ). (5 11)


ErM r0
The quantity D(r) which we denote by SD, in (5-8) and (5-10) is the

generalized Sharpe Ratio for the master fund. It shows what increase in excess return

can be obtained by increasing the deviation of the asset by 1. In the classical portfolio

theory, master fund has the highest Sharpe Ratio among all assets. The same result holds

in the generalized setting as we show next.

Lemma 2. For the case of the master fund of positive ';'. the master fund has the

highest generalized Shapre Ratio in the econ.-'i,, i.e.

ErM ro Eri ro
> ,i ... n. (51 2)
D(rD)- (r )

Proof: Consider generalized CAPM relations

cov(- r, Q )
Er ro = (r) [Er- ro]

for some asset i > 0. The generalized Sharpe Ratio for the master fund is strictly positive
Er ro
SM = D(r > 0 since E, 7 ro > 0 and D( 7) > 0.

Eri ro
If cov(-r, QD) 0, then Eri = ro, therefore D = 0, and (5-12) holds.

Eri ro
If cov(-ri, QD) < 0, then Er, < ro, therefore < 0, and (5-12) holds.
D(r)
If cov(-ri, QD) > 0, then according to the dual representation of D(ri), we have


D(ri) max cov(-ri, Q) > cov(-rr, Q ) > 0,
QEQ









where Q is the risk envelope for the deviation measure D, and QM e Q. Dividing both

sides of generalized CAPM relations by cov(-r>, QM), we get

E, ro Eri ro Eri ro
D(r") cov r, Q ) > (ri)



Formulas (5-8) and (5-10) imply that the risk adjustment is determined by the

correlation of the asset rate of return with the risk identifier of the master fund.

To gain a better intuition about the meaning of this form of risk adjustment, we

compare the classical CAPM formula with the generalized CAPM relations for the

CVaR-deviation D(X) CVaRf (X) CVaR(X EX).

First note, that more valuable assets are those with lower returns. When pricing two

assets with the same expected return, investors will p .iv higher price for a more valuable

asset, therefore its return will be lower than that of the less valuable asset.

We begin by analyzing the classical CAPM formula written in the form


Er, r + covr, M (ErM ), (5-13)


where the left-hand side of the equation is the asset return. The return is governed by the

correlation of the asset rate of return with the market portfolio rate of return, i.e. by the

quantity cov(ri, rM). Assets with higher return correlation with the market portfolio have

higher expected returns, and vice versa. Formula (5-14) implies that assets with lower

correlation with the market are more valuable. There is the following intuition behind

this result. Investors hold the market portfolio and the risk-free asset; the proportions

of holdings depend on the target expected portfolio return. The only source of risk of

such investments is introduced by the performance of the market portfolio. The most

undesirable states of future are those where market portfolio returns are low. The assets

with higher p li-off in such states would be more valued, since they serve as insurance

against poor performance of the market portfolio. Therefore, the lower the correlation of









an asset return with the market portfolio return, the more protection against undesirable

states of the world the asset offers, and the more valuable the asset is. The assets with

higher correlation with the market would have higher returns, and vice versa.

Now consider the case of CVaR-deviation, D(X) = CVaR,(X EX). Investors

measuring uncertainty of the portfolio performance by this deviation measure are

concerned about the value of the average of the (' worst returns relative to the mean

of the return distribution.

We consider generalized CAPM relations for the CVaR-deviation in the case of the

master fund of positive type.

Er = ro + si(E ,- ro), (5-14)

cov(-r Q7)
where 3 = -D(rQ) and the risk identifier Qj is given by


Q< Q(w) < a-1, EQ 1,

SQ() 0 when r{(w) > -VaR( ), (5-15)

Q )= a-1 when r'(w) < -VaR,(r').

If prob{, = VaR(r)} 0, then

E[Er, r r-r < -VaR,(r)]1
=M (5-16)
E[Er' r < I r < -VaR(r' (516)

For further discussion, assume a = 1('". Then the numerator of (5-16) is the expected

underperformance of the asset rate of return with respect to its average rate of return,

conditional on the master fund being in its 1('. lowest values. The denominator of (5-16)

is the the same quantity for the master fund. An investor holds the master fund and the

risk-free asset in his portfolio. The portfolio risk is introduced by the performance of

the master fund. Formula (5-16) -ir-.-. -I; that assets are valued based on their relative

performance versus the master fund performance in those future states where the master









fund is in its 10' lowest values. The most valued assets, i.e. assets with lowest returns,

would have the lowest betas. Low betas correspond to relatively high asset returns (small

values of Er ri) compared to the master fund returns (values of ErM rD), when rM is

among 10',. its lowest values.

From the general portfolio theory point of view, the value of the asset is, therefore,

determined by the extent to which this asset provides protection against poor master fund

performance. Depending on the specific form of the deviation measure, the need for this

protection corresponds to different parts of the return distribution of the master fund.

Most valuable assets drastically differ in performance from the master fund in those cases

when protection is needed the most.

5.3 Stochastic Discount Factors in General Portfolio Theory

5.3.1 Basic Facts from Asset Pricing Theory.

The concept of a stochastic discount factor appears in the classical Asset Pricing

Theory (see Cochrane (2001)). Under certain assumptions (stated below), there exists

a random variable m, called the (stochastic) discount factor or the pricing kernel, which

relates asset p ,voffs (i to prices 7r as follows.


i = E[m (], i =0,...,n. (5-17)


The discount factor is of fundamental importance to asset pricing. Below, we present two

theorems due to Ross (1978), and Harrison and Kreps (1979) which emphasize connections

between the discount factor and assumptions of absence of arbitrage and linearity of

pricing. In the narration, we follow Cochrane (2001), ('! Ilpter 4.

Let X be the space of all p ',offs an investor can form using all available instruments.

We will consider two assumptions, the portfolio formation assumption (Al) and the

law of one price assumption (A2).

(Al) If (' E X, (" E X, then a(' + b(" E X for any a, b E R.

Let Price(() be the price of p ,,off (.









(A2) If Price((') -= and Price((") = r", then Price(a(' + b(") = ar' + br" for any

a, b R.

Under assumption (Al), the 1 i off space X is defined as follows.


X = {( ( = ao + a,1( + ... + a,,(, a E R, i = 0,..., n},

where (i is the p i-off of asset i, i = 0,..., 1.

Theorem 3. (1) The existence of a discount factor implies the law of one price A2. (2)

Given portfolio formation Al and the law of one price A2, there exists a unique J.';.''f

* E X such that the price ~r of i,., y'.n.;.'f ( E X is given by ~ = E[(*(].

The second theorem has to do with absence of arbitrage, which is defined as follows.

Absence of Arbitrage: The 1p 'off space X and the pricing function Price(.) leave no

arbitrage opportunities if every 1p ioff ( that is ahv--l- non-negative, ( > 0 (almost surely),

and positive, ( > 0 with some positive probability, has positive price, Price(() > 0.

Theorem 4. (1) Existence of a strictly positive discount factor implies absence of

arbitrage opportunities. (2) No arbitrage implies the existence of a strictly positive discount

factor, m > 0, = E[,,(] for i,.:; ( E X.

These theorems for the case of assets with continuous 1p loffs are given in Hansen and

Richard (1987).

From the perspective of discount factors, a complete market is characterized by a

unique discount factor; in an incomplete market there exists an infinite number of discount

factors and each discount factor produces the same prices of all assets with ip .ioffs in X

through (5-17). More details on pricing assets in compete and incomplete markets will be

provided later on. Important implications of these theorems are as follows.

There exists a strictly positive discount factor m > 0, and such factor might not be

unique.

In the space of p 'loffs X there exists only one discount factor (* e X, which may or

may not be strictly positive.









In a complete market with no arbitrage opportunities, the unique discount factor lies

in the p .ioff space X and is strictly positive.

In an incomplete market with no arbitrage opportunities, all discount factors can be

generated as m = (* + F, where (* is the discount factor (unique) in the 1p ioff space

X, and c is a random variable, orthogonal to X, E[(] = 0 V( E X.

The discount factor (* is a projection of any discount factor m on X. For any asset,

S= [,.1(] = E[(proj(m|X) + E)(] E[proj(mX) ].

It should be mentioned that the existence of so-called I:-1l-neutral" measure is

justified by the existence of a strictly positive discount factor. Indeed, we can rewrite

(5-17) as follows.

S= r[ii] /n m()((u)dP(u) =- (()dQ(u), (5-18)
ct l+ o Jo

where dQ(u) = (1 + ro)m(u)dP(u). Since expectation of (1 + ro)m equals to one1

and m > 0, dQ(w) can be treated as a probability measure. It is usually called the

i,-I:-neutral" probability measure; the risk-neutral pricing form of (5-17) is

1
1 + ro

where EQ[.] denotes expectation with respect to the risk-neutral measure.

If one picks a discount factor m, which is not strictly positive, the transformation

(5-18) will lead to the pricing equation r = f ((u()dQ(wu) that correctly prices all assets

with 1p ,ioffs in X. However, dQ(w) will not be a probability measure.



1 Application of (5-17) to the risk-free rate gives 1 = E[m(1 + ro)].









Now consider application of the formula 7 = F[,,,(] for pricing new assets2 in

complete and incomplete markets.

In a complete market, the p ., off of any new asset lies in X, therefore any new asset

will be uniquely priced by the law of one price (alternatively, since the discount factor is

unique, there exists only one price, Frew = E[,gew], for a new asset with p ,',off (g, e

In an incomplete market, two cases are possible. (1) The p 'ioff of a new asset belongs

to X; its price is uniquely determined by the law of one price (alternatively, the formula

7rnew = E[,,e,,w] will give the same price regardless of which discount factor m is used).

(2) The p ',-off of a new asset does not belong to X, i.e. the new asset cannot be replicated

by the existing ones. In this case, one cannot decide upon a single price of the asset. Let

(new be the p .,ioff of a new asset. Upper r,,,w and lower ,wr prices (forming the range of

non-arbitrage prices [e,,w,e]) of this asset can be defined as follows.


TTnew sup E[m (ew], ,new inf E[mnew,], (519)
mEb mEb

where = {-m I m(w) > 0 with probability 1}. Including only strictly positive discount

factors to the set + leads to arbitrage-free prices given by formula 7 = [,,,(].



2 Originally, we assumed that the market consists of n + 1 assets with rates of returns
ro, rl, ..., rT. Any other asset is considered to be new to the market. A new asset may be
replicable by the existing assets (in which case its p 'ioff will belong to X) or may not be
(then its p i, off will not belong to X).









5.3.2 Derivation of Discount Factor for Generalized CAPM Relations

We begin by rewriting CAMP-like relations as follows.


cov(-rn, QM)
Er ro= c ( ) [Er r

E( 1 cov(-(, M) 2 ro
rEr roD D
E (ro + l)7r = DM 0 (E(EQM E[( QM]),

7i t / Ei ErE ro E[(QMiP
S+ ro ^ D(rf) [M



i = 0,1, ..., n.


Letting



we arrive at


rn(P) (QM ) )ErM + 1

the pricing formula in the form (5-17)


i = E[m' (i], i 0,1,...,n. (5-22)

The discount factor corresponding to the deviation measure D is given by (5-21).

Pricing formulas (5-22) corresponding to different deviation measures D will yield the

same prices for assets ri, i = 0, ..., n, and their combinations (defined by portfolio

formation assumption Al), but will produce different prices of new assets, whose p lioffs

cannot be replicated by p ,lioffs of existing n + 1 assets. Each deviation measure D has the

corresponding discount factor mW, which is used in (5-22) to determine a unique price of

a new asset. An investor has risk related to imperfect replication of the lp i-off of a new

asset, and specifies his risk preferences by choosing a deviation measure in pricing formula

(5-22).


ErrM ro EQ,


+1) 1 ,


(5-20)


(5-21)









5.3.3 Geometry of Discount Factors for Generalized CAPM Relations

Consider two deviation measures, D' and D". Both measures provide the same pricing

of assets i = 0,..., n: 7r = E[m',(i] and 7, = E[mD",,(]. Subtracting these equations

yields E[(mD, mD,)(i] = 0, i =, ..., n. The difference of discount factors for any two

deviation measures is orthogonal to the p .'off space X. It follows that discount factor mD

for any D can be represented as mD = m* + ED, where m* E X is the projection of all

discount factors mD on the p .,ioff space X, and ED is orthogonal to X. We call m* pricing

generator for the general portfolio theory.

The pricing generator m* coincides with the discount factor for the standard deviation

D = a, since
1 r' (u) Er"' Er, ro
,() 1 (lr ) -ErM Er r (5-23)
1 + ro o(r) ao (r )

together with i E X imply m, E X.

For a given p ,,off space X, discount factors m'D for all D form a subset of all discount

factors corresponding to X.

5.3.4 Strict Positivity of Discount Factors Corresponding to Deviation
Measures

We now examine strict positivity of discount factors corresponding to general

deviation measures.

The strict positivity condition ma(w) > 0 (a.s.) can be written as
S1) E ro 1 >0
((Qro M"() 1i) 1 >

QfM )> > U ) (5-24)


Note that the left-hand side of condition (5-24) contains a random variable, while the

right-hand side is a constant, and the inequality between them should be satisfied with

probability one. Scaling the deviation measure D by some A > 0 will change the value of

the left-hand side. We show next that it does not change meaning of the condition (5-24).

Lemma 3. Condition (5-24) is invariant with respect to re--. Al1.:, deviation measure D.









Proof: Condition (5-24) can be expressed as


1
Q0(w)>1 -SD
SM


where SV Consider a re-scaled deviation measure D
S F _-ro


AD, A > 0. Let S


be the Sharpe Ratio corresponding to D. Since master funds for ED and ED are the same,

M AM

Since the risk envelopes Q and Q for deviation measures D and ED are related as


(1 A) + AQ,


the risk identifiers Q(rm) and Q(r') will be related in the same way, as shown next.


Q(I ,0)


argmin cov(-
QEQ


I ,,Q)


argmin cov(-r, Q)
Qe(1-A)+AQ
(1 A) + A argmin cov(
QEQ
(1 A) + A argmin cov(
QEQ


' (1

-I Q)


A) + AQ)


S (1 A)+ AQ(, ,).


Finally, if (5-25) holds for D, it holds for AD as well, since


(1 A) + AQv()

AQv(Lw)


QVM(L)


> 1


> 1


> A-A
> A
>
S3


(5-25)









Next, we show that the pricing generator m* is not strictly positive. Indeed, the

corresponding risk identifier is given by

Q ((w) =1 M


Condition M(w) > 0 takes the form

r' (u) Er' P(r,)
1- > Er ro



17(r'T < Er' ro
u2(rU
r' (a) < Er +
Er( ro

The last inequality is violated with positive probability, for instance, for normally

distributed random variables.

Consider an alternative representation of mD(w) in (5 21). Letting S = ,-, we
)we
get

1 1
(w ((Q()) 1)S + 1) ((Q (w)Sm + (1 S)) (5-26)
1 + ro 1 + ro

In Lemma 1 we showed that risk identifiers Q( a ) and Q( T) for deviation measures

D = AD (A > 0) and D, respectively, are related as
Q(0 ,) -(1- A)+AQ(,).



This allows to rewrite the expression for the discount factor as follows,


1 + ro

where QjM (w) is a risk identifier for the deviation measure DM = S -* D. Strict positivity

of a discount factor is then equivalent to strict positivity of the risk identifier QM (u().









5.4 Calibration of Deviation Measures Using Market Data

5.4.1 Identification of Risk Preferences of Market Participants

As was discussed earlier, if a general portfolio problem is posed for a set of basic

assets ro, ri, ..., rn, then deviation measures gives the same prices to these assets, as

well as to any assets whose p li-offs can be replicated by p ,lioffs of the basic assets. In

this section, we examine v-- i of estimation (calibration) of the deviation measure ED

in the general portfolio theory from market data. Numerical methods supporting the

proposed algorithms are not considered in this work; we concentrate on the meaning

of the calibration methods, their advantages and drawbacks, and their limitations in

determination of risk preferences of market participants. Essentially, calibration of

deviation measures is done by adjusting it until the generalized CAPM relations

cov(-r Q{)
Er = ro + fI [[E, ro], i 1= ,...,n. (5-27)


provide the most accurate asset pricing. We could either take a set of asset returns ri

from the market and estimate the master fund rm, or treat the master fund as given by

the market and estimate expected returns Eri. The obtained quantities, the master fund

return or expected returns of the assets, will depend on the deviation measure D, which

can be calibrated by comparing estimated quantities to their market values.

We limit our consideration to the case of known master fund; the method based on

estimation of a master fund given a set of assets is more computationally difficult, because

the generalized portfolio problem should be solved for each choice of D.

Assumption that a master fund can be obtained from the market is justified by the

existence of indices, such as S&P 500, Dow Jones Industrial Average and N -I1 .1 100,

which represent the state of some large part of the market; moreover, investing in these

indices can be thought of as investing in the market.

Any broad-based market index is associated with certain selection of assets; the index

summarizes the behavior of the market of these assets. We could calibrate the generalized









CAPM relations by pricing assets from the index-associated pool, or by pricing foreign

assets to this pool. These two ideas have different meaning as they refer to different v- --

risk preferences are manifested in the general portfolio theory.

The first calibration method is based on pricing assets from the index pool. The index

serve as a master fund in a generalized portfolio problem posed for assets from the pool.

Given a fixed selection of assets, different deviation measures would produce different

master funds. The existence of a particular master fund for these assets in the market can,

therefore, be used as a basis for estimation of a deviation measure. The "b' -1 deviation

measure is the one which yields the best match between the expected returns of assets

from the pool and the index return through the generalized CAPM relations.

The second calibration method is based on pricing assets lying outside of the index

pool. As we discussed earlier, when pricing a new asset whose p i-off does not belong

to the initially considered 1p .ioff space, the price investors would p lv depends on their

risk preferences, defined by the deviation measure. The second method, therefore, uses

prices of ,, .--" assets with respect to the index pool as the basis for estimation of risk

preferences. It should be noted that in the setup of the general portfolio theory the

selection of assets is fixed, and the master fund depends on the deviation measure. In

the present method we assume that the master fund is fixed and change the deviation

measure to obtain the best match between the master fund return and expected returns

of new assets. By doing so, we imply that the choice of the index-associated pool of assets

depends on the deviation measure.

We justify the assumption of a fixed master fund by the observation that master

funds, expected returns of assets, and their generalized betas can be determined from the

market data quite easily, while the selection of assets corresponding to an index can be

determined much more approximately. An index usually represents behavior of a part

of the market consisting of much more instruments that the index is comprised of. With

much certainty, though, we could assume that assets constituting the index belong to the









pool of assets represented by the index. Therefore, the first calibration method can be

based on matching the prices of assets the index consists of.

We also note that implementations of both methods are the same: selecting some

index as a master fund, we adjust the deviation measure until the generalized CAPM

relations provide most accurate pricing of a certain group of assets. We refer to this group

of assets as the target group.

Finally, we discuss the question, should the two calibration methods give the same

results. Generally speaking, for a fixed set of assets, the choice of risk preferences in terms

of a deviation measure determines both the master fund and pricing of new assets with

p ,ioffs outside of the considered p ,i-off space. When the generalized portfolio problem is

posed for the whole market, risk preferences can be determined only through matching

the master fund, since there are no i. .-" assets with respect to the whole market. The

master fund coincides with the market portfolio, i.e. weight of an asset in the master fund

equals the capitalization weight of this asset in the market.

If a certain index is assumed to represent the whole market, then calibration of the

deviation measure based on different target groups of assets (for example, on a group

of stocks and a group of derivatives on these stocks) should give the same result. If the

obtained risk preferences do not agree, this may indicate that either the general portfolio

theory with a single deviation measure is not applicable to the market or that the index

does not adequately represent the market.

If indices track performance of some parts of the market, the two methods are

not, generally -I'" i1:i:- expected to give the same results. Market prices of assets not

belonging to an index group may not be directly influenced by risk preferences of investors

holding the index in their portfolios. For example, it does not make sense to calibrate

risk preferences by taking one index as a master fund and assets from another index

as a target set of assets. However, it is reasonable to suppose that prices of derivatives

(for example, options) on the assets belonging to an index group are formed by risk









preferences of investors holding this index. Derivatives on assets have non-linear p li-offs

and cannot be replicated by p ,lioffs of these assets. The second calibration method applied

to pricing derivatives on some stocks is expected to give similar risk preferences as the first

calibration method applied to pricing the same stocks, where the master fund is taken to

be the index representing these stocks. If the so-obtained risk preferences do not agree,

either the general portfolio theory is does not adequately represent the chosen part of the

market or option prices are significantly influenced by factors, not captured by the risk

preferences of investors holding the corresponding index in their portfolios.

5.4.2 Notations

We consider two implementations of calibration methods. We assume that the

index-associated group of assets consists of n assets with rates of return rl, ..., r,, the

master fund associated with the deviation measure D is a portfolio of these assets and the

risk-free asset with the rate of return ro; the rate of return of the master fund is r%. The

target group of assets consists of k assets with rates of return r',..., r'. The target assets

may or may not belong to the index-associated group.

For the purposes of calibration, we assume a parametrization of a deviation measure

S= D,, where a = (a, ..., al) is a vector of parameters.

5.4.3 Implementation I of Calibration Methods

The first implementation is based on direct estimation of expected returns of target

assets and minimization of the estimation error with respect to parameters d. Let

Ef'(a) = (Er(a),..., Er'(a)) be a vector of expected returns of target assets estimated

using the deviation measure D,, Er' = (Er,..., Er') be a vector of the true expected

rates of return, and Dist(Er', Ef'(a)) be a measure of distance between the two vectors.

The parameters a the deviation measure can be calibrated by solving the following









optimization problem.


minj Dist(Er', EP'(a))

cov(-rI, &Q^)
s.t. Er(a) 0 ro + zv [Er(M[ r0], i = ..., k,
D(rM)

where covariances cov(-r, Q M,) are calculated from the market.

5.4.4 Implementation II of Calibration Methods

The second implementation is common in the literature of restoring risk preferences.

It is based on estimating the ratio of risk-neutral and actual distributions of the

master fund. We adapt the procedure to the setup of the general portfolio theory.

The target group of assets now consists of European call options on the master fund.

The implementation differs from the previous one as follows. Expected returns of

target assets and the rate of return of the master fund are replaced by the risk-neutral

density of the master fund and the actual density of the master fund, respectively;

the match between these densities is optimized with respect to parameters a. The

mentioned risk-neutral density can actually be referred to as ", i-ly only under certain

assumptions, ensuring that this function is non-negative. However, methods of estimating

this function from market data usually assume its non-negativity. The use of European

options in this implementation is essential, therefore it can be an implementation of the

second calibration method only. Options have non-linear p .,offs with respect to the

underlying assets and do not belong to the pool of assets associated with the chosen index.

Assume that the probability measure P in the market has a density function p(w).

We consider the generalized CAPM relations in the form (5-22) and transform them as

follows (Q denotes the complete set of future events u).


w = ER] = I(()'(u)p(udu = t)( (()(1 + ro)mD(uw)p(u)du, (5-28)
7 I1 + r o "









where 7 and ( are the price and the p ,,off of an asset. Letting


qD(o) = (1 + ro)mn'()p(w), (5-29)

we get
t= j ()D(w)dw. (5-30)
1 + ro J
As we discussed above, if the discount factor mr(w) is strictly positive, the function q' (w)

could be called the i -:-neutral" density function.

The future event w consists of future returns of all assets in the market and can be

represented as a = (r r, ..., r', f), where f represents rates of returns of the rest of assets

in the market.3

Now consider integrating relationship (5-29) with respect to r ',..., r, r.

qD ( /,..., r, r)dr...drdr (1+ro) jM (r, r,... r, r)p(r, r,..., r, r)dr...drdr.
Ja Jo
(5-31)

Let
(~) q( ,ri,.. ,r)d...drIddrr.

If ma was strictly positive, qD( 7) would be a risk-neutral marginal distribution of the

master fund. To simplify the right-hand side of (5-31), note that the discount factor m is

a linear transformation of the risk identifier Q' (both anD and Q' are random variables

and are function of w). Due to the representation


Qc E QD = argminE[rDQ],
QEQ



3 The master fund is not an asset but a portfolio of assets with rates of return ri,..., r..
The future state w is initially represented as a = (r,..., r,, r, ..., r ,, ). Assuming that
the asset r, is represented in the master fund with non-zero coefficient, we can represent w
as Lc (r, r,.., r,, ..., r, r ). After including r2,..., r, to f, we get the representation
= Iwk









the risk identifier QV(wo) depends on u through rv, Qv(Wo) = Q(r' (w)). For example,
for the case of standard deviation D = a

r', (L.) E r'
rM()- Er
QM P( () 1 (

so Q(w( = Q((r ,(w)).
Equation (5-31) becomes


q"(ry) = (1 + rno m(r')p(r, r..., r', )dr ...dr dr,
4"(r') = (1 + ro)m'(r) p(r'), rf, ..., r ', )dr ...dr di,

q(r') (1 + ro)M(r)p(), (5-32)

where p(r) f,, p(r', r', ..., r, f)dr[...dr'dr is the actual marginal distribution of the

master fund.

Relationship (5-29) is now transformed into

q(i7) (1 + ro)n(ir)p( r), (5 33)

where = 7 ,(w). This relationship provide the basis for calibration of D. Let q(1' T)
denote the true risk-neutral distribution of the master fund. Both functions q(rv) and

(i 7,) can be estimated from market data; the error in estimation of q(i 7) by q4rv) in
(5 33) is minimized with respect to D.
First, we consider estimation of q(r ). Let q(u) be the true market risk-neutral

distribution, q(r') = q(r', r ...., r', r)dr...dr'dr. Applying formula (5-30) with q(w)
for pricing an option on the master fund, we get

1 f t P)41
tc r= (cr'd.)q( ,r'I ..drdr I \k i ,)dr( if M
1 + ro r 1 + ro

where r, and (c are the price and the 1 ,-off of the option.









To simplify notations, we will consider applying the above formula for a call option

on with price C, strike K, time to expiration T.4 The option is written on a master fund

with current price So and price S = S(w) at expiration of the option. The derivation

below concerns estimation of the function 4(S), and q(r{) = q(S/So).


p+oo +00
rTc= [S K]+q(S)dS (S- K)4(S)dS
(5-34)
= Sq(S)dS K q(S)dS.
K K
Differentiating (5-34) with respect to K, we get

r = -K(K) (S)dS + Kq(K) q(S)dS.
OK JK JK

Differentiating (5-34) twice with respect to K, we arrive at the formula for estimating

risk-neutral density q from cross section of option prices


e q 4(S)dS = q(K),
OK2 8K JK

or in the most common form

4(S) -= e (5-35)
OK2 K=S
Formula (5-35) allows to estimate the function q( ,) when the cross-section of prices

of options written on the master fund is available. It is worth mentioning that this method
estimates q(r\) at a given point in time; it is based on options prices at this time.

Now consider estimation of the marginal probability density p(r'). The most

common way to estimate this density is to use kernel density estimation based on certain

period of historical data. However, this method assumes that the density does not

change over time. When time dependence is taken into account, we are left with only one



4 For options with time to maturity T, the discount coefficient is e'T rather than 1 + ro.









realization of this density at each point in time; direct estimation of the density is not

possible.

However, the formula (5-32) provides a way of estimating p(rM) for a specific date,

if the function m'(r) is known. This idea is utilized in the utility estimation algorithm

-, .-. -- I.1 in Bliss and Panigirtzoglou (2001). We develop a modification of this method to

calibrate the deviation measure, as follows.

Assume the parametrization D = D,. Also assume that the master fund is known

from the market and therefore is fixed, its rate of return is denoted by rM. For each date

t ...,T, we estimate the function m'(rM) using (5 21). Quantities D,(rM), Erv", and

QVM in the definition of ma(rM) are calculated based on a certain period of historical

returns the index. Also, we estimate functions qt(rM), t = 1, ..., T, using (5-35). Formula

(5-33) allows to estimate function qt(rM) for each parametrization of D The parameters

a can be calibrated by hypothesizing that qt(rM) (rM) for t = 1, ..., T (which holds

if D is the correct deviation measure in the market) and maximizing the p-value of an

appropriate statistic.

This hypothesis is further transformed as follows. Using the true risk-neutral

distributions qt(rM), the actual distributions pt(rM) are estimated using (5-33),


4t(rM)
<(rM) (1 + ro) (FM)'

t 1, ..., T. We then test the null hypothesis that risk-neutral distributions pf(rM),

t = 1,..., T, equal to the true risk-neutral distributions t(rM), t =1, ..., T.

For each time t = 1, ..., T, only one realization rM(t) of the master fund is available;

the value rM(t) is a single sample from the true density Pt(rM). Under the null hypothesis

p(rM) = pt(M), therefore random variables yt defined by

I rM(t) l
t f 0C tp r)dr,
-o00

for t = 1, ..., T, are i.i.d. Uniform[0, 1] random variables.









Joint uniformity and independence of y', t = ..., T, can be tested using Berkowitz

LR3 statistic (see Berkowitz (2001)), which has the chi-squared distribution with 3 degrees

of freedom X2(3) under the null hypothesis. The deviation measure D, can be calibrated

by maximizing the p-value of the LRs statistic with respect to parameters a.

5.4.5 Discussion of Implementation Methods

Both considered implementations are based on the same idea (fitting the generalized

CAPM relations to market data) but algorithmically are quite different.

Implementation I requires calculation expected returns of assets and estimation of

actual distribution of the master fund. These quantities can be found from the market

data quite easily and accurately. However, the results of this implementation depend on

a particular choice of the objective function Dist(-, .). It can be argued that the choice of

the objective should depend on a the parametrization D, of the deviation measure being

calibrated. For example, if the deviation measure is calibrated in the form of the mixed

CVaR-deviation, then Dist(-, -) should be based on the CVaR-deviation, rather than on the

standard deviation. Another drawback of implementation I is that the financial literature

did not use similar algorithms for calibration of utility functions. When risk preferences

are estimated using this implementation for the general portfolio theory are compared

with risk preferences estimated in financial literature for the utility theory, the results may

differ just due to differences in numerical procedures, underlying the two estimations.

Implementation II is widely used in financial literature for estimation of risk-aversion

coefficients of utility functions. However, this implementation suffers from some numerical

challenges related to evaluation of the actual density in the formula

4t (rM)
prm) (t + ro) Ma

The first challenge is estimation of risk-neutral distributions qt(rM). There are several

methods of estimation the risk-neutral distribution from the cross-section of options prices

used in the literature, but the results of estimations are sensitive to the data and may









significantly depend on a method used. Second challenge is estimation of the function

F(rTM). Discount factors ma(rM) may be close to zero for some values of rM, which makes

accuracy of estimation of 4t(rM) crucial for calculation of densities pF(rM) and even more

crucial for calculation of y4, t = 1, ..., T. It follows that risk preferences obtained using

implementation II can only be trusted if the underlying numerical methods are very

reliable.

There is one more drawback of this implementation when it is in the general portfolio

theory. When using numerical estimation of risk-neutral densities, we have to assume

that no arbitrage opportunities exist in the prices of options from the cross-section. This

implies that only strictly positive discount factors m'(rM) should be used for calibration.

Indeed, if m'(rM(u)) < 0 with positive probability, then estimates of the risk-neutral

density (T(rM) = (1 + ro)m (rM)p(rM) can be negative, and the hypothesis that the true

risk-neutral densities qt(rM(u)) > 0 (estimated from options cross-section) are equal to

the estimated densities ff(rM) does not make sense. However, it is not clear at this point,

which deviation measures have the property that mS(w) > 0 with probability 1.

Finally, there is an issue relevant to estimation of return distributions of assets based

on their historical returns. Historical data may contain outliers or effects of rare historical

events. After such "cleaning historical data may provide more reliable conclusions.

However, filtering historical data from historical effects is an open question.

5.5 Coherence of Mixed CVaR-Deviation

One of the flexible parameterizations of a deviation measure is mixed CVaR-deviation

of gains and losses. One of desirable properties of deviation measures is coherence.

Coherent deviation measures express risk preferences which are more appealing from the

point of view financial intuition and optimization than the deviation measures lacking

coherence. In this section, we examine coherence of the mixed CVaR-deviation of gains

and losses









The mixed-CVaR deviation of gains and losses is defined as follows.


D(X) -= 7,CVaR,(X EX) + i j6CVaR 3(-X + EX),
i j1 (5-36)

+ Y 6= 1, ai > 0,j > 0 for all i, j.
i= 1 j= 1

We will refer to deviation measure CVaR,(X EX) as CVaR deviation of gains, to

deviation measure CVaR3(-X + EX) as CVaR deviation of losses.

Risk identifier for a convex combination of deviation measures is a convex combination

of their risk identifiers. Risk identifier for CVaR deviation of losses was derived in

Rockafellar et al. (2006). Below, we derive the risk identifier for CVaR deviation of

losses and mixed-CVaR deviation of losses and examine coherence of these deviation

measures.

The following lemma will help to find risk-identifier for D(X) = CVaR,(-X + EX).

Lemma 4. Consider the deviation measure D and let D be the r fl. u. .; of D, i.e.

D(X) = D(-X). Let Q and Q be risk envelopes and Q(X) and Q(X) be risk .:l. ,,.:/i. rs

for the random variable X for deviation measures D and D, ,, "i.. /,'. l; Then


S= 2 Q, (5-37)


and

Q(X) 2- Q(-X). (5-38)

Proof:

To verify (5-37) we need to prove the dual representation D(X) = EX -

infO E[XQ] and also show that Q satisfies properties (Q1) (Q3). The dual representation

is correct since,


EX inf E[XQ] = EX inf E[X(2 Q)] = E[-X] inf E[-XQ] = D(-X) = D(X).
QEo Q








Properties (Q1) and (Q3) of Q follow immediately from properties of Q. To prove
property (Q2) we need to show that for each non-constant X there exists Q E Q so
that E[XQ] < EX. Indeed, fix a non-constant X. According to property (Q2) of the
risk envelope Q stated for the random variable -X there exists Q' E Q such that
E[-XQ'] < E[-X]. The property (Q2) will hold with Q' 2 Q', since

E[XQ'] = E[X(2 Q')] = 2EX + E[-XQ'] < 2EX + E[-X] = EX.


To prove (5-38), we will use the formula OD(X)

aD(X) = -aD(-X).

1 Q(X) = 9D(X) = -D(-X) -

Q(X) 2- Q(-X).


1 Q(X) and the fact that



- + Q(-X)


From (5-38), the risk envelope for the deviation measure D(X) = CVaRg(-X + EX)


Q= Q 2-a-1

To find the risk identifier Q(X), consider the risk identifier Q(X) for CVaR deviation of

gains D(X) = CVaRa(X EX), given by


Q(wU) = Q-1,

Q e Q(X) < 0 Q(w ) a-1,

Q(L) 0,


when X(w) <

when X() =

when X(w) >


s The risk envelope for D(X) = CVaR,(X EX) is Q = Q 0 < Q < a- 1, EQ = 1}.


VaR (X)

VaR (X)

VaR, (X).









Therefore Q E Q(-X) is equivalent to having

Q(w) = a-, when -X(w) < -VaR,(-X)

<0 < Q(w) < a-1, when -X() -VaR,(-X)

Q(w) = 0, when -X(w) > -VaR,(-X),

or

Q(w) = a-, when X(w) > VaR,(-X)

0 < Q(L) < a-1, when X(u) = VaR,(-X)

Q(w) 0, when X(w) < VaR,(-X),

and the risk identifier Q(X) is given by


Q(wj) 2- 3-1 when X(w) > VaR3(-X)

Q Q(X) 2 j- < 2- 3- < Q( ) < 2, when X(w) VaR,(-X) (5-39)

Q(w) = 2, when X(w) < VaR,(-X).

Next, we examine coherence of CVaR and mixed-CVaR deviations of losses.

Coherence of a deviation measure D is equivalent to having Q > 0 for all Q E Q,

where Q is a risk envelope for the deviation measure D. We will now show that it suffices

to check the non-negativity of all risk identifiers Q(X) for all random variables X.

Lemma 5. Let D be a deviation measure, Q be an associated risk envelope, Q(X) be the

risk .1,. ,/.,l: rfor the r.v. X. Then D is coherent if and only if

Q > 0 for all Q e Q(X) for all X. (5-40)

Proof: If D is coherent, then (5-40) holds since Q(X) E Q for any X.

To prove the converse statement, we need to show that (5-40) implies Q > 0 for all

Q E Q. Suppose this is not true, namely there exists Q E Q, such that Q(w) < 0 on some
set S C Q. Since Q is convex, there exists a subset Qg c Q with the property Q(w) < 0

on S for all Q E Qg. Consider a random variable X such that X() = 1 if w E S, and









X() = 0 otherwise. Then,


Q(X) =argminE[XQ] argminE[ls Q] argminE[ls Q] C Q, (5-41)
QEQ QEQ QEQs

which contradicts with the condition Q > 0 for all Q E Q(X), as required by (5-40). This

concludes the proof.


Risk identifiers (5-39) implies that the deviation measure Q3(X) = CVaR3(-X +

EX) is coherent if 2 0-1 > 0, which is equivalent to having f > 1/2.

Now consider the mixed-CVaR measure


Dz31...,3(X) Z- CVaR(-X + EX)
i= 1
and examine its coherence. The risk identifier for this measure given by


.Q.. ...,. (X) = Q (X),
i= 1

where Q(X) are risk identifiers for measures CVaR,(-X + EX). Assume for further

analysis that si > 302 > ... > 3,n, then VaR p(-X) < VaR p(-X) < ... < VaRp, (-X).

The graph of members of Q .....,, (X) are step functions decreasing at the breakpoints

VaR 3(-X), so that having Q E Q~,..... (X) means that

EQ =1,

Q(w) = 2, when X(w) < VaRP(-X),


O(w) c [2 Etj 1(y/j), 2- Ei 1(7y/ly)], when X(w) VaR (-X), k > 2,

Q(uw) E [2, 2 7//3], when X(w) = VaR3,(-X).
(5-42)









The measure D3 1.... ,(X) is coherent if the lowest value of members of Qap..., p (X) are

greater than zero, i.e.

a < 2.
i= 1
It is important to mention that a mixed measure D3, ..... (X) can be coherent even if

the some of its components are not. For example, combining the non-coherent measure

CVaR45%(-X+EX) and a coherent one CVaRa(-X+EX), f > 1/2, with equal weights,

we get a coherent mixed measure

1 1
,. < CVaR45%(-X +EX) + -CVaR(-X + EX),
2 2

when 3 > 9/16.

5.6 Conclusions

Discount factors corresponding to generalized CAPM relations exist and depend on

risk identifiers for master funds. The projection of these discount factors on the space of

asset p .,voffs coincides with the discount factor corresponding to the standard deviation.

It is possible to calibrate the deviation measure in the general portfolio theory from

market data if a parametrization of the deviation measure is assumed. One of candidate

parameterizations is mixed-CVaR deviation of gains and losses. The risk identifier of

CVaR and mixed-CVaR deviations of losses are derived and coherence of these deviation

measures is examined.









REFERENCES


Ait-Sahalia, Y., Duarte, J. (2003) Non-parametric option pricing under shape
restrictions. Journal of Econometrics 116, 9-47

Avouyi-Govi, S., Morin, A., and Neto, D. (2004) Optimal Asset Allocation With
Omega Function. Technical report, Benque de France.

Berkowitz, J. (2001) Testing Density Forecasts with Applications to Risk
Management. Journal of Business and Economic Statistics 19, 46574.

Bertsimas, D., and A. Lo (1998) Optimal Control of Execution Costs. Jourmal of
Financial Markets 1, 1-50.

Bertsimas, D., Kogan, L., Lo, A. (2001) Pricing and Hedging Derivative Securities in
Incomplete Markets: An e-Arbitrage Approach, Operations Research 49, 372-397.

Bertsimas, D., Popescu, I. (1999) On the Relation Between Option and Stock Prices:
A Convex Optimization Approach, Operations Research 50(2), 358-374, 2002.

Black, F., Scholes, M. (1973) The Pricing of Options and Corporate Liabilities,
Journal of Policital Ecorni. i, 81(3), 637-654.

Bliss, R. R. and Panigirtzoglou, N. (2001) Recovering Risk Aversion From Options.
Federal Reserve Bank of Chicago, Working Paper No. 2001-15. Available at SSRN:
http://ssrn.com/abstract 294941.

Bouchaud, J-P., Potters, M. (2000) Theory of Financial Risk and Derivatives
Pricing: From Statistical Physics to Risk Management, Ciai,.:,ll',: U.Uu,.:,'; : Press.

Boyle, P. (1977) Options: A Monte Carlo Approach, Journal of Financial Economics
4(4) 323-338.

Boyle, P., Broadie, M. and Glasserman, P. (1997) Monte Carlo Methods for Security
Pricing, Journal of Economic Diiiii:.. and Control 21(8/9), 1276-1321.

Broadie, M., Detemple, J. (2004) Option Pricing: Valuation Models and
Applications, Mfr..ir, i,, u.i Science 50(9), 1145-1177.

Broadie, M., Glasserman, P (2004) Stochastic Mesh Method for Pricing
High-Dimensional American Options, Journal of Computational Finance 7(4),
35-72.

Carriere, L. (1996) Valuation of the Early-Exercise Price for Options Using
Simulations and Nonparametric Regression, Insurance, Mathematics, Economics
19, 19-30.

Cochrane, J. (2001) Asset Pricing. Princeton University Press.









Coleman, T., Kim, Y., Li, Y., Patron, M, (2004) Robustly hedging variable
annuities with guarantees under jump and volatility risks, Technical Report, Cornell
University.

Dembo, R., Rosen D. (1999) The Practice of Portfolio Replication, Annals of
Operations Research 85, 267-284.

Dempster, M., Hutton, J. (1999) Pricing American Stock Options by Linear
Programming, Math. Finance 9, 229-254.

Dempster, M,. Hutton, J., Richards, D. (1998) LP Valuation of Exotic American
Options Exploiting Structure, The Journal of Computational Finance 2(1), 61-84.

Dempster, M., Thompson, G. (2001) Dynamic Portfolio Replication Using Stochastic
Programming. In Dempster, M.A.H. (ed.): Risk i,.i.i.r, i,, ,n value at risk and
17. ;;. ;./ Cambridge: Cambridge University Press, 100-128

Dennis P. (2001) Optimal Non-Arbitrage Bounds on S&P 500 Index Options and the
Volatility Smile, Journal of Futures Markets 21, 1151-1179.

Duffie, D. and H. Richardson (1991) Mean-Variance Hedging in Continuous Time,
The Annals of Applied P,.. l.al..:.; 1, 1-15.

Edirisinghe, C,. Naik, V., Uppal, R. (1993) Optimal replication of options with
transactions costs and trading restrictions, The Journal of Financial and Quantita-
tive Al.i.;-. 28 (\1 wr., 1993), 372-397.

Fedotov, S., Mikhailov, S. (2001) Option Pricing for Incomplete Markets via
Stochastic Optimization: Transaction Costs, Adaptive Control and Forecast,
International Journal of Theoretical and Applied Finance 4(1), 179-195.

Follmer, H., Schied, A. (2002) Stochastic Finance: An Introduction to Discrete
Time. Walter de Gruyter Inc.

Fllmer, H., Schweizer, M. (1989) Hedging by Sequential Regression: An Introduction
to the Mathematics of Option Ti I'.1- ASTIN Bulletin 18, 147-160.

Gilli, M., Kellezi, E., and Hysi, H. (2006) A Data-Driven Optimization Heuristic for
Downside Risk Minimization. The Journal of Risk 8(3), 1-18.

Glasserman, P. (2004) Monte-Carlo Method in Financial Engineering, Springer-
Verlag, New-York.

Gotoh, Y., Konno, H. (2002) Bounding Option Price by Semi-Definite Programming,
Mlr,.,, I., ,,: Science 48(5), 665-678.

Hansen, L. P. and Richard, S. F. (1987) The Role of Conditioning Information
in Deducing Testable Restrictions Implied by Dynamic Asset Pricing Models.
Econometrica 55, 587-614.









Harrison, J. M. and Kreps, D. M. (1979) Martingales and Arbitrage in Multiperiod
Securities Markets. Journal of Economic Theory 20, 381-408.

Jackwerth, J. C. (2000) Recovering Risk Aversion From Option Prices and Realized
Returns. The Review of Financial Studies 13(2), 433-451

Joy, C., Boyle, P., and Tan, K.S.(1996) Quasi Monte Carlo Methods in Numerical
Finance, .i.ri,, ,, .ul. Science 42, 926-936.

King, A. (2002) Duality and Martingales: A Stochastic Programming Perspective on
Contingent Claims, Mathematical P,..j,.ir ,,,,.:,. 91, 5 ;.i.'

Konishi, H., and N. Makimoto (2001) Optimal Slice of a Block Trade. Jourmal of
Risk 3(4), 33-51.

Konishi, H. (2002) Optimal Slice of a VWAP Trade. Jourmal of Financial Markets
5, 197-221.

Levy, H. (1985) Upper and Lower Bounds of Put and Call Option Value: Stochastic
Dominance Approach, Journal of Finance 40, 1197-1217.

Longstaff, F., Schwartz, E. (2001) Valuing American Options by Simulation: A
Simple Least-Squares Approach, A Review of Financial Studies 14(1), 113-147.

Luenberger, D.G. (1998) Investment Science, Oxford University Press, Oxford, New
York.

Madgavan, A. (2002) VWAP Strategies, Technical Report, Available at
http://www.itginc.com/.

Markowitz, H.M. (1959) Portfolio Selection, Efficient Diversification of Investments,
Wiley, New York.

Merton, R. (1973) Theory of Rational Options Pricing, Bell Journal of Economics
4(1), 141-184.

Mausser, H., Saunders, D., and Seco, L. (2006) Optimizing Omega. Risk M.I,,r. .,
November 2006.

Naik, V., Uppal, R. (1994) Leverage constraints and the optimal hedging of stock
and bond options, Journal of Financial and Quantitative A,.ibl;-.: 29(2), 199223.

Passow, A. (2005) Omega Portfolio Construction With Johnson Distributions. Risk
MIrj.~ ..:,,: Limited 18(4), pp. 85-90.

Perrakis, S., Ryan, P. J. (1984) Option Pricing Bounds in Discrete Time, Journal of
Finance 39, 519-525.

Ritchken, P. H. (1985) On Option Pricing Bounds, Journal of Finance 40,
1219-1233.









Rockafellar, R.T., Uryasev, S. (2000) Optimization of Conditional Value-at-Risk,
Journal of Risk 2, 2142.

Rockafellar, R.T., Uryasev, S. (2002a) Conditional Value-at-Risk for General Loss
Distributions, Journal of B.ii1..:, and Finance 26, 14431471.

Rockafellar, R.T., Uryasev, S., Zabarankin, M. (2002b) Deviation Measures in
Generalized Linear Regression. Research Report 2002-9. ISE Dept., University of
Florida.

Rockafellar, R.T., Uryasev, S., Zabarankin, M. (2005a) Generalized Deviations in
Risk Analysis, Finance and Stochastics 10(1), 51-74.

Rockafellar, R.T., Uryasev, S., Zabarankin, M. (2005b) Master Funds in Portfolio
Analysis with General Deviation Measures, Journal of Banking and Finance 30(2),
743-778.

Rockafellar, R.T., Uryasev, S., Zabarankin, M. (2006) Optimality Conditions in
Portfolio Analysis with Generalized Deviation Measures, Mathematical P,.;,.i,,I,,,:.,j
108(2-3), 515-540.

Ross, S.A. (1978) A Simple Approach to the Valuation of Risky Streams, Journal of
Business 51, 453-475.

Schweizer, M. (1991) Option Hedging for Semi-martingales, Stochastic Processes and
their Applications 37, 339-363.

Schweizer, M. (1995) Variance-Optimal Hedging in Discrete Time, Mathematics of
Operations Research 20, 1-32.

Schweizer, M. (2001) A Guided Tour Through Quadratic Hedging Approaches. In E.
Jouini, J. Cvitanic, and M. Musiela (Eds.), Option Pricing, Interest Rates and Risk
Management, 538-574. C.,,l.,.:n/ll C.i,,l.,.:ll' Ui,;, rl;i, Press.

Shadwick, W. and Keating, C. (2002) A Universal Performance Measure. Journal of
Portfolio Mi,...r. in. ,.I Spring 2002, 59-84.

Tsitsiklis, J., Van Roy, B. (2001) Regression Methods for Pricing Complex
American-Style Options, IEEE Trans. Neural Networks 12(4) (Special Issue on
computational finance), 694-703.

Tsitsiklis, J., Van Roy, B. (1999) Optimal Stopping of Markov Processes:
Hilbert space theory, approximation algorithms, and an application to pricing
high-dimensional financial derivatives, IEEE Trans. Automatic Control 44,
1840-1851.

Wu, J., Sen, S. (2000) A Stochastic Programming Model for Currency Option
Hedging, Annals of Operations Research 100, 227-249.









BIOGRAPHICAL SKETCH

Sergey Sarykalin was born in 1982, in Voronezh, Russia. In 1999, he completed his

high school education in High School #15 in Voronezh. He received his bachelor's degree

in applied mathematics and physics from Moscow Institute of Physics and Technology in

Moscow, Russia, in 2003. In August 2003, he began his doctoral studies in the Industrial

and Systems Engineering Department at the University of Florida. He finished his Ph.D.

in industrial and systems engineering in December 2007.





PAGE 1

1

PAGE 2

2

PAGE 3

3

PAGE 4

IwanttothankmyadvisorProf.StanUryasevforhisguidancesupport,andenthusiasm.Ilearnedalotfromhisdeterminationandexperience.IwanttothankmycommitteemembersProf.JasonKarseski,Prof.FaridAitSahlia,andProf.R.TyrrellRockafellarfortheirconcernandinspiration.IwanttothankmycollaboratorsVladBugeraandValeriyRyabchekno,whowerealwaysgreatpleasuretoworkwith.Iwouldliketoexpressmydeepestappreciationtomyfamilyandfriendsfortheirconstantsupport. 4

PAGE 5

page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 11 2TRACKINGVOLUMEWEIGHTEDAVERAGEPRICE ............ 13 2.1Introduction ................................... 13 2.2BackgroundandPreliminaryRemarks .................... 15 2.3GeneralDescriptionofRegressionModel ................... 18 2.3.1Mean-AbsoluteError .......................... 18 2.3.2CVaR-objective ............................. 19 2.3.3MixedObjective ............................. 20 2.4ExperimentsandAnalysis ........................... 21 2.4.1ModelDesign .............................. 21 2.4.2NearestSample ............................. 23 2.4.3DataSet ................................. 23 2.4.4EvaluationofModelPerformance ................... 23 2.5ExperimentsandResults ............................ 25 2.6Conclusions ................................... 27 3PRICINGEUROPEANOPTIONSBYNUMERICALREPLICATION ..... 32 3.1Introduction ................................... 32 3.2FrameworkandNotations ........................... 37 3.2.1PortfolioDynamicsandSquaredError ................ 37 3.2.2HedgingStrategy ............................ 38 3.3AlgorithmforPricingOptions ......................... 41 3.3.1OptimizationProblem ......................... 41 3.3.2FinancialInterpretationoftheObjective ............... 44 3.3.3Constraints ............................... 45 3.3.4TransactionCosts ............................ 45 3.4JusticationOfConstraintsOnOptionValuesAndStockPositions .... 46 3.4.1ConstraintsforPutOptions ...................... 46 3.4.2JusticationofConstraintsonOptionValues ............. 47 3.4.3JusticationofConstraintsonStockPosition ............. 55 3.5CaseStudy ................................... 58 5

PAGE 6

.................................. 59 3.5.2PricingEuropeanoptionsonS&P500Index ............. 59 3.5.3DiscussionofResults .......................... 60 3.6ConclusionsandFutureResearch ....................... 63 4METHODSOFREDUCINGMAXIMIZATIONOFOMEGAFUNCTIONTOLINEARPROGRAMMING ............................. 73 4.1Introduction ................................... 73 4.2OmegaOptimization .............................. 75 4.2.1DenitionofOmegaFunction ..................... 75 4.2.2GeneralProblem ............................ 77 4.2.3TwoReductionTheorems ....................... 78 4.3ProofsOfReductionTheoremsForOmegaOptimizationProblem ..... 81 4.4ApplicationsofReductionTheoremstoProblemswithLinearConstraints 84 4.5Example:ResourceAllocationProblem .................... 85 4.6Conclusions ................................... 88 5CALIBRATIONOFGENERALDEVIATIONMEASURESFROMMARKETDATA ......................................... 89 5.1Introduction ................................... 89 5.1.1DenitionsandNotations ........................ 89 5.1.2GeneralPortfolioTheory ........................ 90 5.1.3GeneralizedCAPMrelationsandPricingEquilibrium ........ 91 5.2IntuitionBehindGeneralizedCAPMRelations ................ 92 5.2.1TwoWaystoAccountForRisk .................... 92 5.2.2PricingFormsofGeneralizedCAPMRelations ............ 93 5.3StochasticDiscountFactorsinGeneralPortfolioTheory .......... 98 5.3.1BasicFactsfromAssetPricingTheory. ................ 98 5.3.2DerivationofDiscountFactorforGeneralizedCAPMRelations ... 102 5.3.3GeometryofDiscountFactorsforGeneralizedCAPMRelations .. 103 5.3.4StrictPositivityofDiscountFactorsCorrespondingtoDeviationMeasures ................................. 103 5.4CalibrationofDeviationMeasuresUsingMarketData ........... 106 5.4.1IdenticationofRiskPreferencesofMarketParticipants ....... 106 5.4.2Notations ................................ 109 5.4.3ImplementationIofCalibrationMethods ............... 109 5.4.4ImplementationIIofCalibrationMethods .............. 110 5.4.5DiscussionofImplementationMethods ................ 115 5.5CoherenceofMixedCVaR-Deviation ..................... 116 5.6Conclusions ................................... 121 REFERENCES ....................................... 122 BIOGRAPHICALSKETCH ................................ 126 6

PAGE 7

Table page 2-1Performanceoftrackingmodels:stockvs.stock+index,fullhistoryregression 28 2-2Performanceoftrackingmodels:stockvs.stock+index,bestsampleregression 28 2-3Performanceoftrackingmodels:mixedobjective,changingsizeofhistoryandbestsample ...................................... 29 2-4Performanceoftrackingmodels:CVaRdeviation,changingsizeofhistoryandbestsample ...................................... 29 2-5Performanceoftrackingmodels:mixedobjective ................. 29 3-1PricesofoptionsonthestockfollowingthegeometricBrownianmotion:calculatedversusBlack-Scholesprices. ............................. 68 3-2S&P500ptionsdataset. ............................... 69 3-3PricingoptionsonS&P500index:100paths ................... 70 3-4PricingoptionsonS&P500index:20paths .................... 71 3-5SummaryofcashowdistributionsforobitainedhedgingstrategiespresentedonFigures 3.6 3.6 3.6 ,and 3.6 ............................ 72 3-6Calculationtimesofthepricingalgorithm. ..................... 72 3-7Numericalvaluesofinexionpointsofthestockpositionasafunctionofthestockpriceforsomeoptions. ............................. 72 4-1Optimalallocation .................................. 88 7

PAGE 8

Figure page 2-1Percentagesofremainingvolumevs.percentagesoftotalvolume ........ 30 2-2MAD,CVaR,andmixeddeviations ........................ 30 2-3Dailyvolumedistributions .............................. 31 3-1Impliedvolatilityvs.strike:CalloptionsonS&P500indexpricedusing100samplepaths ..................................... 64 3-2Impliedvolatilityvs.strike:PutoptionsonS&P500indexpricedusing100samplepaths ..................................... 64 3-3Impliedvolatilityvs.strike:CalloptionsinBlack-Scholessettingpricedusing200samplepaths ................................... 65 3-4Impliedvolatilityvs.strike:PutoptionsinBlack-Scholessettingpricedusing200samplepaths ................................... 65 3-5Black-Scholescalloption:distributionofthetotalexternalnancingonsamplepaths .......................................... 66 3-6Black-Scholescalloption:distributionofdiscountedinows/outowsatre-balancingpoints ......................................... 66 3-7SPXcalloption:distributionofthetotalexternalnancingonsamplepaths .. 67 3-8SPXcalloption:distributionofdiscountedinows/outowsatre-balancingpoints 67 8

PAGE 9

Ourstudydevelopednovelapproachestosolvingandanalyzingchallengingproblemsofnancialengineeringincludingoptionspricing,marketforecasting,andportfoliooptimization.Wealsomakeconnectionsoftheportfoliotheorywithgeneraldeviationmeasurestoclassicalportfolioandassetpricingtheories. WeconsideraproblemfacedbytraderswhoseperformanceisevaluatedusingtheVWAPbenchmark.Ecienttradingmarketordersincludepredictingfuturevolumedistributions.SeveralforecastingalgorithmsbasedonCVaR-regressionweredevelopedforthispurpose. Next,weconsiderassumption-freealgorithmforpricingEuropeanOptionsinincompletemarkets.Anon-self-nancingoptionreplicationstrategywasmodelledonadiscretegridinthespaceoftimeandthestockprice.Thealgorithmwaspopulatedbyhistoricalsamplepathsadjustedtocurrentvolatility.Hedgingerroroverthelifetimeoftheoptionwasminimizedsubjecttoconstraintsonthehedgingstrategy.Theoutputofthealgorithmconsistsoftheoptionpriceandthehedgingstrategydenedbythegridvariables. AnotherconsideredproblemwasoptimizationoftheOmegafunction.HedgefundsoftenusetheOmegafunctiontorankportfolios.WeshowthatmaximizingOmegafunctionofaportfoliounderpositivelyhomogeneousconstraintscanbereducedtolinearprogramming. 9

PAGE 10

10

PAGE 11

Fastdevelopmentofnancialindustrymakeshighdemandsofriskmanagementtechniques.Successofnancialinstitutionsoperatinginmodernmarketsislargelyaectedbytheabilitytodealwithmultiplesourcesofuncertainty,formalizeriskpreferences,anddevelopappropriateoptimizationmodels.Recently,thesynthesisofengineeringintuitionandmathematicsledtothedevelopmentofadvancedriskmanagementtools.Thetheoryofriskanddeviationmeasureshasbeencreated,withitsapplicationstoregression,portfoliooptimization,andassetpricing;whichencouragedtheuseofnovelriskmanagementmethodsinacademiaandindustryandstimulatedalotorresearchintheareaofmodellingandformalizingriskpreferences.Ourstudymakesaconnectionbetweennancialapplicationsofthetheoryofgeneraldeviationmeasuresandclassicalassetpricingtheory.Wealsodevelopnovelapproachestosolvingandanalyzingchallengingproblemsofnancialengineeringincludingoptionspricing,marketforecasting,andportfoliooptimization. Chapter2considersabrokerwhoissupposedtotradeaspeciednumberofsharesovercertaintimeinterval(marketorder).PerformanceofthebrokerisevaluatedbyVolumeWeightedAveragePrice(VWAP),whichrequirestradingtheorderaccordingtothemarketvolumedistributionduringthetradingperiod.Acommonapproachtothistaskistotradetheorderfollowingtheaveragehistoricalvolumedistribution.Weintroduceadynamictradingalgorithmbasedonforecastingmarketvolumedistributionusingtechniquesofgeneralizedlinearregression. Chapter3presentsanalgorithmforpricingEuropeanOptionsinincompletemarkets.Thedevelopedalgorithm(a)isfreefromassumptionsonthestockprocess;(b)achieves0.5%-3%pricingerrorforEuropeanin-andat-the-moneyoptionsonS&P500Index;(c)closelymatchesthemarketvolatilitysmile;(d)isabletopriceoptionsusing20-50samplepaths.Weusereplicationideatondoptionprice,howeverweallowthehedgingstrategy 11

PAGE 12

Chapter4provestworeductiontheoremsfortheOmegafunctionmaximizationproblem.Omegafunctionisacommoncriterionforrankingportfolios.Itisequaltotheratioofexpectedoverperformanceofaportfoliowithrespecttoabenchmark(hurdlerate)toexpectedunderperformanceofaportfoliowithrespecttothesamebenchmark.TheOmegafunctionisanon-linearfunctionofaportfolioreturn;however,itispositivelyhomogeneouswithrespecttoinstrumentexposuresinaportfolio.ThispropertyallowstransformationoftheOmegamaximizationproblemwithpositivelyhomogeneousconstraintsintoalinearprogrammingprobleminthecasewhentheOmegafunctionisgreaterthanoneatoptimality. Chapter5looksattheportfoliotheorywithgeneraldeviationmeasuresfromtheperspectiveoftheclassicalassetpricingtheory.Inparticular,weanalyzethegeneralizedCAPMrelations,whichcomeoutasanecessaryandsucientconditionsforoptimalityinthegeneralportfoliotheory.WederivepricingformsofthegeneralizedCAPMrelationsandshowhowthestochasticdiscountfactoremergesinthegeneralizedportfoliotheory.Wedevelopmethodsofcalibratingdeviationmeasuresfrommarketdataanddiscussapplicabilityofthesemethodstoestimationofriskpreferencesofmarketparticipants. 12

PAGE 13

ThereareseveraltypesofbenchmarkssimilartoVWAP.VWAP,asitisdenedabove,isreasonableforevaluationofrelativelysmallordersofliquidstocks.VWAPexcludingowntransactionsisappropriatewhenthetotalvolumeoftransactionsconstitutesasignicantportionofthemarket'sdailyvolume.Forhighlyvolatilestocks,value-weightedaveragepriceisalsoused,wherepricesoftransactionsareweightedbydollarvaluesofthistransactions.VWAPbenchmarksarewidespreadmostlyoutsideUSA,forexample,inJapan. ThepurposeoftheVWAPtradingistoobtainthevolume-weightedpriceoftransactionsasclosetothemarketVWAPaspossible.AninvestormayactdierentlywhenseekingforVWAPexecutionofhisorder.HecanmakeacontractwithabrokerwhoguaranteessellingorbuyingordersatthedailyWVAP.SincethebrokerassumesalltheriskoffailingtoachievetheaveragepricebetterthanVWAPandisusuallyriskaverse,commissionsarequitelarge. ThischapterisbasedonjointworkwithVladimirBugeraandStanUryasev. 13

PAGE 14

Aninvestorwithdirectaccesstothemarketmaytradehisorderdirectly.ButsinceVWAPevaluationmotivatestodistributetheorderoverthetradingperiodandtradebysmallportions,thisalternativeisnotpreferableduetointensityoftradingandthepresenceoftransactioncosts. ThemostrecentapproachtoVWAPtradingisparticipatinginVWAPautomatedtrading,whereatradingperiodisbrokenupintosmallintervalsandtheorderisdistributedascloselyaspossibletothemarket'sdailyvolumedistribution,thatistradedwiththeminimalmarketimpact.Thisstrategyprovidesagoodapproximationtomarket'sVWAP,althoughitgenerallyfailstoreachthebenchmark.Moreintelligentsystemsperformcarefulprojectionsofthemarketvolumedistributionandexpectedpricemovementsandusethisinformationintrading.AmoredetailedsurveyofVWAPtradingcanbefoundinMadgavan(2002). AlthoughVWAP-benchmarkhasgainedpopularity,veryfewstudiesconcerningVWAPstrategiesareavailable.Severalstudies,BertsimasandLo(1998),KonishiandMakimoto(2001)havebeendoneaboutblocktradingwhereoptimalsplittingoftheorderinordertooptimizetheexpectedexecutioncostisconsidered.Inthesetupofblocktrading,onlypricesareuncertain,whereasthepurposeofVWAPtradingistoachieveaclosematchofthemarketVWAP,whichimpliesdealingwithstochasticvolumesaswell.Konishi(2002)developsastaticVWAPtradingstrategythatminimizestheexpectedexecutionerrorwithrespecttothemarketrealizationofVWAP.Astaticstrategyisdeterminedforthewholetradingperiodanddoesnotchangeasnewinformationarrives. 14

PAGE 15

InthischapterwedevelopdynamicVWAPstrategies.Weconsiderliquidstocksandsmallorders,thatmakenegligibleimpactonpricesandvolumesofthemarket.Theforecastofvolumedistributionisthetarget;thestrategyconsistsintradingtheorderproportionallytoprojectedmarketdailyvolumedistribution.Wesplitatradingdayintosmallintervalsandestimatethemarketvolumeconsecutivelyforeachintervalusinglinearregressiontechniques. IfatradingdayissplitintoNequalintervalsf(tn1;tn]jn=1;::;Ng,tn=(n=N)T,whereTisthelengthoftheday,thenthecorrespondingexpressionforthedailyVWAPisgivenby where isthevolumetradedduringtimeperiod(tn1;tn], canbethoughtofasanaveragemarketpriceduringthenthinterval. 15

PAGE 16

Valuesofxnareassumedtobenonnegative(i.e.thetraderisnotallowedtobuystocks). Weconstructthedynamictradingstrategybyforecastingthevolumesofstocktradedinthemarketduringeachintervalofatradingday.Weassumethatduringasmallinterval(about5min)wecanperformtransactionsattheaveragemarketpriceduringthisinterval.Then,from( 2{5 )itfollowsthatapossiblewaytomeetthemarketVWAPistotradetheorderproportionallytothemarketvolumeduringeachinterval,yieldingthesamedailydistributionofthetradedvolumeasthemarket'sone.Foreachintervalofadaywemakeaforecastofthemarketvolumethatwillbetradedduringthisintervalandthentradeaccordingtothisforecast.Attheendofthedayweobtaintheforecastofthefulldailyvolumedistribution;theorderistradedaccordingtothisdistribution. Thewayofdynamiccomputingofthedistributionshouldbediscussedrst.Directestimationsofproportionsofthemarketvolumev1;:::;vNdoesnotguaranteethatthe 16

PAGE 17

2-1 demonstratesthetworepresentationsofthevolumedistribution.Note,thatwNisalwaysequalto1.Thereisaone-to-onecorrespondencebetweenrepresentations(v1;:::;vN)and(w1;:::;wN);thetransitionsbetweenthemaregivenbyformulas and Thelastequationsfollowfromthefactthatwi(1wi1):::(1wim)=Vi 17

PAGE 18

ConsiderthegeneralregressionsettingwherearandomvariableYisapproximatedbyalinearcombination ofindicatorvariablesX1;:::;Xn.Inourstudythevariablesaremodelledbyasetofscenarios Forascenariostheapproximationerroris Weconsiderourregressionmodelasanoptimizationproblemofminimizingtheaggregatedapproximationerror.Belowwedescribepenaltyfunctionsweuseastheobjective. 2{8 ) 18

PAGE 19

2{9 ),theoptimizationproblemis minDMAD=1 2{10 ),howeverourintentionpenalizethelargest(bytheabsolutevalue)outcomesoftheerror.TogiveamoreformaldenitionoftheCVaR-objectiveandshowtherelevanceofusingitinregressionproblems,weeneedtorefertothenewlydevelopedtheoryofdeviationmeasuresandgeneralizedlinearregression,seeRockafellaretal.(2002b). CVaR-objectiveconsistsoftwoCVaR-deviations(Rockafellaretal.(2005a))andpenalizesthe-highestandthe-lowestoutcomesoftheestimationerror( 2{10 )foraspeciedcondencelevel(isusuallyexpressedinpercentages).WewilluseacombinationofCVaR-deviationsasanobjective: (2{13) =CVaR()+CVaR(): 2{8 )andtheminimization( 2{13 )determinestheoptimalvaluesofvariablesc1;:::;cnonly.Theoptimalvalueofthetermdcanbefoundfromdierentconsiderations;weusetheconditionthattheestimator( 2{8 )isnon-biased. 19

PAGE 20

minc;dCVaRYY+CVaRYYs:t:EY=E[Y]Y=Pni=1ciXi+d:(2{14) Since 1E[X];(2{15) optimizationprogram( 2{14 )becomes: minc;dCVaRYY+(1)CVaR1YYs:t:EY=E[Y]Y=Pni=1ciXi+d:(2{16) ThetermE[YY]isnotincludedintotheobjectivefunctionsinceE[YY]=0duetotherstconstraint. Forthecaseofscenarios( 2{9 )theoptimizationproblem( 2{16 )canbereducedtothefollowinglinearprogrammingproblem. min++(1)s:t:PSs=1[Pni=1ciXsi+d]=PSs=1Ys+1 (1)SPSs=1zs1zsYs(Pni=1ciXsi+d)zs1Ys(Pni=1ciXsi+d)1Variables:ci;d2Rfori=1;:::;n;;12R;zs;zs10fors=1;:::;S.(2{17) 2{14 )byDCVaR,thentheproblemwiththemixedobjectiveis 20

PAGE 21

minDMAD+IXi=1iDiCVaRsubjecttoconstraintsin( 2{17 ); wherei2[0;1];i=1;:::;I,+PIi=1i=1. Inourexperiments,weusedconvexcombinationsoftwoCVaR-objectives,onewiththecondencelevel50%: minD50%CVaR+(1)DCVaRsubjecttoconstraintsin( 2{17 ); andofthemean-absoluteerrorfunctionandtheCVaR-deviation: minDMAD+(1)DCVaRsubjecttoconstraintsin( 2{17 )withouttherstone; wherethebalancecoecient2[0;1].Forcomparison,dierenttypesofdeviationsarepresentedonFigure 2-2 2.4.1ModelDesign Weconsiderthefollowingregressionmodel 21

PAGE 22

2{21 ). Valuesofthecorrespondingparameterspj(kl);sandfractionsoftheremainingvolumewks,s=1;:::;S,i=1;:::;L,j=1;:::;P,arecollectedfromtheprecedingSdaysofthehistory.Thus,wehavethesetofscenarios Denotethelinearcombination as^wks,thecollectionofikjas~. Inourstudyweconsiderthefollowingoptimizationproblems: min~Ejwk^wkj;(2{24) min~CVaR(wk^wk)+CVaR(^wkwk)s:t:E[wk]=E[^wk](2{25) min~Ejwk^wkj+(1)CVaR(wk^wk)+CVaR(^wkwk)(2{26) min~CVaR50%(wk^wk)+CVaR50%(^wkwk)++(1)CVaR(wk^wk)+CVaR(^wkwk)s:t:E[wk]=E[^wk];(2{27) 22

PAGE 23

Bysolvingtheseproblems,theoptimalvalueof~isobtained.Theforecastofw0kisthenmadebytheexpression( 2{21 ). AftercalculatingdistancestoallSscenarios,wechooseSbestclosestscenarioscorrespondingtolowestvaluesofDiin( 2{28 ).Bydoingso,weeliminate"outliers"withunusual,withrespecttothecurrentday,behaviorofthemarketwhichfavorstheaccuracyofforecasting. 23

PAGE 24

theforecastedvolumesare thentheestimationerroris Wealsocalculatedanothererror Asabenchmarkmeasuringtherelativeaccuracyofthemodel"averagedailyvolumes"(ADV)strategywasused.ThisverysimplestrategyprovidesagoodapproximationtoVWAP.Supposeasetofhistoricalvolumesofthemarket: Denote Vn=SXs=1Vns;Vtotal=NXn=1Vn:(2{34) Thentheaveragevolumedistributionis (v1;:::;vN);vn=Vn AnexampleofaveragevolumedistributionversustheactualvolumeevolutionispresentedinFigure 2-3 .Itcanbeseenthatdailyvolumeexhibitsthe"U-shape"andthattheaveragedistributionprovidesagoodapproximationtothedailyvolumeevolution. 24

PAGE 25

2{32 ).Therelativegaininaccuracyoftheregressionalgorithmwasjudgedbythevalueof GMAD=MADADVMAD MADADV100%:(2{36) Relativegaininstandarddeviationis GSD=SDADVSD SDADV100%:(2{37) Withrespecttotheparameters( 2{21 )wetookfromeachinterval,theexperimentsweredividedintotwogroups. Intherstgroup,theexperimentswerebasedonusingonlypricesandvolumesofthestockasusefulinformation.Namely,fromeachintervalweusedthefollowinginformation: lnVandlnPclose whereVismarketvolumeduringtheinterval,PopenandPcloseareopenandclosepricesoftheinterval.R=Pclose=Popenis,therefore,thereturnduringtheinterval.Logarithmswereusedtotakeintoaccountthepossibilitythattheratiosofreturnsandvolumes,asidefromreturnsandvolumesthemselves,containsomeinformationaboutthefuturevolume.Alinearcombinationoflogarithmsofparameterscanberepresentedasalinearcombinationoftheparametersandtheirratios. 25

PAGE 26

lnV;lnPclose Theideaofusingindexinformationcomesfromthefactthatevolutionsofindexandstockarecorrelatedandthattheratiosofreturnsandpricesofstockandindexmayalsocontainusefulinformation. Tables 2-1 2-2 showtheresultsforthemean-absolutedeviationusedasanobjectiveanddierentvaluesofL;SandSbest.ThesetablesshowthatincludingINDEXdatadoesnotimprovetheaccuracyofprediction.Also,asonecannotice,thereisabalancebetweenthenumberoftermsNterm=LPinthelinearcombination( 2{21 )andthenumberofscenarios(Sbest)usedintheregressionmodel.AsNtermincreases,themodelbecomesmoreexibleandmorescenariosareneededtoachievethesamelevelofaccuracy.Forexample,thebesttwomodelsthatusestockdata(P=2),havevaluesofSbestandNtermequal450and4,200and2,respectively.Also,whentheindexdataisused,thenumberofparametersPdoubles,andthenumberofscenariosinthebestmodelsincreasesto700800forthesameregressionlengthL. InthecaseofCVaR-objectiveandmixedobjective(Tables 2-3 2-4 ),dierentvaluesofL;SandSbestyieldedasimilarorderofsuperiorityasinthecaseofthemean-absolutedeviation. Twomorefactscanbeseenfromtheresults.First,thatthemostsuccessfulmodelsuseinformationonlyfromthelastoneortwointervals,whichmeansthattheinformationaboutthefuturevolumeisconcentratedinthepastfewminutes.Second,theideaofchoosingtheclosestscenariosfromtheprecedinghistorydoeswork,especiallywhenasmallportionofnon-similardays(50or100outof500or800potentialscenarios)isexcluded.Thisagreeswiththeobservationthatmostofthedaysare"regular"enoughtobeusedfortheestimationofthefuture. 26

PAGE 27

2-5 wechangedtheformofthemixedobjective,thatis,dieredandin( 2{17 ).WefoundthatthebestmodelshaveallweightputontheCVaRobjectiveandforaxedbalancethemodelswithsmallvaluesofaresuperior. ThemostaccuratemodelturnedouttobetheonewithCVaR-objectivehavingtherelativegain4%. 27

PAGE 28

Performanceoftrackingmodels:stockvs.stock+index,fullhistoryregression SSbestLMAD,%SD,%GMAD,%GSD,% STOCK500500234.041.13.53.2500500134.141.43.42.5500500334.241.33.02.8800800234.342.12.70.9800800134.442.52.6-0.1800800334.442.22.40.5STOCK+INDEX500500134.141.03.23.5500500234.241.03.13.5800800134.241.23.13.1800800234.240.83.13.9800800334.340.72.74.0500500334.441.22.43.0 Table2-2. Performanceoftrackingmodels:stockvs.stock+index,bestsampleregression SSbestLMAD,%SD,%GMAD,%GSD,% STOCK500450234.040.83.74.0500200134.039.43.67.1500450134.041.03.63.3500400234.040.53.64.6500400134.041.03.53.3800500234.241.03.13.4500450334.241.13.03.1800700234.342.12.90.9STOCK+INDEX500450134.140.63.34.4500450234.140.13.25.4800750134.240.83.23.8800750234.240.83.24.0800700134.240.93.13.7800700234.240.53.14.6500400134.341.03.03.5500400234.341.12.83.3 28

PAGE 29

Performanceoftrackingmodels:mixedobjective,changingsizeofhistoryandbestsample SSbestL,%,%MAD,%SD,%GMAD,%GSD,% STOCK5004502305034.040.83.74.05002001305034.039.43.67.25004501305034.041.03.63.45004002305034.040.53.64.65004001305034.041.03.53.35005002305034.041.13.53.25005001305034.141.43.42.58005002305034.241.03.03.4 Table2-4. Performanceoftrackingmodels:CVaRdeviation,changingsizeofhistoryandbestsample SSbestL,%,%MAD,%SD,%GMAD,%GSD,% STOCK50040023010033.940.74.04.150020013010033.939.73.96.650020023010033.939.43.97.150045023010033.940.83.94.050048023010033.940.83.83.050040033010034.040.43.74.950040013010034.041.13.73.350045013010034.041.03.73.4 Table2-5. Performanceoftrackingmodels:mixedobjective SSbestL,%,%MAD,%SD,%GMAD,%GSD,% STOCK50045022010033.940.74.04.250045023010033.939.63.94.450045021010033.939.83.94.0500450253033.940.63.94.55004502103033.940.73.94.15004502203034.040.73.84.15004502510034.041.63.84.45004502303034.041.73.84.2 29

PAGE 30

Percentagesofremainingvolumevs.percentagesoftotalvolume MAD,CVaR,andmixeddeviations 30

PAGE 31

Dailyvolumedistributions 31

PAGE 32

Below,werefertooptionpricingmethodsdirectlyrelatedtoouralgorithm.AlthoughthispaperconsidersEuropeanoptions,somerelatedpapersconsiderAmericanoptions. Replicationoftheoptionpricebyaportfolioofsimplerassets,usuallyoftheunderlyingstockandarisk-freebond,canincorporatevariousmarketfrictions,suchastransactioncostsandtradingrestrictions.Forincompletemarkets,replication-basedmodelsarereducedtolinear,quadratic,orstochasticprogrammingproblems,see,forinstance,BouchaudandPotters(2000),Bertsimasetal.(2001),DemboandRosen(1999),Colemanetal.(2004),NaikandUppal(1994),Dennis(2001),DempsterandThompson ThischapterisbasedonthepaperRyabchenko,V.,Sarykalin,S.,andUryasev,S.(2004)PricingEuropeanOptionsbyNumericalReplication:QuadraticProgrammingwithConstraints.Asia-PacicFinancialMarkets,11(3),301-333. 32

PAGE 33

Analyticalapproachestominimizationofquadraticriskareusedtocalculateanoptionpriceinanincompletemarket,seeDueandRichardson(1991),FollmerandSchied(2002),FollmerandSchweizer(1989),Schweizer(1991,1995,2001). Anothergroupofmethods,whicharebasedonasignicantlydierentprinciple,incorporatesknownpropertiesoftheshapeoftheoptionpriceintothestatisticalanalysisofmarketdata.Ait-SahaliaandDuarte(2003)incorporatemonotonicandconvexpropertiesofEuropeanoptionpricewithrespecttothestrikepriceintoapolynomialregressionofoptionprices.Inouralgorithmwelimitthesetoffeasiblehedgingstrategies,imposingconstraintsonthehedgingportfoliovalueandthestockposition.ThepropertiesoftheoptionpriceandthestockpositionandboundsontheoptionpricehasbeenstudiedboththeoreticallyandempiricallybyMerton(1973),PerrakisandRyan(1984),Ritchken(1985),BertsimasandPopescu(1999),GotohandKonno(2002),andLevi(85).Inthispaper,wemodelstockandbondpositionsonatwo-dimensionalgridandimposeconstraintsonthegridvariables.Theseconstraintsfollowundersomegeneralassumptionsfromnon-arbitrageconsiderations.SomeoftheseconstraintsaretakenfromMerton(1973). Monte-CarlomethodsforpricingoptionsarepioneeredbyBoyle(1977).Theyarewidelyusedinoptionspricing:Joyetal.(1996),BroadieandGlasserman(2004),LongstaandSchwartz(2001),Carriere(1996),TsitsiklisandVanRoy(2001).ForasurveyofliteratureinthisareaseeBoyle(1997)andGlasserman(2004).Regression-basedapproachesintheframeworkofMonte-CarlosimulationwereconsideredforpricingAmericanoptionsbyCarriere(1996),LongstaandSchwartz(2001),TsitsiklisandVanRoy(1999,2001).BroadieandGlasserman(2004)proposedstochasticmeshmethodwhichcombinedmodellingonadiscretemeshwithMonte-Carlosimulation.Glasserman(2004),showedthatregression-basedapproachesarespecialcasesofthestochasticmeshmethod. 33

PAGE 34

Thepricingalgorithmdescribedinthispapercombinesthefeaturesoftheaboveapproachesinthefollowingway.Weconstructahedgingportfolioconsistingoftheunderlyingstockandarisk-freebondanduseitsvalueasanapproximationtotheoptionprice.Weaimedatmakingthehedgingstrategyclosetoreal-lifetrading.Theactualtradingoccursatdiscretetimesandisnotself-nancingatre-balancingpoints.Theshortageofmoneyshouldbecoveredatanydiscretepoint.Largeshortagesareundesirableatanytimemoment,evenifself-nancingispresent.Weconsidernon-self-nancinghedgingstrategies.Externalnancingoftheportfolioorwithdrawalisallowedatanyre-balancingpoint.Weuseasetofsamplepathstomodeltheunderlyingstockbehavior.Thepositioninthestockandtheamountofmoneyinvestedinthebond(hedgingvariables)aremodelledonnodesofadiscretegridintimeandthestockprice.Twomatricesdeningstockandbondpositionsongridnodescompletelydeterminethehedgingportfolioonanypricepathoftheunderlyingstock.Also,theydetermineamountsofmoneyaddedto/takenfromtheportfolioatre-balancingpoints.Thesumofsquaresofsuchadditions/subtractionsofmoneyonapathisreferredtoasthesquarederroronapath. Thepricingproblemisreducedtoquadraticminimizationwithconstraints.Theobjectiveistheaveragedquadraticerroroverallsamplepaths;thefreevariablesarestockandbondpositionsdenedineverynodeofthegrid.Theconstraints,limitingthefeasiblesetofhedgingstrategies,restricttheportfoliovaluesestimatingtheoptionpriceandstockpositions.Werequiredthattheaverageoftotalexternalnancingoverallpathsequalstozero.Thismakesthestrategy"self-nancingonaverage".Weincorporatedmonotonic, 34

PAGE 35

Weperformedtwonumericaltestsofthealgorithm.First,wepricedoptionsonthestockfollowingthegeometricBrownianmotion.StockpriceismodelledbyMonte-Carlosample-paths.CalculatedoptionpricesarecomparedwiththeknownpricesgivenbytheBlack-Scholesformula.Second,wepricedoptionsonS&P500Indexandcomparedtheresultswithactualmarketprices.Bothnumericaltestsdemonstratedreasonableaccuracyofthepricingalgorithmwitharelativelysmallnumberofsample-paths(consideredcasesinclude100and20sample-paths).Wecalculatedoptionpricesbothindollarsandintheimpliedvolatilityformat.TheimpliedvolatilitymatchesreasonablywelltheconstantvolatilityforoptionsintheBlack-Scholessetting.TheimpliedvolatilityforS&P500indexoptions(pricedwith100sample-paths)trackstheactualmarketvolatilitysmile. Theadvantageofusingthesquarederrorasanobjectivecanbeseenfromthepracticalperspective.Althoughweallowsomeexternalnancingoftheportfolioalongthepath,theminimizationofthesquarederrorensuresthatlargeshortagesofmoneywillnotoccuratanypointoftimeiftheobtainedhedgingstrategyispracticallyimplemented. Anotheradvantageofusingthesquarederroristhatthealgorithmproducesahedgingstrategysuchthatthesumofmoneyaddedto/takenfromthehedgingportfolioonanypathisclosetozero.Also,theobtainedhedgingstrategyrequireszeroaverageexternalnancingoverallpaths.Thisjustiesconsideringtheinitialvalueofthehedgingportfolioasapriceofanoption.Weusethenotionof"apriceofanoptioninthepracticalsetting"whichisthepriceatraderagreestobuy/selltheoption.Intheexample 35

PAGE 36

Weassumeanincompletemarketinthispaper.Weusetheportfoliooftwoinstruments-theunderlyingstockandabond-toapproximatetheoptionpriceandconsidermanysamplepathstomodelthestockpriceprocess.Asaconsequence,thevalueofthehedgingportfoliomaynotbeequaltotheoptionpayoatexpirationonsomesamplepaths.Also,thealgorithmisdistribution-free,whichmakesitapplicabletoawiderangeofunderlyingstockprocesses.Therefore,thealgorithmcanbeusedintheframeworkofanincompletemarket. Usefulnessofouralgorithmshouldbeviewedfromtheperspectiveofpracticaloptionspricing.Commonlyusedmethodsofoptionspricingaretime-continuousmodelsassumingspecictypeoftheunderlyingstockprocess.Iftheprocessisknown,thesemethodsprovideaccuratepricing.Ifthestockprocesscannotbeclearlyidentied,thechoiceofthestockprocessandcalibrationoftheprocesstotmarketdatamayentailsignicantmodellingerror.Ouralgorithmissuperiorinthiscase.Itisdistribution-freeandisbasedonrealisticassumptions,suchasdiscretetradingandnon-self-nancinghedgingstrategy. Anotheradvantageofouralgorithmislowback-testingerrors.Time-continuousmodelsdonotaccountforerrorsofimplementationonhistoricalpaths.Theobjectiveinouralgorithmistominimizetheback-testingerrorsonhistoricalpaths.Therefore,thealgorithmhasaveryattractiveback-testingperformance.Thisfeatureisnotsharedbyanyoftime-continuousmodels. 36

PAGE 37

3.2.1PortfolioDynamicsandSquaredError Thepriceoftheoptionattimetjisapproximatedbythepricecjofahedgingportfolioconsistingoftheunderlyingstockandarisk-freebond.Thehedgingportfolioisrebalancedattimestj,j=1;:::;N1.Supposethatatthetimetj1thehedgingportfolioconsistsofuj1sharesofthestockandvk1dollarsinvestedinthebond tobenon-zero.Thevalueajistheexcess/shortfallofthemoneyinthehedgingportfolioduringtheinterval[tj1;tj].Inotherwords,ajistheamountofmoneyaddedto(ifaj0)orsubtractedfrom(ifaj<0)theportfolioduringtheinterval[tj1;tj].Thus,theinow/outowofmoneyto/fromthehedgingportfolioisallowed. 37

PAGE 38

Thenon-self-nancingportfoliodynamicsisgivenby wheretheportfoliovalueattimetjiscj=ujSj+vj;j=0;:::;N. Thedegreetowhichaportfoliodynamicsdiersfromaself-nancingoneisanimportantcharacteristic,essentialtoourapproach.Inthispaper,wedeneasquarederroronapath, tomeasurethedegreeof\non-self-nancity".Thereasonsforchoosingthisparticularmeasurewillbedescribedlateron. 38

PAGE 39

[Ukj]=266666664U10U11:::U1NU20U21:::U2N............UK0UK1:::UKN377777775;[Vkj]=266666664V10V11:::V1NV20V21:::V2N............VK0VK1:::VKN377777775(3{4) arereferredtoasahedgingstrategy.Thesematricesdeneportfoliomanagementdecisionsonthediscretesetofthegridnodes.Inordertosetthosedecisionsonanypath,notnecessarilygoingthroughgridpoints,approximationrulesaredened. Wemodelthestockpricedynamicsbyasetofsamplepaths whereS0istheinitialprice.Letvariablesupjandvpjdenethecompositionofthehedgingportfolioonpathpattimetj,wherep=1;:::;P,j=0;:::;N.ThesevariablesareapproximatedbythegridvariablesUkjandVkjasfollows.SupposethatfS0;Sp1;:::;SpNgisarealizationofthestockprice,whereSpjdenotesthepriceofthestockattimetjonpathp,j=0;:::;N,p=1;:::;P.Letupjandvpjdenotetheamountsofthestockandthebond,respectively,heldinthehedgingportfolioattimetjonpathp.VariablesupjandvpjarelinearlyapproximatedbythegridvariablesUkjandVkjasfollows 39

PAGE 40

3{1 ),wedenetheexcess/shortageofmoneyinthehedgingportfolioonpathpattimetjbyapj=upj+1Spj+1+vpj+1(upjSpj+1+(1+r)vpj): WedenetheaveragesquarederrorEonthesetofpaths( 3{5 )asanaverageofsquarederrorsEpoverallsamplepaths( 3{5 ) E=1 Thematrices[Ukj]and[Vkj]andtheapproximationrule( 3{6 )specifythecompositionofthehedgingportfolioasafunctionoftimeandthestockprice.Foranygivenstockpricepathonecanndthecorrespondingportfoliomanagementdecisionsf(uj;vj)jj=0;:::;N1g,thevalueoftheportfoliocj=Sjuj+vjatanytimetj,j=0;:::;N,andtheassociatedsquarederror. Thevalueofanoptioninquestionisassumedtobeequaltotheinitialvalueofthehedgingportfolio.Firstcolumnsofmatrices[Ukj]and[Vkj],namelythevariablesUk0andVk0,k=1;:::;K;determinetheinitialvalueoftheportfolio.Ifoneoftheinitialgridnodes,forexamplenode(0;~k);correspondstothestockpriceS0,thenthepriceoftheoptionisgivenbyU~k0S0+V~k0:Iftheinitialpoint(t=0;S=S0)ofthestockprocessfallsbetweentheinitialgridnodes(0;k),k=1;:::;K,thenapproximationformula( 3{6 )withj=0andSp0=S0isusedtondtheinitialcomposition(u0;v0)oftheportfolio.Then,thepriceoftheoptionisfoundasu0S0+v0. 40

PAGE 41

minE=1 3{6 ),constraints( 3{10 )-( 3{18 )(denedbelow)forcalloptions,orconstraints( 3{19 )-( 3{27 )(denedbelow)forputoptions,freevariables:Ukj;Vkj;j=0;:::;N;k=1;:::;K: 3{9 )istheaveragesquarederroronthesetofpaths( 3{5 ).Therstconstraintrequiresthattheaveragevalueoftotalexternalnancingoverallpathsequalstozero.Thesecondconstraintequatesthevalueoftheportfolioandtheoptionpayoatexpiration.FreevariablesinthisproblemarethegridvariablesUkjandVkj;thepathvariablesupjandvpjintheobjectiveareexpressedintermsofthegridvariablesusingapproximation( 3{6 ).Thetotalnumberoffreevariablesintheproblemisdeterminedbythesizeofthegridandisindependentofthenumberofsample-paths.Aftersolvingtheoptimizationproblem,theoptionvalueattimetjforthestockpriceSjisdenedbyujSj+vj,whereujandvjarefoundfrommatrices[Ukj]and[Vkj],respectively,using 41

PAGE 42

3{6 ).Thepriceoftheoptionistheinitialvalueofthehedgingportfolio,calculatedasu0S0+v0. Thefollowingconstraints( 3{10 )-( 3{18 )forcalloptionsor( 3{19 )-( 3{27 )forputoptionsimposerestrictionsontheshapeoftheoptionvaluefunctionandonthepositioninthestock.Theserestrictionsreducethefeasiblesetofhedgingstrategies.Subsection3.3discussesthebenetsofinclusionoftheseconstraintsintheoptimizationproblem. Below,weconsidertheconstraintsforEuropeancalloptions.Theconstraintsforputoptionsaregiveninthenextsection,togetherwithproofsoftheconstraints.Mostoftheconstraintsarejustiedinaquitegeneralsetting.Weassumenon-arbitrageandmake5additionalassumptions.Proofsoftwoconstraintsonthestockposition(horizontalmonotinicityandconvexity)inthegeneralsettingwillbeaddressedinsubsequentpapers.InthispaperwevalidatetheseinequalitiesintheBlack-Scholescase. ThenotationCkjstandsfortheoptionvalueinthenode(j;k)ofthegrid,Ckj=Ukj~Skj+Vkj: 3{10 )coincideswiththeimmediateexercisevalueofanAmericanoptionhavingthecurrentstockprice~SkjandthestrikepriceXer(Ttj):

PAGE 43

Thisconstraintsboundsensitivityofanoptionpricetochangesinthestockprice. 0. Verticalmonotonicity.Foranyxedtime,thepriceofanoptionisanincreasingfunctionofthestockprice. ~Skj 0. Horizontalmonotonicity.Thepriceofanoptionisadecreasingfunctionoftime. 0Ukj1;j=0;:::;N;k=1;:::;K:(3{15) 43

PAGE 44

(1k+1j)Uk+2j+k+1jUkjUk+1j;ifk>^k;(1k1j)Uk2j+k1jUkjUk1j;ifk^k;whereljissuchthat~Slj=lj~Sl1j+(1lj)~Sl+1j;l=(k+1);(k1): 3.3.4 ). Theexpectedhedgingerrorisanestimateof\non-self-nancity"ofthehedgingstrategy.Thepricingalgorithmseeksastrategyascloseaspossibletoaself-nancingone,satisfyingtheimposedconstraints.Ontheotherhand,fromatrader'sviewpoint,theshortageofmoneyatanyportfoliore-balancingpointcausestheriskassociatedwiththehedgingstrategy.Theaveragesquarederrorcanbeviewedasanestimatorofthisriskonthesetofpathsconsideredintheproblem. Thereareotherwaystomeasuretheriskassociatedwithahedgingstrategy.Forexample,Bertsimasetal.(2001)considersaself-nancingdynamicsofahedgingportfolioandminimizesthesquaredreplicationerroratexpiration.Inthecontextofourframework,dierentestimatorsofriskcanbeusedasobjectivefunctionsintheoptimizationproblem( 3{9 )and,therefore,producedierentresults.However,consideringotherobjectivesisbeyondthescopeofthispaper. 44

PAGE 45

3{10 )-( 3{14 )forcalloptionsand( 3{19 )-( 3{23 )forputoptionsfollowunderquitegeneralassumptionsfromthenon-arbitrageconsiderations.Thetypeoftheunderlyingstockpriceprocessplaysnoroleintheapproach:thesetofsamplepaths( 3{5 )speciesthebehavioroftheunderlyingstock.Forthisreason,theapproachisdistribution-freeandcanbeappliedtopricinganyEuropeanoptionindependentlyofthepropertiesoftheunderlyingstockpriceprocess.Also,asshowninsection5presentingnumericalresults,theinclusionofconstraintstoproblem( 3{9 )makesthealgorithmquiterobusttothesizeofinputdata. Thegridstructureisconvenientforimposingtheconstraints,sincetheycanbestatedaslinearinequalitiesonthegridvariablesUkjandVkj.Animportantpropertyofthealgorithmisthatthenumberofthevariablesinproblem( 3{9 )isdeterminedbythesizeofthegridandisindependentofthenumberofsamplepaths. 3{16 )-( 3{18 )requiringmonotonicityandconcavityofthestockpositionwithrespecttothestockpriceandmonotonicityofthestockpositionwithrespecttotime(constraints( 3{25 )-( 3{27 )forputoptionsarepresentedinthenextsection).Thegoalistolimitthevariabilityofthestockpositionwithrespecttotimeandstockprice,whichwouldleadtosmallertransactioncostsofimplementingahedgingstrategy.Theminimizationoftheaveragesquarederroris 45

PAGE 46

3.4.1ConstraintsforPutOptions 3{9 )forpricingEuropeanputoptions. 0. Verticalmonotonicity. 0. Horizontalmonotonicity. 46

PAGE 47

3{9 )weusedthefollowingconstraintsholdingforoptionsinquiteageneralcase.Weassumenon-arbitrageandmaketechnicalassumptions1-5(usedbyMerton(1973)forderivingpropertiesofcallandputoptionvalues.SomeoftheconsideredpropertiesofoptionvaluesareprovedbyMerton(1973).Otherinequalitiesareprovedbytheauthors. Therestofthesectionisorganizedasfollows.First,weformulateandproveinequalities( 3{10 )-( 3{14 )forcalloptions.Someoftheconsideredpropertiesofoption 47

PAGE 48

3{9 ),theyareusedinproofsofsomeofconstraints( 3{10 )-( 3{14 ).Inparticular,weakandstrongscalingpropertiesandtwoinequalitiesprecedingproofsofoptionpricesensitivityconstraintsandconvexityconstraintsarenotincludedinthesetofconstraints. Second,weconsiderinequalities( 3{19 )-( 3{23 )forputoptions.Weprovideproofsofverticalandhorizontaloptionpricemonotonicity;proofsofotherinequalitiesaresimilartothoseforcalloptions. Weusethefollowingnotations.C(St;T;X)andP(St;T;X)denotepricesofcallandputoptions,respectively,withstrikeX,expirationT,whenthestockpriceattimetisSt.Whenappropriate,weuseshorternotationsCtandPttorefertotheseoptions. SimilartoMerton(1973),wemakethefollowingassumptionstoderiveinequalities( 3{10 )-( 3{14 )and( 3{19 )-( 3{23 ). Belowaretheproofsofinequalities( 3{10 )-( 3{14 ). 1."Immediateexercise"constraints.Merton(1973)Ct[StXer(Tt)]+:

PAGE 49

Foranyk>0considertwostockpriceprocessesS(t)andkS(t).Fortheseprocesses,thefollowinginequalityisvalidC(kSt;T;kX)=kC(St;T;X);whereStisthevalueoftheprocessS(t)attimet. Underassumptions4and5,thecalloptionpriceC(S;T;X)ishomogeneousofdegreeoneinthestockpricepershareandexerciseprice.Inotherwords,ifC(S;T;X)andC(kS;T;kX)areoptionpricesonstockswithinitialpricesSandkSandstrikesXandkX,respectively,thenC(kS;T;kX)=kC(S;T;X): NowconsideranoptionCwiththestrikeX1writtenononeshareofthestock1.DenoteitspricebyC1(S1;T;X1):OptionsAandChaveequalinitialpricesS1=1 49

PAGE 50

ForanyX1,X2suchthat0X1X2,thefollowinginequalityholdsC(St;T;X1)C(St;T;X2)+(X2X1)er(Tt):

PAGE 51

FortwooptionswithstrikeXandinitialpricesS1andS2,S2S1,thereholdsC(S1;T;X)S1 X1;T;X):BysettingX1=S1 LetC(t;S;T;X)denotethepriceofaEuropeancalloptionwithinitialtimet;initialpriceattimetequaltoS;timetomaturityT;andstrikeX:Undertheassumptions1,2and3foranyt,u,t
PAGE 52

multiplyingbothsidesofthepreviousinequalitybyS3givesS3C(1;T;X3)S1C(1;T;X1)+(1)S2C(1;T;X2):Further,usingtheweakscalingproperty,wegetC(S3;T;S3X3)C(S1;T;S1X1)+(1)C(S2;T;S2X2):UsingdenitionsofX1andX2andexpandingS3X3asS3(X1+(1)X2)=S3X0B@ S1+1 S21CA==S3X0B@S1 S3+1 S31CA=X;

PAGE 53

1.\Immediateexercise"constraints. Foranyk>0,considertwostockpriceprocessesS(t)andkS(t).Fortheseprocessesthefollowinginequalityholds:P1(kSt;T;kX)=kP2(St;T;X);whereP1andP2areoptionsontherstandthesecondstocksrespectively. Undertheassumptions4and5,putoptionvalueP(S;T;X)ishomogeneousofdegreeoneinthestockpriceandthestrikeprice,i.e.,foranyk>0;P(kS;T;kX)=kP(S;T;X): 53

PAGE 54

Underassumptions1,2,and3,foranyinitialtimestandu,t
PAGE 55

3{15 )-( 3{18 )and( 3{24 )-( 3{27 )onthestockposition.Stockpositionboundsandverticalmonotonicityareproveninthegeneralcase(i.e.underassumptions1-5andthenon-arbitrageassumption);horizontalmonotonicityandconvexityarejustiedundertheassumptionthatthestockprocessfollowsthegeometricBrownianmotion. ThenotationC(S;T;X)(P(S;T;X))standsforthepriceofacall(put)optionwiththeinitialpriceS,timetoexpirationT,andthestrikepriceX.Thecorrespondingpositioninthestock(forbothcallandputoptions)isdenotedbyU(S;T;X). First,wepresenttheproofsofinequalities( 3{15 )-( 3{18 )forcalloptions. 1.Verticalmonotonicity(Calloptions). 0U(S;T;X)1SincetheoptionpriceC(S;t;X)isanincreasingfunctionofthestockpriceS,itfollowsthatU(S;t;X)=C0s(S;t;X)0. NowweneedtoprovethatU(S;t;X)1.WewillassumethatthereexistssuchSthatC0s(S)forsome>1andwillshowthatthisassumptioncontradictstheineqiality

PAGE 56

3)Horizontalmonotonicity(Calloptions) 2dZ;(3{29) andd1andd2aregivenbyexpressionsd1=1 2p 2p

PAGE 57

2)+ln(S X)2 X 2: X:F(S)0(implyingU0t(S;T;X)0)whenSLandF(S)0(implyingU0t(S;T;X)0)whenSL,whereL=XeT(r+2=2): 3-7 ). TheError(%)columncontainserrorsofapproximatinginexionpointsbystrikeprices.Theseerrorsdonotexceed3%forabroadrangeofparameters.Weconcludethatinexionpointscanbeapproximatedbystrikepricesforoptionsconsideredinthecasestudy. 3{24 )-( 3{27 )forputoptions. 57

PAGE 58

58

PAGE 59

3-1 3-3 ,and 3-4 report\relative"valuesofstrikesandoptionprices,i.e.strikesandpricesdividedbytheinitialstockprice.Pricesofoptionsarealsogivenintheimpliedvolatilityformat,i.e.,foractualandcalculatedpriceswefoundthevolatilityimpliedbytheBlack-Scholesformula. 3-1 Table1showsquitereasonableperformanceofthealgorithm:theerrorsintheprice(Err(%),Table 3-1 )arelessthan2%formostofcalculatedputandcalloptions. Also,itcanbeseenthatthevolatilityisquiteatforbothcallandputoptions.Theerrorofimpliedvolatilitydoesnotexceed2%formostcallandputoptions(Vol.Err(%),Table 3-1 ).Thevolatilityerrorslightlyincreasesforout-of-the-moneyputsandin-the-moneycalls. 3-2 .Theactualmarketpriceofanoptionisassumedtobetheaverageofitsbidandaskprices.ThepriceoftheS&P500indexwasmodelledbyhistoricalsample-paths.Non-overlappingpathsoftheindexweretakenfromthehistoricaldatasetandnormalizedsuchthatallpathshavethesameinitialpriceS0.Then,thesetofpathswas\massaged"tochangethespreadofpathsuntiltheoptionwiththeclosesttoat-the-moneystrikeispricedcorrectly.Thissetofpathswiththeadjustedvolatilitywasusedtopriceoptionswiththeremainingstrikes. Table 3-3 displaystheresultsofpricingusing100historicalsample-paths.Thepricingerror(seeErr(%),Table 3-3 )isaround1:0%forallcallandputoptionsandincreases 59

PAGE 60

3-4 showsthatin-the-moneyS&P500indexoptionscanbepricedquiteaccuratelywith20sample-paths.)Atthesametime,themethodisexibleenoughtotakeadvantageofspecicfeaturesofhistoricalsample-paths.WhenappliedtoS&P500indexoptions,thealgorithmwasabletomatchthevolatilitysmilereasonablywell(Figures 3.6 3.6 ).Atthesametime,theimpliedvolatilityofoptionscalculatedintheBlack-Scholessettingisreasonablyat(Figures 3.6 3.6 ).Therefore,onecanconcludethattheinformationcausingthevolatilitysmileiscontainedinthehistoricalsample-paths.Thisobservationisinaccordancewiththepriorknownfactthatthenon-normalityofassetpricedistributionisoneofcausesofthevolatilitysmile. Figures 3.6 3.6 3.6 ,and 3.6 presentdistributionsoftotalexternalnancing(PNj=1apjerj)onsamplepathsanddistributionsofdiscountedmoneyinows/outows(apjerj)atre-balancingpointsforBlack-ScholesandSPXcalloptions.WesummarizestatisticalpropertiesofthesedistributionsinTable( 3-5 ). Figures 3.6 3.6 3.6 ,and 3.6 alsoshowthattheobtainedpricessatisfythenon-arbitragecondition.Withrespecttopricingasingleoption,thenon-arbitrageconditionisunderstoodinthefollowingsense.Iftheinitialvalueofthehedgingportfolioisconsideredasapriceoftheoption,thenatexpirationthecorrespondinghedgingstrategyshouldoutperformtheoptionpayoonsomesamplepaths,andunderperformtheoptionpayoonsomeothersamplepaths.Otherwise,thefreemoneycanbeobtainedbyshortingtheoptionandbuyingthehedgingportfolioorviseversa.Thealgorithmproducesthepriceoftheoptionsatisfyingthenon-arbitrageconditioninthissense.Thevalueofexternal 60

PAGE 61

Thepricingproblemisreducedtoquadraticprogramming,whichisquiteecientfromthecomputationalstandpoint.ForthegridconsistingofProws(thestockpriceaxis)andNcolumns(thetimeaxis),thenumberofvariablesintheproblem( 3{9 )is2PNandthenumberofconstraintsisO(NK),regardlessofthenumberofsamplepaths.Table 3-6 presentscalculationtimesfordierentsizesofthegridwithCPLEX9.0quadraticprogrammingsolveronPentium4,1.7GHz,1GBRAMcomputer. Inordertocompareouralgorithmwithexistingpricingmethods,weneedtoconsideroptionspricingfromthepracticalperspective.Pricingofactuallytradedoptionsincludesthreesteps. Mostcommonlyusedapproachforpracticalpricingofoptionsistimecontinuousmethodswithaspecicunderlyingstockprocess(Black-Scholesmodel,stochasticvolatilitymodel,jump-diusionmodel,etc).Wewillrefertothesemethodsasprocess-specicmethods.Inordertojudgetheadvantagesoftheproposedalgorithmagainsttheprocess-specicmethods,weshouldcomparethemstepbystep. 61

PAGE 62

Ouralgorithmdoesnotrelyonsomespecicmodelanddoesnothaveerrorsrelatedtothechoiceofthespecicprocess.Also,wehaverealisticassumptions,suchasdiscretetrading,non-self-nancinghedgingstrategy,andpossibilitytointroducetransactioncosts(thisfeatureisnotdirectlypresentedinthepaper). Calibrationofprocess-specicmethodsusuallyrequireasmallamountofmarketdata.Ouralgorithmcompeteswellinthisrespect.Weimposeconstraintsreducingfeasiblesetofhedgingstrategies,whichallowspricingwithverysmallnumberofsamplepaths. Themajoradvantageofouralgorithmisthattheerrorsofback-testinginourcasecanbemuchlowerthantheerrorsofprocess-specicmethods.Thereasonbeing,theminimizationoftheback-testingerroronhistoricalpathsistheobjectiveinouralgorithm.Minimizationofthesquarederroronhistoricalpathsensuresthattheneedofadditionalnancingtopracticallyhedgetheoptionisthelowestpossible.Noneoftheprocess-specicmethodspossessthisproperty. 62

PAGE 63

Thispaperistherstintheseriesofpapersdevotedtoimplementationofthedevelopedalgorithmtovarioustypesofoptions.OurtargetispricingAmerican-styleandexoticoptionsandtreatmentactualmarketconditionssuchastransactioncosts,slippageofhedgingpositions,hedgingoptionswithmultipleinstrumentsandotherissues.Inthispaperweestablishedbasicsofthemethod;thesubsequentpaperswillconcentrateonmorecomplexcases. 63

PAGE 64

Impliedvolatilityvs.strike:CalloptionsonS&P500indexpricedusing100samplepaths.BasedonpricesincolumnsCalc.Vol(%)andAct.Vol(%)ofTable 3-3 .CalculatedVol(%)=impliedvolatilityofcalculatedoptionsprices(100sample-paths),ActualVol(%)=impliedvolatilityofmarketoptionsprices,strikepriceisshiftedleftbythevalueoftheloweststrike. Figure3-2. Impliedvolatilityvs.strike:PutoptionsonS&P500indexpricedusing100samplepaths.BasedonpricesincolumnsCalc.Vol(%)andAct.Vol(%)ofTable 3-3 .CalculatedVol(%)=impliedvolatilityofcalculatedoptionsprices(100sample-paths),ActualVol(%)=impliedvolatilityofmarketoptionsprices,strikepriceisshiftedleftbythevalueoftheloweststrike. 64

PAGE 65

Impliedvolatilityvs.strike:CalloptionsinBlack-Scholessettingpricedusing200samplepaths.BasedonpricesincolumnsCalc.Vol(%)andB-S.Vol(%)ofTable 3-1 .CalculatedVol(%)=impliedvolatilityofcalculatedoptionsprices(200sample-paths),ActualVol(%)=atvolatilityimpliedbyBlack-Scholesformula,strikepriceisshiftedleftbythevalueoftheloweststrike. Figure3-4. Impliedvolatilityvs.strike:PutoptionsinBlack-Scholessettingpricedusing200samplepaths.BasedonpricesincolumnsCalc.Vol(%)andB-S.Vol(%)ofTable 3-1 .CalculatedVol(%)=impliedvolatilityofcalculatedoptionsprices(200sample-paths),ActualVol(%)=atvolatilityimpliedbyBlack-Scholesformula,strikepriceisshiftedleftbythevalueoftheloweststrike. 65

PAGE 66

Black-Scholescalloption:distributionofthetotalexternalnancingonsamplepaths.Initialprice=$62,strike=$62timetoexpiration=70,risk-freerate=10%,volatility=20%.Stockpriceismodelledwith200Monte-Carlosamplepaths. Figure3-6. Black-Scholescalloption:distributionofdiscountedinows/outowsatre-balancingpoints.Initialprice=$62,strike=$62timetoexpiration=70,risk-freerate=10%,volatility=20%.Stockpriceismodelledwith200Monte-Carlosamplepaths. 66

PAGE 67

SPXcalloption:distributionofthetotalexternalnancingonsamplepaths.Initialprice=$1183:77,strikeprice=$1190timetoexpiration=49days,risk-freerate=2:3%.Stockpriceismodelledwith100samplepaths. Figure3-8. SPXcalloption:distributionofdiscountedinows/outowsatre-balancingpoints.Initialprice=$1183:77,strikeprice=$1190timetoexpiration=49days,risk-freerate=2:3%.Stockpriceismodelledwith100samplepaths. 67

PAGE 68

PricesofoptionsonthestockfollowingthegeometricBrownianmotion:calculatedversusBlack-Scholesprices. StrikeCalc.B-SErr(%)Calc.Vol.(%)B-S.Vol.(%)Vol.Err(%) Calloptions1.1450.00370.0038-3.7819.6320.00-1.861.1130.00750.00741.3519.9120.00-0.461.0810.01340.01330.6519.8720.00-0.651.0480.02260.0227-0.0419.7920.00-1.041.0160.03640.03610.8019.9420.00-0.281.0000.04460.04450.1919.8220.00-0.920.9680.06510.06480.4719.9420.00-0.310.9350.08910.0892-0.0819.5920.00-2.070.9030.11660.1168-0.1119.2920.00-3.560.8710.14640.1465-0.0718.7120.00-6.44Putoptions1.1450.12740.1276-0.1619.7320.00-1.361.1130.09950.09940.0420.0320.000.171.0810.07380.07380.0520.0220.000.121.0480.05140.0514-0.1019.9720.00-0.161.0160.03340.03320.7120.1420.000.681.0000.02580.02580.1520.0220.000.110.9680.01470.01441.8220.1920.000.930.9350.00700.0071-1.6019.8920.00-0.560.9030.00290.0031-5.7719.7120.00-1.450.8710.00100.0011-12.8819.5220.00-2.41 Initialprice=$62,timetoexpiration=69days,risk-freerate=10%,volatility=20%,200samplepathsgeneratedbyMonte-Carlosimulation. Strike($)=optionstrikeprice,Calc.=obtainedoptionprice(relative),BS=Black-Scholesoptionprice(relative),Err=(FoundBS)=BS,Calc.Vol.=obtainedoptionpriceinvolatilityform,BS.Vol.(%)=Black-Scholesvolatility,Vol.Err(%)=(Calc:Vol:BS.Vol.)=BS.Vol.

PAGE 69

S&P500ptionsdataset. StrikeBidAskPriceRel.Pr StrikeBidAskPriceRel.Pr CalloptionsPutoptions1500N/A0.5N/AN/A 1500311.3313.3312.30.263813250.30.50.40.0003 1300112.7114.7113.70.096013000.450.80.6250.0005 127588.890.889.80.075912751.151.651.40.0012 122546.948.947.90.040512503.74.23.950.0033 121036.938.937.90.032012258.69.69.10.0077 120031.033.032.00.0270121013.214.814.00.0118 119026.128.127.10.0229120017.518.918.20.0154 117519.821.420.60.0174119022.124.123.10.0195 115012.514.013.250.0112117530.832.831.80.0269 11258.09.08.50.0072115048.050.049.00.0414 11005.15.95.50.0046112568.369.568.90.0582 10753.34.13.70.0031110090.292.291.20.0770 10502.23.02.60.0022500682.1684.1683.10.5771 10251.552.051.80.0015 Strike($)=optionstrikeprice,Bid($)=optionbidprice,Ask($)=optionaskprice,Price($)=optionprice(averageofbidandaskprices),Rel.Pr=relativeoptionprice 69

PAGE 70

PricingoptionsonS&P500index:100paths StrikeCalc.ActualErr(%)Calc.Vol.(%)Act.Vol.(%)Vol.Err(%) Calloptions1.1190.00020.0003-40.0013.1714.14-6.821.0980.00050.0005-5.2812.8012.92-0.901.0770.00130.001211.5712.7012.402.421.0560.00350.00335.7013.0312.801.781.0350.00790.00773.1513.3813.181.521.0220.01170.0118-0.7513.4313.49-0.481.0140.01560.01541.3213.9113.771.031.0050.01950.01950.0114.0714.060.010.9930.02690.02690.1814.6314.600.230.9710.04160.04140.5015.5715.401.090.9500.05890.05821.1216.8116.134.250.9290.07750.07700.6218.0417.353.940.4220.57890.57710.3369.39N/AN/APutoptions1.2670.26330.2638-0.2022.5029.02-22.441.0980.09560.0960-0.4713.8815.14-8.351.0770.07560.0759-0.3613.7114.18-3.321.0350.04060.04050.3314.2214.110.771.0220.03190.0320-0.2514.2914.35-0.401.0140.02740.02701.2614.7514.511.621.0050.02290.0229-0.0114.8914.90-0.010.9930.01760.01741.3815.4715.301.100.9710.01110.0112-0.5216.4316.47-0.280.9500.00700.0072-1.9517.5817.72-0.790.9290.00450.0046-3.4218.8419.05-1.090.9080.00280.0031-10.0020.0220.57-2.680.8870.00150.0022-32.2720.4622.24-7.990.8660.00110.0015-26.0022.4623.78-5.54 Initialprice=$1183:77,timetoexpiration=49days,risk-freerate=2:3%.Stockpriceismodelledwith100samplepaths.Griddimensions:P=15,N=49. Strike=optionstrikeprice(relative),Calc.=calculatedoptionprice(relative),Actual=actualoptionprice(relative),Err=(Calc:Actual)=Actual,Calc.Vol.=calculatedoptionpriceinvolatilityform,Act.Vol.(%)=actualoptionpriceinvolatilityterms,Vol.Err(%)=(Calc:Vol:Act:Vol:)=Act:Vol:

PAGE 71

PricingoptionsonS&P500index:20paths StrikeCalc.ActualErr(%)Calc.Vol.(%)Act.Vol.(%)Vol.Err(%) Calloptions1.1190.00050.000345.0014.9514.145.781.0980.00100.000588.8014.4812.9212.091.0770.00200.001266.8613.9512.4012.501.0560.00470.003341.8014.3912.8012.381.0350.00920.007719.8414.4313.189.421.0220.01320.011811.4114.4713.497.261.0140.01600.01544.0314.2013.773.131.0050.01950.01950.0014.0614.060.000.9930.02640.0269-1.6614.2814.60-2.150.9710.03930.0414-5.0113.6715.40-11.230.9500.05480.0582-5.7612.0116.13-25.520.9290.07370.0770-4.358.3917.35-51.650.4220.57900.57710.34N/AN/AN/APutoptions1.2670.26330.2638-0.1923.4529.02-19.161.0980.09590.0960-0.1314.8215.14-2.111.0770.07620.07590.4014.6714.183.451.0350.04150.04052.4914.9214.115.721.0220.03320.03203.6915.2014.355.931.0140.02780.02702.7415.0314.513.541.0050.02290.02290.0114.9014.900.010.9930.01680.0174-3.3114.9015.30-2.630.9710.00890.0112-20.7214.5816.47-11.480.9500.00300.0072-58.7312.9917.72-26.730.9290.00000.0046-100.004.3819.05-77.000.9080.00000.0031-100.006.0720.57-70.500.8870.00000.0022-100.007.6822.24-65.480.8660.00000.0015-100.008.9823.78-62.21 Initialprice=$1183:77,timetoexpiration=49days,risk-freerate=2:3%.Stockpriceismodelledwith20samplepaths.Griddimensions:P=15,N=49. Strike=optionstrikeprice(relative),Calc.=calculatedoptionprice(relative),Actual=actualoptionprice(relative),Err=(Calc:Actual)=Actual,Calc.Vol.=calculatedoptionpriceinvolatilityform,Act.Vol.(%)=actualoptionpriceinvolatilityterms,Vol.Err(%)=(Calc:Vol:Act:Vol:)=Act:Vol:

PAGE 72

SummaryofcashowdistributionsforobitainedhedgingstrategiespresentedonFigures 3.6 3.6 3.6 ,and 3.6 TotalnancingRe-bal.cashowTotalnancingRe-bal.cashow Black-ScholesCallSPXCallmean0.00.00.00.0st.dev.0.62740.044916.15491.2730median0.0770-0.00080.2695-0.0314 Totalnancing($)=thesumofdiscountedinows/outowsofmoneyonapath;Re-bal.cashow($)=discountedinow/outowofmoneyonre-balancingpoints. Black-ScholesCall:Initialprice=$62,strike=$62,timetoexpiration=70,risk-freerate=10%,volatility=20%.Stockpriceismodelledwith200Monte-Carlosamplepaths. SPXCall:Initialprice=$1183:77,strikeprice=$1190,timetoexpiration=49days,risk-freerate=2:3%.Stockpriceismodelledwith100samplepaths. Table3-6. Calculationtimesofthepricingalgorithm. #ofpathsPNBuildingtime(sec)CPLEXtime(sec)Totaltime(sec) 2020490.88.29.010025491.612.614.220025705.531.737.2 CalculationsaredoneusingCPLEX9.0onPentium4,1.7GHz,1GBRAM. #ofpaths=numberofsample-paths,P=verticalsizeofthegrid,N=horizontalsizeofthegrid,Buildingtime=timeofbuildingthemodel(preprocessingtime),CPLEXtime=timeofsolvingoptimizationproblem,Totaltime=totaltimeofpricingoneoption. Table3-7. Numericalvaluesofinexionpointsofthestockpositionasafunctionofthestockpriceforsomeoptions. Expir.(days)Strike($)Inexion($)Error(%) 06260.1263.02356261.0561.52696261.9750.0405452.3683.02355453.1781.52695453.9740.0507168.8553.02357169.9191.52697170.9670.05 Expir.(days)=timetoexpiration,Strike($)=strikepriceoftheoption,Inexion($)=inexionpoint,Error(%)=(Strike-Inexion)/Strike. 72

PAGE 73

Oneofthealternativestothemean-varianceapproachistheOmegafunction,recentlyintroducedinShadwickandKeating(2002).Omegafunctionr(rh)istheratiooftheupperandthelowerpartialmomentsofanassetrateofreturnragainstthebenchmarkrateofreturnrh.Theupperpartialmomentistheexpectedoutperformanceofanassetoverabenchmark;lowerpartialmomentistheexpectedunderperformanceofanassetwithrespecttothebenchmark.TheOmegafunctionhasseveralattractivefeatureswhichmadeitapopulartoolinriskmeasurement.First,ittakesthewholedistributionintoaccount.Asinglevaluer(rh)containstheimpactofallmomentsofthedistribution.Acollectionofr(rh)forallpossiblerhfullydescribesthereturndistribution.Second, 73

PAGE 74

ThechoiceoftheOmega-optimalportfoliowithrespecttoaxedbenchmarkwithlinearconstraintsonportfolioweightsleadstoanon-linearoptimizationproblem.Severalapproachestosolvingthisproblemhasbeenproposed,amongwhicharetheglobaloptimizationapproachinAvouyi-Govietal.(2004)andparametricapproachemployingthefamilyofJohnsondistributionsinPassow(2005).Mausseretal.(2006)proposesreductionoftheOmegamaximizationproblemtolinearproblemusingchangeofvariables.ThesuggestedreductionispossibleiftheOmegafunctionisgreaterthan1atoptimality,severalnon-linearmethodsaresuggestedotherwise. ThispaperinvestigatesreductionoftheOmega-basedportfoliooptimizationproblemwithxedbenchmarktolinearprogramming.WeconsideramoregeneralproblemthanMausseretal.(2006)byallowingshortpositionsinportfolioinstrumentsandconsideringconstraintsofthetypeh(x)0withthepositivelyhomogeneousfunctionh(),insteadoflinearconstraintsinMausseretal.(2006).WeprovethattheOmega-maximizingproblemcanbereducedtotwodierentproblems.Therstproblemhastheexpectedgainasanobjective,andhasaconstraintonthelowpartialmoment.Secondproblemhasthelowpartialmomentasanobjectiveandaconstraintontheexpectedgain.IftheOmegafunctionisgreaterthan1atoptimality,theOmegamaximizationproblemcanbereducedlinearprogrammingproblem.IftheOmegafunctionislowerthan1atoptimality,theproposedreductionmethodsleadtotheproblemeitherofmaximizingaconvexfunction,orwithlinearobjectiveandanon-convexconstraint. 74

PAGE 75

4.2.1DenitionofOmegaFunction Letxibetheexposureininstrumentiintheportfolio;thecorrespondingweightsarewi=xi=PNi=1xi,i=1;:::;N. ThelossfunctionmeasuringunderperformanceoftheportfoliowithrespecttothehurdlerateattimetisdenedbyL(t;x)=NXi=1(rhrti)xi:

PAGE 76

TheOmegafunctionistheratioofthetwopartialmoments(x)=(x) 76

PAGE 77

ItisnotnecessaryforvariablesinproblemP0tobeweights.Notethatthefunction(x)isinvarianttoscalingitsargument,since(x)=1+q(x) foranyfeasiblexand>0.Moreover,ifconstraintshk(x)0,k=1;:::;Kholdforsomex,theyalsoholdforx,>0. ConsiderthefollowingalternativetoP0.(P00)max(x)=1+q(x)

PAGE 78

Conversely,supposexistheoptimalsolutiontoP00.ThentheobjectivefunctioninP0areboundedfromaboveby(x).Take=(PIi=1xi)1,thenxisfeasiblepointinP0,and(x)=(x),thereforew=xistheoptimalsolutiontoP0. EquivalenceofproblemsP1andP2isdenotedbyP1()P2. 78

PAGE 79

and(P1)maxKTD1q(x): and(Pq1)maxKTDq1(x): 79

PAGE 80

maxq(w)s.t.hk(w)0;k=1;:::;K;PIi=1wi=1;wi2R;i=1;:::;I:(4{1) Ifq(x)>0atoptimalityin( 4{1 ),thenproblemP0canbereducedtoP1(orPq1),otherwiseitcanbereducedtoP1(orPq1).Thealternativetosolving( 4{1 ),onecouldsolve maxq(x)s.t.hk(x)0;k=1;:::;K;PIi=1xi0;xi2R;i=1;:::;I;(4{2) wherethevariablesarenotrestrictedtobeweights.Ifq(x)>0atoptimalityin( 4{2 ),thenq(x)>0for=1=(PNi=1xi),wherexisafeasiblepointinP0.Ifq(x)0in( 4{2 ),thenq(x)0forallfeasiblepointsin( 4{1 ),sincethefeasibleregionin( 4{2 )containsthefeasibleregionin( 4{1 ). AnotherprescriptiontodetermineifDq+TKw=?istosolveP1rst.IfDq+TKw6=0,thenthereductiontoP1iscorrect,andq(x)>0(or(x)>1)atoptimality.IfDq+TKw=0,thenproblemP1hasnosolutionorhavetheoptimalobjectivevalueequaltozero.Toseethis,notethatifDq=0TKw6=?,thenthereexistsapointxsuchthatq(x)=0and(x)=1,thereforetheobjectiveinP1isequaltozeroatoptimality.IfDq=0TKw=?,thenq(x)<0foranyx2Dq.However, 80

PAGE 81

Alternatively,theproblemPq1canbeattempted.IfDq+TKw6=?,thenthesolutiontoPq1afternormalizinggivesthesolutiontoP0.IfDq+TKw=?,theproblemPq1isinfeasible,duetotheconstraintq(x)1. IfKwTD=0=?,feasiblesetsinbothproblemsP0andP1areboundedandclosed,andobjectivefunctionarecontinuous,thereforebothproblemshavenite 81

PAGE 82

(20)LetxbesolutiontomaxDq+(x),xbesolutiontomaxDq+TD=1(x). ThenmaxDq+TD=1(x)(x).Take=1=(x),then(x)=1and(x)=(x),sox=x. (30)maxDq+TD=10B@1+q(x) 11CA=1+maxDq+TD=1q(x). (40)SupposethatxisthesolutiontomaxDq+TD1q(x)and(x)<1.Take=1=(x)>1.Then(x)=1,q(x)=q(x)>q(x),whichisacontradiction.Therefore,(x)=1atoptimalityinproblemP1,andtheequivalence(40)isjustied. NowconsiderthecaseDq+TKw=?.Denitionsoffunctionsq(x)and(x)implythatDqTD=0=?,so(x)>0foranyx2Kw.Bythesameargumentasabove,bothproblemsP0andP1havenitesolutions,andP0$P00. First,considerthecasewhenDq=0TKw6=?.Inthiscase,theoptimalsolutionxtoP0gives(x)=1,andq(x)=0,(x)>0.Taking=1=(x),yieldsq(x)=0,(x)=1,soxxistheoptimalsolutiontoP1,andq(x)=0. IfDq=0TKw=?,thenq(x)<0forallx2Dq.ThefollowingsequenceofreductionsleadstotheproblemP1.P00=maxDq(x)(100)$maxDqTD=1(x)(200),maxDqTD=1q(x)(300),maxDqTD1q(x)=P1: 82

PAGE 83

1 .ConsiderproblemPq1.Take^x2D=0TKw,thenq(^x)>0,(^x)=0.Taking^=1=q(^x),wehaveq(^^x)=1,(^^x)=0.Since(x)0forallx,theoptimalobjectivevalueinproblemPq1iszero. IfDq+TKw6=?andD=0TKw=?,thenproblemP0canbereducedtoproblemP00,aswasshownintheproofofTheorem 1 .ThefollowingsequenceofreductionstransformstheproblemP00intoPq1.P00=maxK(x)(10),maxDq+(x)(20)$maxDq+TDq=1(x)(30),minDq+TDq=1(x)(40),minDq+TDq1(x)=Pq1: 1 (20)LetxbesolutiontomaxDq+(x),xbesolutiontomaxDq+TDq=1(x). ThenmaxDq+TDq=1(x)(x).Take=1=q(x),thenq(x)=1and(x)=(x),sox=x. (30)maxDq+TDq=10B@1+q(x) minDq+TDq=1(x). 83

PAGE 84

NowconsiderthecaseDq+TKw=?andDq=0TKw6=?.Take^x2Dq=0TKw.FromtheassumptionA1,itfollowsthat(^x)>0.SincemaxKw(x)0and(^x)=0,weconcludethattheoptimalobjectivevalueinproblemP0iszero.AsforproblemPq1,pointsoftheform^xfor>0areallfeasible,and(x)!+1as!+1,soproblemPq1isunbounded. IfDq+TKw=?andDq=0TKw=?,thenthefeasibleregionofproblemP0isclosedandbounded,andtheobjectivefunctioniscontinuous,therefore,problemP0hasasolution.AccordingtoLemma 1 ,P0canbereducedtoP00.Considerthefollowingsequenceofreductions.P00=maxDq(x)(100)$maxDqTDq=1(x)(200),maxDqTDq=1(x)(300),maxDqTDq1q(x)=Pq1: ThenmaxDqTDq=1(x)(x).Take=1=q(x)>0,thenq(x)=1and(x)=(x),sox=x. (00)maxDqTDq=10B@1+q(x) maxDqTDq=1(x). (300)SupposethatxisthesolutiontomaxDqTDq1(x)andq(x)>1.Take=1=q(x)>1.Thenq(x)=1,(x)=(x)>(x),whichisacontradiction.Therefore,q(x)=1atoptimalityinproblemPq1,andtheequivalence(300)isjustied. 84

PAGE 85

canbewrittenintheform( 4{3 )bytakingh(x)=bNXi=1xiAx Inthissubsection,wediscussapplicationofTheorems1and2toproblemswithlinearconstraints.InthecasewhenDq+TKw6=?(alternatively,(x)>1atoptimality),theproblemP0canbereducedtoP1orPq1.InproblemP1,theconstraint(x)1canbereducedtolinearprogramming.Recallthat(x)=1 TheproblemPq1cansimilarlybereducedtolinearprogramming.Theminimizationoftheconvexfunction(x)canbereducedtomaximizationofPTt=1ztwithadditionalconstraintsztL(t;x);zt0,t=1;:::;T. If(x)1atoptimality,theproblemP0isreducedtoP1orPq1.Bothoftheseproblemscannotbereducedtolinearprogrammingduetothepresenceoftheconstraint(x)1inP1ormaximizationoftheconvexobjective(x)inPq1. 85

PAGE 86

max(w)=1+q(w) TheconstraintPIi=1xi=1allowstorewritethesetofconstraintsblmXj2Jmxjbum;m=1;:::;M; inthefollowingformblmIXi=1xiXj2JmxjbumIXi=1xi;m=1;:::;M; Foranyxsatisfying( 4{7 )-( 4{8 ),xfor>0willalsosatisfy( 4{7 )-( 4{8 ).Therefore,constraints( 4{7 )-( 4{8 )arespecialcaseoftheconstraintsoftype( 4{3 ). AccordingtoTheorem1,theproblem( 4{4 )canbereducedtothefollowingproblem. 86

PAGE 87

wherethelossconstraintis(x)1ifq(x)>0((x)>1)atleastatonefeasiblepointoftheproblem( 4{4 ),and(x)1otherwise. Wesolvetheaboveproblemwiththeconstraint(x)1byreducingittothefollowinglinearprogrammingprogram.Inthecasestudydescribedbelowthesolutiongives(x)>1atoptimality,implyingthatthereductiontoLPisvalid.ThefollowingLPformulationusesexplicitexpressionsoffunctionsq(x)and(x). maxPTt=1PIi=1(rhrit)xis.t.PTt=1zt1;ztPIi=1(rhrit)xi;t=1;:::;T;blmPIi=1xiPj2JmxjbumPIi=1xi;m=1;:::;MliPIi=1xixiuiPIi=1xi;(whereli>),i=1;:::;Izt0;t=1;:::;T:xi2R;i=1;:::;I:(4{10) Wesolvetheallocationproblem 4{10 foraportfolioconsistingof10strategies.WeusedhistoricaldailyratesofreturnforthefundsfromOctober1,2003,toMarch17,2006.Thedailyhurdlerateissettorh=0:00045.Theoptimalsolutionxto( 4{10 )gives(x)=1:164>1,whichindicatesthatthereductionof( 4{4 )to( 4{10 )iscorrect.Thesolutionto( 4{4 )isobtainedbynormalizingthesolutionx.TheoptimalallocationisgivenintheTable 4-1 87

PAGE 88

Optimalallocation StrategyAllocation(%) Manager110.00Manager220.00Manager37.50Manager42.22Manager50.00Manager67.50Manager70.00Manager812.78Manager920.00Manager1020.00 88

PAGE 89

ThispapermakesaconnectionbetweenthegeneralportfoliotheoryandtheclassicalassetpricingtheorybyexaminationofgeneralizedCAPMrelations.Inparticular,wederivediscountfactors,correspondingtotheCAPM-likerelationsandconsiderpricingformsofgeneralizedCAPMrelations.Weproposeamethodofcalibratingdeviationmeasuresfrommarketdataanddiscusswaysofidentifyingriskpreferencesofinvestorsinthemarketwithintheframeworkofthegeneralportfoliotheory. 89

PAGE 90

Investorssolvethefollowingportfoliooptimizationproblem. minD(x0r0+x1r1+:::+xnrn) (5{1) s.t.E(x0r0+x1r1+:::+xnrn)r0+x0+x1+:::+xn=1xi2R;i=0;:::;n: 5{1 )hasthreedierenttypesofsolutiondependingonthemagnitudeoftherisk-freerate,correspondingtocasesofthemasterfundofpositivetype,themasterfundofnegativetype,andthemasterfundofthresholdtype.Masterfundofpositivetypeistheonemostcommonlyobservedinthemarket,whenreturnofthemarketportfolioisgreaterthantheriskfreerate,andinvestorswouldtakelongpositionsinthemasterfundwhenformingtheirportfolios. Inthispaper,weconsiderthecaseofmasterfundsofpositivetypeandthecorrespondingCAPM-relations 90

PAGE 91

5{2 )closelyresembletheclassicalCAPMformula.However,generalizedCAPMrelationscannotplaythesameroleinthegeneralportfolioframeworkasCAPMformulaplaysintheclassicaltheory,asdiscussedinRockafellaretal.(2005b).ThegroupofinvestorsusingthedeviationmeasureDisviewedonlyasasubgroupofalltheinvestors.generalizedCAPMrelationsdonotnecessarilyrepresentthemarketequilibrium,astheclassicalCAPMformuladoes,andthereforecannotbereadilyusedasatoolforassetpricing.Anotherdicultywithusingrelations( 5{2 )forassetpricingisthatneitherthemasterfundnortheassetbetaforaxedmasterfundcanbeuniquelydetermined. ForthepricingusingthegeneralizedCAPMrelationstomakesense,wemakethefollowingassumptions. (A1)AllinvestorsintheconsideredeconomyusethesamedeviationmeasureD. (A2)Themasterfundcanbeidentiedinthemarket(orsomeproxyforthemasterfundexists).Ifthesetofriskidentiersforthemasterfundisnotasingleton,thechoiceofaparticularriskidentierfromthissethasnegligibleeectonassetpricesobtainedthoughthegeneralizedCAPMrelations.Therefore,wecanxaparticularriskidentierforthepurposeofassetpricing. AssumptionA2makessensebecauseformostbasicdeviationmeasuresmembersoftheriskidentiersetQD(rDM)foragivenmasterfundrDMdieronasetoftheformfrDM=Cg,whereCisaconstant.FordeviationmeasuresconsideredinRockafellaretal.(2006),theriskidentiersetforstandarddeviationandsemideviationsisasingleton;C=VaR(X)forCVaR-deviationwithcondencelevel;C=ErDMformeanabsolutedeviationandsemideviations.SinceassetpricesingeneralizedCAPM( 5{2 )dependontheriskidentierQDMthough(ri;QDM),assumptionA2suggeststhatProbfrDM=Cg=0. 91

PAGE 92

AssumptionA2cannotbesatisedforworst-casedeviationandsemideviations,seeRockafellaretal.(2006) UnderassumptionsA1andA2,allquantitiesingeneralizedCAPMrelationsarexedandwell-dened,andtherelationsrepresentpricingequilibrium.InfurtherchapterswewillcloselyexaminegeneralizedCAPMrelationsundertheseassumptions. 5.2.1TwoWaystoAccountForRisk 1+r0E[];(5{3) wherer0isarisk-freerateofreturn.Thepriceofanassetisthediscountedexpectedvalueofitsfuturepayo.Theassetwithrandompayowouldhavethesamepriceasanassetwithpays^=E[]withprobability1inthefuture. Iftheriskispresent,thepriceofanassetpaying^withcertaintyinfuturewould,generallyspeaking,dierfromthepriceoftheassethavingrandompayo,suchthatE[]=^.Theformula( 5{3 )needstobecorrectedforrisk.Therearetwowaystodoit. Therstwayistomodifythediscountedquantity: 1+r0(asset);(5{4) where(asset)iscalledthecertaintyequivalent.Itisafunctionofassetparametersandisequaltothepayoofarisk-freeassethavingthesamepriceastheriskyassetwithpayo. 92

PAGE 93

1+rra(asset)E[];rra(asset)6=1:(5{5) whererra(asset)istherisk-adjustedrateofreturn. PricingformsoftheclassicalCAPM(see,forexample,Luenberger(1998))areasfollows. CertaintyequivalentformofCAPM: 1+r0E[]cov(;rM)(ErMr0) Risk-adjustedformofCAPM: 1+r0+(ErMr0)E[]:(5{7) Hereassetbeta=cov(r;rM) Relevanttofurtherdiscussion,thereisameasureofassetqualityknownastheShapreRatioS=E[r]r0 5{2 ),weget 93

PAGE 94

1+r00B@Ei+ErDMr0 Pricingformula( 5{8 )thecertaintyequivalentpricingformofgeneralizedCAPMrelations( 5{2 )(compareitto( 5{4 )),wherethecertaintyequivalent (i)=Ei+ErDMr0 isthepayoofarisk-freeassethavingthesamepricei. Wecouldrearrangetheformula( 5{2 )inadierentway,namely whenErDMr0

PAGE 95

5{10 )istherisk-adjustedpricingformofgeneralizedCAPMrelations( 5{2 )(comparewith( 5{10 )),wheretherisk-adjustedrateofreturnis ThequantityErDMr0 5{8 )and( 5{10 )isthegeneralizedSharpeRatioforthemasterfund.Itshowswhatincreaseinexcessreturncanbeobtainedbyincreasingthedeviationoftheassetby1.Intheclassicalportfoliotheory,masterfundhasthehighestSharpeRatioamongallassets.Thesameresultholdsinthegeneralizedsettingasweshownext. forsomeasseti>0.ThegeneralizedSharpeRatioforthemasterfundisstrictlypositiveSDM=ErDMr0 Ifcov(ri;QDM)=0,thenEri=r0,thereforeErir0 5{12 )holds. Ifcov(ri;QDM)<0,thenEri0,thenaccordingtothedualrepresentationofD(ri),wehaveD(ri)=maxQ2Qcov(ri;Q)cov(ri;QDM)>0;

PAGE 96

5{8 )and( 5{10 )implythattheriskadjustmentisdeterminedbythecorrelationoftheassetrateofreturnwiththeriskidentierofthemasterfund. Togainabetterintuitionaboutthemeaningofthisformofriskadjustment,wecomparetheclassicalCAPMformulawiththegeneralizedCAPMrelationsfortheCVaR-deviationD(X)=CVaR(X)CVaR(XEX). Firstnote,thatmorevaluableassetsarethosewithlowerreturns.Whenpricingtwoassetswiththesameexpectedreturn,investorswillpayhigherpriceforamorevaluableasset,thereforeitsreturnwillbelowerthanthatofthelessvaluableasset. WebeginbyanalyzingtheclassicalCAPMformulawrittenintheform wheretheleft-handsideoftheequationistheassetreturn.Thereturnisgovernedbythecorrelationoftheassetrateofreturnwiththemarketportfoliorateofreturn,i.e.bythequantitycov(ri;rM).Assetswithhigherreturncorrelationwiththemarketportfoliohavehigherexpectedreturns,andviceversa.Formula( 5{14 )impliesthatassetswithlowercorrelationwiththemarketaremorevaluable.Thereisthefollowingintuitionbehindthisresult.Investorsholdthemarketportfolioandtherisk-freeasset;theproportionsofholdingsdependonthetargetexpectedportfolioreturn.Theonlysourceofriskofsuchinvestmentsisintroducedbytheperformanceofthemarketportfolio.Themostundesirablestatesoffuturearethosewheremarketportfolioreturnsarelow.Theassetswithhigherpayoinsuchstateswouldbemorevalued,sincetheyserveasinsuranceagainstpoorperformanceofthemarketportfolio.Therefore,thelowerthecorrelationof 96

PAGE 97

NowconsiderthecaseofCVaR-deviation,D(X)=CVaR(XEX).Investorsmeasuringuncertaintyoftheportfolioperformancebythisdeviationmeasureareconcernedaboutthevalueoftheaverageofthe%worstreturnsrelativetothemeanofthereturndistribution. WeconsidergeneralizedCAPMrelationsfortheCVaR-deviationinthecaseofthemasterfundofpositivetype. wherei=cov(ri;QDM) IfprobfrDM=VaR(rDM)g=0,then Forfurtherdiscussion,assume=10%.Thenthenumeratorof( 5{16 )istheexpectedunderperformanceoftheassetrateofreturnwithrespecttoitsaveragerateofreturn,conditionalonthemasterfundbeinginits10%lowestvalues.Thedenominatorof( 5{16 )isthethesamequantityforthemasterfund.Aninvestorholdsthemasterfundandtherisk-freeassetinhisportfolio.Theportfolioriskisintroducedbytheperformanceofthemasterfund.Formula( 5{16 )suggeststhatassetsarevaluedbasedontheirrelativeperformanceversusthemasterfundperformanceinthosefuturestateswherethemaster 97

PAGE 98

Fromthegeneralportfoliotheorypointofview,thevalueoftheassetis,therefore,determinedbytheextenttowhichthisassetprovidesprotectionagainstpoormasterfundperformance.Dependingonthespecicformofthedeviationmeasure,theneedforthisprotectioncorrespondstodierentpartsofthereturndistributionofthemasterfund.Mostvaluableassetsdrasticallydierinperformancefromthemasterfundinthosecaseswhenprotectionisneededthemost. 5.3.1BasicFactsfromAssetPricingTheory. Thediscountfactorisoffundamentalimportancetoassetpricing.Below,wepresenttwotheoremsduetoRoss(1978),andHarrisonandKreps(1979)whichemphasizeconnectionsbetweenthediscountfactorandassumptionsofabsenceofarbitrageandlinearityofpricing.Inthenarration,wefollowCochrane(2001),Chapter4. LetX Wewillconsidertwoassumptions,theportfolioformationassumption(A1)andthelawofonepriceassumption(A2). (A1)If02X LetPrice()bethepriceofpayo. 98

PAGE 99

Underassumption(A1),thepayospaceX Fromtheperspectiveofdiscountfactors,acompletemarketischaracterizedbyauniquediscountfactor;inanincompletemarketthereexistsaninnitenumberofdiscountfactorsandeachdiscountfactorproducesthesamepricesofallassetswithpayosinX 5{17 ).Moredetailsonpricingassetsincompeteandincompletemarketswillbeprovidedlateron.Importantimplicationsofthesetheoremsareasfollows. 99

PAGE 100

5{17 )asfollows. 1+r0Z(!)dQ(!);(5{18) wheredQ(!)=(1+r0)m(!)dP(!).Sinceexpectationof(1+r0)mequalstoone 5{17 )is=1 1+r0EQ[]; Ifonepicksadiscountfactorm,whichisnotstrictlypositive,thetransformation( 5{18 )willleadtothepricingequation=R(!)dQ(!)thatcorrectlypricesallassetswithpayosinX 5{17 )totherisk-freerategives1=E[m(1+r0)]. 100

PAGE 101

Inacompletemarket,thepayoofanynewassetliesinX Inanincompletemarket,twocasesarepossible.(1)ThepayoofanewassetbelongstoX new])ofthisassetcanbedenedasfollows. where=fmjm(!)>0withprobability1g:Includingonlystrictlypositivediscountfactorstothesetleadstoarbitrage-freepricesgivenbyformula=E[m]. 101

PAGE 102

1+r0Ei+ErDMr0 1+r0Ei(QDM1)ErDMr0 1+r0(QDM(!)1)ErDMr0 wearriveatthepricingformulaintheform( 5{17 ) ThediscountfactorcorrespondingtothedeviationmeasureDisgivenby( 5{21 ).Pricingformulas( 5{22 )correspondingtodierentdeviationmeasuresDwillyieldthesamepricesforassetsri,i=0;:::;n;andtheircombinations(denedbyportfolioformationassumptionA1),butwillproducedierentpricesofnewassets,whosepayoscannotbereplicatedbypayosofexistingn+1assets.EachdeviationmeasureDhasthecorrespondingdiscountfactormD,whichisusedin( 5{22 )todetermineauniquepriceofanewasset.Aninvestorhasriskrelatedtoimperfectreplicationofthepayoofanewasset,andspecieshisriskpreferencesbychoosingadeviationmeasureinpricingformula( 5{22 ). 102

PAGE 103

X .ItfollowsthatdiscountfactormDforanyDcanberepresentedasmD=m+"D,wherem2 istheprojectionofalldiscountfactorsmDonthepayospace X ,and"Disorthogonalto X .Wecallmpricinggeneratorforthegeneralportfoliotheory. ThepricinggeneratormcoincideswiththediscountfactorforthestandarddeviationD=,since 1+r01rM(!)ErM togetherwithrM2 implym2X Foragivenpayospace X ,discountfactorsmDforallDformasubsetofalldiscountfactorscorrespondingto X ThestrictpositivityconditionmD(!)>0(a.s.)canbewrittenas 1+r0(QDM(!)1)ErDMr0 Notethattheleft-handsideofcondition( 5{24 )containsarandomvariable,whiletheright-handsideisaconstant,andtheinequalitybetweenthemshouldbesatisedwithprobabilityone.ScalingthedeviationmeasureDbysome>0willchangethevalueoftheleft-handside.Weshownextthatitdoesnotchangemeaningofthecondition( 5{24 ). 5{24 )isinvariantwithrespecttore-scalingdeviationmeasureD.

PAGE 104

5{24 )canbeexpressedas whereSDM=D(rDM) Sincetheriskenvelopes^QandQfordeviationmeasures^DandDarerelatedas^Q=(1)+Q; 5{25 )holdsforD,itholdsforDaswell,since^Q^DM(!)11 ^S^DM(1)+QDM(!)1 SDMQDM(!) SDMQDM(!)11

PAGE 105

ConsideranalternativerepresentationofmD(!)in( 5{21 ).LettingSDM=ErDMr0 1+r0(QDM(!)1)SDM+1=1 1+r0(QDM(!)SDM+(1SDM):(5{26) InLemma1weshowedthatriskidentiers^Q(rDM)andQ(rDM)fordeviationmeasures^D=D(>0)andD,respectively,arerelatedas^Q(rDM)=(1)+Q(rDM): 1+r0QDMM(!); 105

PAGE 106

5.4.1IdenticationofRiskPreferencesofMarketParticipants providethemostaccurateassetpricing.WecouldeithertakeasetofassetreturnsrifromthemarketandestimatethemasterfundrDM,ortreatthemasterfundasgivenbythemarketandestimateexpectedreturnsEri.Theobtainedquantities,themasterfundreturnorexpectedreturnsoftheassets,willdependonthedeviationmeasureD,whichcanbecalibratedbycomparingestimatedquantitiestotheirmarketvalues. Welimitourconsiderationtothecaseofknownmasterfund;themethodbasedonestimationofamasterfundgivenasetofassetsismorecomputationallydicult,becausethegeneralizedportfolioproblemshouldbesolvedforeachchoiceofD. Assumptionthatamasterfundcanbeobtainedfromthemarketisjustiedbytheexistenceofindices,suchasS&P500,DowJonesIndustrialAverageandNasdaq100,whichrepresentthestateofsomelargepartofthemarket;moreover,investingintheseindicescanbethoughtofasinvestinginthemarket. Anybroad-basedmarketindexisassociatedwithcertainselectionofassets;theindexsummarizesthebehaviorofthemarketoftheseassets.Wecouldcalibratethegeneralized 106

PAGE 107

Therstcalibrationmethodisbasedonpricingassetsfromtheindexpool.Theindexserveasamasterfundinageneralizedportfolioproblemposedforassetsfromthepool.Givenaxedselectionofassets,dierentdeviationmeasureswouldproducedierentmasterfunds.Theexistenceofaparticularmasterfundfortheseassetsinthemarketcan,therefore,beusedasabasisforestimationofadeviationmeasure.The\best"deviationmeasureistheonewhichyieldsthebestmatchbetweentheexpectedreturnsofassetsfromthepoolandtheindexreturnthroughthegeneralizedCAPMrelations. Thesecondcalibrationmethodisbasedonpricingassetslyingoutsideoftheindexpool.Aswediscussedearlier,whenpricinganewassetwhosepayodoesnotbelongtotheinitiallyconsideredpayospace,thepriceinvestorswouldpaydependsontheirriskpreferences,denedbythedeviationmeasure.Thesecondmethod,therefore,usespricesof\new"assetswithrespecttotheindexpoolasthebasisforestimationofriskpreferences.Itshouldbenotedthatinthesetupofthegeneralportfoliotheorytheselectionofassetsisxed,andthemasterfunddependsonthedeviationmeasure.Inthepresentmethodweassumethatthemasterfundisxedandchangethedeviationmeasuretoobtainthebestmatchbetweenthemasterfundreturnandexpectedreturnsofnewassets.Bydoingso,weimplythatthechoiceoftheindex-associatedpoolofassetsdependsonthedeviationmeasure. Wejustifytheassumptionofaxedmasterfundbytheobservationthatmasterfunds,expectedreturnsofassets,andtheirgeneralizedbetascanbedeterminedfromthemarketdataquiteeasily,whiletheselectionofassetscorrespondingtoanindexcanbedeterminedmuchmoreapproximately.Anindexusuallyrepresentsbehaviorofapartofthemarketconsistingofmuchmoreinstrumentsthattheindexiscomprisedof.Withmuchcertainty,though,wecouldassumethatassetsconstitutingtheindexbelongtothe 107

PAGE 108

Wealsonotethatimplementationsofbothmethodsarethesame:selectingsomeindexasamasterfund,weadjustthedeviationmeasureuntilthegeneralizedCAPMrelationsprovidemostaccuratepricingofacertaingroupofassets.Werefertothisgroupofassetsasthetargetgroup. Finally,wediscussthequestion,shouldthetwocalibrationmethodsgivethesameresults.Generallyspeaking,foraxedsetofassets,thechoiceofriskpreferencesintermsofadeviationmeasuredeterminesboththemasterfundandpricingofnewassetswithpayosoutsideoftheconsideredpayospace.Whenthegeneralizedportfolioproblemisposedforthewholemarket,riskpreferencescanbedeterminedonlythroughmatchingthemasterfund,sincethereareno\new"assetswithrespecttothewholemarket.Themasterfundcoincideswiththemarketportfolio,i.e.weightofanassetinthemasterfundequalsthecapitalizationweightofthisassetinthemarket. Ifacertainindexisassumedtorepresentthewholemarket,thencalibrationofthedeviationmeasurebasedondierenttargetgroupsofassets(forexample,onagroupofstocksandagroupofderivativesonthesestocks)shouldgivethesameresult.Iftheobtainedriskpreferencesdonotagree,thismayindicatethateitherthegeneralportfoliotheorywithasingledeviationmeasureisnotapplicabletothemarketorthattheindexdoesnotadequatelyrepresentthemarket. Ifindicestrackperformanceofsomepartsofthemarket,thetwomethodsarenot,generallyspeaking,expectedtogivethesameresults.Marketpricesofassetsnotbelongingtoanindexgroupmaynotbedirectlyinuencedbyriskpreferencesofinvestorsholdingtheindexintheirportfolios.Forexample,itdoesnotmakesensetocalibrateriskpreferencesbytakingoneindexasamasterfundandassetsfromanotherindexasatargetsetofassets.However,itisreasonabletosupposethatpricesofderivatives(forexample,options)ontheassetsbelongingtoanindexgroupareformedbyrisk 108

PAGE 109

Forthepurposesofcalibration,weassumeaparametrizationofadeviationmeasureD=D,where=(1;:::;l)isavectorofparameters. 109

PAGE 110

AssumethattheprobabilitymeasurePinthemarkethasadensityfunctionp(!).WeconsiderthegeneralizedCAPMrelationsintheform( 5{22 )andtransformthemasfollows(denotesthecompletesetoffutureevents!). 1+r0Z(!)(1+r0)mD(!)p(!)d!;(5{28) 110

PAGE 111

weget 1+r0Z(!)qD(!)d!:(5{30) Aswediscussedabove,ifthediscountfactormD(!)isstrictlypositive,thefunctionqD(!)couldbecalledthe\risk-neutral"densityfunction. Thefutureevent!consistsoffuturereturnsofallassetsinthemarketandcanberepresentedas!=(rDM;r01;:::;r0k;r),whererrepresentsratesofreturnsoftherestofassetsinthemarket. 5{29 )withrespecttor01;:::;r0k;r. Let~qD(rDM)=ZqD(rDM;r01;:::;r0k;r)dr01:::dr0kdr: 5{31 ),notethatthediscountfactormDisalineartransformationoftheriskidentierQDM(bothmDandQDMarerandomvariablesandarefunctionof!).DuetotherepresentationQDM2QDM=argminQ2QE[rDMQ]; 111

PAGE 112

Equation( 5{31 )becomes ~qD(rDM)=(1+r0)RmD(rDM)p(rDM;r01;:::;r0k;r)dr01:::dr0kdr;~qD(rDM)=(1+r0)mD(rDM)Rp(rDM;r01;:::;r0k;r)dr01:::dr0kdr;~qD(rDM)=(1+r0)m(rDM)~p(rDM); where~p(rDM)=Rp(rDM;r01;:::;r0k;r)dr01:::dr0kdristheactualmarginaldistributionofthemasterfund. Relationship( 5{29 )isnowtransformedinto ~qD(rDM)=(1+r0)mD(rDM)~p(rDM);(5{33) whererDM=rDM(!).ThisrelationshipprovidethebasisforcalibrationofD.Let~q(rDM)denotethetruerisk-neutraldistributionofthemasterfund.Bothfunctions~q(rDM)and~p(rDM)canbeestimatedfrommarketdata;theerrorinestimationof~q(rDM)by~qDrDM)in( 5{33 )isminimizedwithrespecttoD. First,weconsiderestimationof~q(rDM).Letq(!)bethetruemarketrisk-neutraldistribution,~q(rDM)=Rq(rDM;r01;:::;r0k;r)dr01:::dr0kdr.Applyingformula( 5{30 )withq(!)forpricinganoptiononthemasterfund,wegetc=1 1+r0Zc(rDM)q(rDM;r01;:::;r0k;r)drDMdr01:::dr0kdr=1 1+r0Zc(rDM)~q(rDM)drDM; 112

PAGE 113

Dierentiating( 5{34 )withrespecttoK,wegeterT@C @K=K~q(K)Z+1K~q(S)dS+K~q(K)=Z+1K~q(S)dS: 5{34 )twicewithrespecttoK,wearriveattheformulaforestimatingrisk-neutraldensityqfromcrosssectionofoptionpriceserT@2C @K2=@ @KZ+1K~q(S)dS=~q(K); ~q(S)=erT@2C @K2K=S:(5{35) Formula( 5{35 )allowstoestimatethefunction~q(rDM)whenthecross-sectionofpricesofoptionswrittenonthemasterfundisavailable.Itisworthmentioningthatthismethodestimates~q(rDM)atagivenpointintime;itisbasedonoptionspricesatthistime. Nowconsiderestimationofthemarginalprobabilitydensity~p(rDM).Themostcommonwaytoestimatethisdensityistousekerneldensityestimationbasedoncertainperiodofhistoricaldata.However,thismethodassumesthatthedensitydoesnotchangeovertime.Whentimedependenceistakenintoaccount,weareleftwithonlyone 113

PAGE 114

However,theformula( 5{32 )providesawayofestimating~p(rDM)foraspecicdate,ifthefunctionmD(rDM)isknown.ThisideaisutilizedintheutilityestimationalgorithmsuggestedinBlissandPanigirtzoglou(2001).Wedevelopamodicationofthismethodtocalibratethedeviationmeasure,asfollows. AssumetheparametrizationD=D.Alsoassumethatthemasterfundisknownfromthemarketandthereforeisxed,itsrateofreturnisdenotedbyrM.Foreachdatet=1;:::;T,weestimatethefunctionmt(rM)using( 5{21 ).QuantitiesD(rM),ErDM,andQDMinthedenitionofmt(rM),arecalculatedbasedonacertainperiodofhistoricalreturnstheindex.Also,weestimatefunctions~qt(rM),t=1;:::;T,using( 5{35 ).Formula( 5{33 )allowstoestimatefunction~qt(rM)foreachparametrizationofD.Theparameterscanbecalibratedbyhypothesizingthat~qt(rM)=~qt(rM)fort=1;:::;T(whichholdsifDisthecorrectdeviationmeasureinthemarket)andmaximizingthep-valueofanappropriatestatistic. Thishypothesisisfurthertransformedasfollows.Usingthetruerisk-neutraldistributions~qt(rM),theactualdistributions~pt(rM)areestimatedusing( 5{33 ),~pt(rM)=~qt(rM) (1+r0)mt(rM);t=1;:::;T.Wethentestthenullhypothesisthatrisk-neutraldistributions~pt(rM),t=1;:::;T,equaltothetruerisk-neutraldistributions~pt(rM),t=1;:::;T. Foreachtimet=1;:::;T,onlyonerealizationrM(t)ofthemasterfundisavailable;thevaluerM(t)isasinglesamplefromthetruedensity~pt(rM).Underthenullhypothesis~pt(rM)=~pt(rM),thereforerandomvariablesytdenedbyyt=ZrM(t)~pt(r)dr; 114

PAGE 115

ImplementationIrequirescalculationexpectedreturnsofassetsandestimationofactualdistributionofthemasterfund.Thesequantitiescanbefoundfromthemarketdataquiteeasilyandaccurately.However,theresultsofthisimplementationdependonaparticularchoiceoftheobjectivefunctionDist(;).ItcanbearguedthatthechoiceoftheobjectiveshoulddependonatheparametrizationDofthedeviationmeasurebeingcalibrated.Forexample,ifthedeviationmeasureiscalibratedintheformofthemixedCVaR-deviation,thenDist(;)shouldbebasedontheCVaR-deviation,ratherthanonthestandarddeviation.AnotherdrawbackofimplementationIisthatthenancialliteraturedidnotusesimilaralgorithmsforcalibrationofutilityfunctions.Whenriskpreferencesareestimatedusingthisimplementationforthegeneralportfoliotheoryarecomparedwithriskpreferencesestimatedinnancialliteraturefortheutilitytheory,theresultsmaydierjustduetodierencesinnumericalprocedures,underlyingthetwoestimations. ImplementationIIiswidelyusedinnancialliteratureforestimationofrisk-aversioncoecientsofutilityfunctions.However,thisimplementationsuersfromsomenumericalchallengesrelatedtoevaluationoftheactualdensityintheformula~pt(rM)=~qt(rM) (1+r0)mt(rM): 115

PAGE 116

Thereisonemoredrawbackofthisimplementationwhenitisinthegeneralportfoliotheory.Whenusingnumericalestimationofrisk-neutraldensities,wehavetoassumethatnoarbitrageopportunitiesexistinthepricesofoptionsfromthecross-section.ThisimpliesthatonlystrictlypositivediscountfactorsmD(rM)shouldbeusedforcalibration.Indeed,ifmD(rM(!))<0withpositiveprobability,thenestimatesoftherisk-neutraldensity~qt(rM)=(1+r0)m(rM)~p(rM)canbenegative,andthehypothesisthatthetruerisk-neutraldensities~qt(rM(!))>0(estimatedfromoptionscross-section)areequaltotheestimateddensities~qt(rM)doesnotmakesense.However,itisnotclearatthispoint,whichdeviationmeasureshavethepropertythatmD(!)>0withprobability1. Finally,thereisanissuerelevanttoestimationofreturndistributionsofassetsbasedontheirhistoricalreturns.Historicaldatamaycontainoutliersoreectsofrarehistoricalevents.Aftersuch\cleaning",historicaldatamayprovidemorereliableconclusions.However,lteringhistoricaldatafromhistoricaleectsisanopenquestion. 116

PAGE 117

(5{36) WewillrefertodeviationmeasureCVaR(XEX)asCVaRdeviationofgains,todeviationmeasureCVaR(X+EX)asCVaRdeviationoflosses. Riskidentierforaconvexcombinationofdeviationmeasuresisaconvexcombinationoftheirriskidentiers.RiskidentierforCVaRdeviationoflosseswasderivedinRockafellaretal.(2006).Below,wederivetheriskidentierforCVaRdeviationoflossesandmixed-CVaRdeviationoflossesandexaminecoherenceofthesedeviationmeasures. Thefollowinglemmawillhelptondrisk-identierforD(X)=CVaR(X+EX). 5{37 )weneedtoprovethedualrepresentation~D(X)=EXinf~Q2~QE[X~Q]andalsoshowthat~Qsatisesproperties(Q1)-(Q3).Thedualrepresentationiscorrectsince,EXinf~Q2~QE[X~Q]=EXinfQ2QE[X(2Q)]=E[X]infQ2QE[XQ]=D(X)=~D(X):

PAGE 118

5{38 ),wewillusetheformula@D(X)=1Q(X)andthefactthat@~D(X)=@D(X):1~Q(X)=@~D(X)=@D(X)=1+Q(X)~Q(X)=2Q(X): 5{38 ),theriskenvelopeforthedeviationmeasure~D(X)=CVaR(X+EX)is 118

PAGE 119

~Q2~Q(X)()8>>>>>><>>>>>>:~Q(!)=21;whenX(!)>VaR(X)21~Q(!)2;whenX(!)=VaR(X)~Q(!)=2;whenX(!)
PAGE 120

whichcontradictswiththeconditionQ0forallQ2Q(~X),asrequiredby( 5{40 ).Thisconcludestheproof. 5{39 )impliesthatthedeviationmeasure~Q(X)=CVaR(X+EX)iscoherentif210,whichisequivalenttohaving1=2: andexamineitscoherence.Theriskidentierforthismeasuregivenby~Q1;:::;n(X)=nXi=1~Qi(X); 120

PAGE 121

2CVaR45%(X+EX)+1 2CVaR(X+EX); 121

PAGE 122

Ait-Sahalia,Y.,Duarte,J.(2003)Non-parametricoptionpricingundershaperestrictions.JournalofEconometrics116,9-47 Avouyi-Govi,S.,Morin,A.,andNeto,D.(2004)OptimalAssetAllocationWithOmegaFunction.Technicalreport,BenquedeFrance. Berkowitz,J.(2001)TestingDensityForecastswithApplicationstoRiskManagement.JournalofBusinessandEconomicStatistics19,46574. Bertsimas,D.,andA.Lo(1998)OptimalControlofExecutionCosts.JourmalofFinancialMarkets1,1-50. Bertsimas,D.,Kogan,L.,Lo,A.(2001)PricingandHedgingDerivativeSecuritiesinIncompleteMarkets:Ane-ArbitrageApproach,OperationsResearch49,372-397. Bertsimas,D.,Popescu,I.(1999)OntheRelationBetweenOptionandStockPrices:AConvexOptimizationApproach,OperationsResearch50(2),358{374,2002. Black,F.,Scholes,M.(1973)ThePricingofOptionsandCorporateLiabilities,JournalofPolicitalEconomy81(3),637-654. Bliss,R.R.andPanigirtzoglou,N.(2001)RecoveringRiskAversionFromOptions.FederalReserveBankofChicago,WorkingPaperNo.2001-15.AvailableatSSRN:http://ssrn.com/abstract=294941. Bouchaud,J-P.,Potters,M.(2000)TheoryofFinancialRiskandDerivativesPricing:FromStatisticalPhysicstoRiskManagement,CambridgeUniversityPress. Boyle,P.(1977)Options:AMonteCarloApproach,JournalofFinancialEconomics4(4)323-338. Boyle,P.,Broadie,M.andGlasserman,P.(1997)MonteCarloMethodsforSecurityPricing,JournalofEconomicDynamicsandControl21(8/9),1276-1321. Broadie,M.,Detemple,J.(2004)OptionPricing:ValuationModelsandApplications,ManagementScience50(9),1145-1177. Broadie,M.,Glasserman,P(2004)StochasticMeshMethodforPricingHigh-DimensionalAmericanOptions,JournalofComputationalFinance7(4),35-72. Carriere,L.(1996)ValuationoftheEarly-ExercisePriceforOptionsUsingSimulationsandNonparametricRegression,Insurance,Mathematics,Economics19,19-30. Cochrane,J.(2001)AssetPricing.PrincetonUniversityPress. 122

PAGE 123

Dembo,R.,RosenD.(1999)ThePracticeofPortfolioReplication,AnnalsofOperationsResearch85,267-284. Dempster,M.,Hutton,J.(1999)PricingAmericanStockOptionsbyLinearProgramming,Math.Finance9,229-254. Dempster,M,.Hutton,J.,Richards,D.(1998)LPValuationofExoticAmericanOptionsExploitingStructure,TheJournalofComputationalFinance2(1),61-84. Dempster,M.,Thompson,G.(2001)DynamicPortfolioReplicationUsingStochasticProgramming.InDempster,M.A.H.(ed.):Riskmanagement:valueatriskandbeyond.Cambridge:CambridgeUniversityPress,100-128 DennisP.(2001)OptimalNon-ArbitrageBoundsonS&P500IndexOptionsandtheVolatilitySmile,JournalofFuturesMarkets21,1151-1179. Due,D.andH.Richardson(1991)Mean-VarianceHedginginContinuousTime,TheAnnalsofAppliedProbability1,1-15. Edirisinghe,C,.Naik,V.,Uppal,R.(1993)Optimalreplicationofoptionswithtransactionscostsandtradingrestrictions,TheJournalofFinancialandQuantita-tiveAnalysis28(Mar.,1993),372-397. Fedotov,S.,Mikhailov,S.(2001)OptionPricingforIncompleteMarketsviaStochasticOptimization:TransactionCosts,AdaptiveControlandForecast,InternationalJournalofTheoreticalandAppliedFinance4(1),179-195. Follmer,H.,Schied,A.(2002)StochasticFinance:AnIntroductiontoDiscreteTime.WalterdeGruyterInc. Fllmer,H.,Schweizer,M.(1989)HedgingbySequentialRegression:AnIntroductiontotheMathematicsofOptionTrading,ASTINBulletin18,147-160. Gilli,M.,Kellezi,E.,andHysi,H.(2006)AData-DrivenOptimizationHeuristicforDownsideRiskMinimization.TheJournalofRisk8(3),1-18. Glasserman,P.(2004)Monte-CarloMethodinFinancialEngineering,Springer-Verlag,New-York. Gotoh,Y.,Konno,H.(2002)BoundingOptionPricebySemi-DeniteProgramming,ManagementScience48(5),665-678. Hansen,L.P.andRichard,S.F.(1987)TheRoleofConditioningInformationinDeducingTestableRestrictionsImpliedbyDynamicAssetPricingModels.Econometrica55,587-614. 123

PAGE 124

Jackwerth,J.C.(2000)RecoveringRiskAversionFromOptionPricesandRealizedReturns.TheReviewofFinancialStudies13(2),433-451 Joy,C.,Boyle,P.,andTan,K.S.(1996)QuasiMonteCarloMethodsinNumericalFinance,ManagementScience42,926-936. King,A.(2002)DualityandMartingales:AStochasticProgrammingPerspectiveonContingentClaims,MathematicalProgramming91,543562. Konishi,H.,andN.Makimoto(2001)OptimalSliceofaBlockTrade.JourmalofRisk3(4),33-51. Konishi,H.(2002)OptimalSliceofaVWAPTrade.JourmalofFinancialMarkets5,197-221. Levy,H.(1985)UpperandLowerBoundsofPutandCallOptionValue:StochasticDominanceApproach,JournalofFinance40,1197-1217. Longsta,F.,Schwartz,E.(2001)ValuingAmericanOptionsbySimulation:ASimpleLeast-SquaresApproach,AReviewofFinancialStudies14(1),113-147. Luenberger,D.G.(1998)InvestmentScience,OxfordUniversityPress,Oxford,NewYork. Madgavan,A.(2002)VWAPStrategies,TechnicalReport,Availableathttp://www.itginc.com/. Markowitz,H.M.(1959)PortfolioSelection,EcientDiversicationofInvestments,Wiley,NewYork. Merton,R.(1973)TheoryofRationalOptionsPricing,BellJournalofEconomics4(1),141-184. Mausser,H.,Saunders,D.,andSeco,L.(2006)OptimizingOmega.RiskMagazine,November2006. Naik,V.,Uppal,R.(1994)Leverageconstraintsandtheoptimalhedgingofstockandbondoptions,JournalofFinancialandQuantitativeAnalysis29(2),199223. Passow,A.(2005)OmegaPortfolioConstructionWithJohnsonDistributions.RiskMagazineLimited18(4),pp.85-90. Perrakis,S.,Ryan,P.J.(1984)OptionPricingBoundsinDiscreteTime,JournalofFinance39,519-525. Ritchken,P.H.(1985)OnOptionPricingBounds,JournalofFinance40,1219-1233. 124

PAGE 125

Rockafellar,R.T.,Uryasev,S.(2002a)ConditionalValue-at-RiskforGeneralLossDistributions,JournalofBankingandFinance26,14431471. Rockafellar,R.T.,Uryasev,S.,Zabarankin,M.(2002b)DeviationMeasuresinGeneralizedLinearRegression.ResearchReport2002-9.ISEDept.,UniversityofFlorida. Rockafellar,R.T.,Uryasev,S.,Zabarankin,M.(2005a)GeneralizedDeviationsinRiskAnalysis,FinanceandStochastics10(1),51-74. Rockafellar,R.T.,Uryasev,S.,Zabarankin,M.(2005b)MasterFundsinPortfolioAnalysiswithGeneralDeviationMeasures,JournalofBankingandFinance30(2),743-778. Rockafellar,R.T.,Uryasev,S.,Zabarankin,M.(2006)OptimalityConditionsinPortfolioAnalysiswithGeneralizedDeviationMeasures,MathematicalProgramming108(2-3),515-540. Ross,S.A.(1978)ASimpleApproachtotheValuationofRiskyStreams,JournalofBusiness51,453-475. Schweizer,M.(1991)OptionHedgingforSemi-martingales,StochasticProcessesandtheirApplications37,339-363. Schweizer,M.(1995)Variance-OptimalHedginginDiscreteTime,MathematicsofOperationsResearch20,1-32. Schweizer,M.(2001)AGuidedTourThroughQuadraticHedgingApproaches.InE.Jouini,J.Cvitanic,andM.Musiela(Eds.),OptionPricing,InterestRatesandRiskManagement,538-574.Cambridge:CambridgeUniversityPress. Shadwick,W.andKeating,C.(2002)AUniversalPerformanceMeasure.JournalofPortfolioManagement,Spring2002,59-84. Tsitsiklis,J.,VanRoy,B.(2001)RegressionMethodsforPricingComplexAmerican-StyleOptions,IEEETrans.NeuralNetworks12(4)(SpecialIssueoncomputationalnance),694-703. Tsitsiklis,J.,VanRoy,B.(1999)OptimalStoppingofMarkovProcesses:Hilbertspacetheory,approximationalgorithms,andanapplicationtopricinghigh-dimensionalnancialderivatives,IEEETrans.AutomaticControl44,1840-1851. Wu,J.,Sen,S.(2000)AStochasticProgrammingModelforCurrencyOptionHedging,AnnalsofOperationsResearch100,227-249. 125

PAGE 126

SergeySarykalinwasbornin1982,inVoronezh,Russia.In1999,hecompletedhishighschooleducationinHighSchool#15inVoronezh.Hereceivedhisbachelor'sdegreeinappliedmathematicsandphysicsfromMoscowInstituteofPhysicsandTechnologyinMoscow,Russia,in2003.InAugust2003,hebeganhisdoctoralstudiesintheIndustrialandSystemsEngineeringDepartmentattheUniversityofFlorida.HenishedhisPh.D.inindustrialandsystemsengineeringinDecember2007. 126