
Citation 
 Permanent Link:
 https://ufdc.ufl.edu/UFE0021770/00001
Material Information
 Title:
 Optimization Methods in Financial Engineering
 Creator:
 Sarykalin, Sergey Vlad
 Place of Publication:
 [Gainesville, Fla.]
Florida
 Publisher:
 University of Florida
 Publication Date:
 2007
 Language:
 english
 Physical Description:
 1 online resource (126 p.)
Thesis/Dissertation Information
 Degree:
 Doctorate ( Ph.D.)
 Degree Grantor:
 University of Florida
 Degree Disciplines:
 Industrial and Systems Engineering
 Committee Chair:
 Uryasev, Stanislav
 Committee Members:
 AitSahlia, Farid
Karceski, Jason J. Rockafellar, Ralph T.
Subjects
 Subjects / Keywords:
 Assets ( jstor )
Call options ( jstor ) Capital asset pricing models ( jstor ) Financial portfolios ( jstor ) Hedging ( jstor ) Market prices ( jstor ) Prices ( jstor ) Pricing ( jstor ) Put options ( jstor ) Stock prices ( jstor ) Industrial and Systems Engineering  Dissertations, Academic  UF capm, deviation, omega, option, trading, vwap
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) borndigital ( sobekcm ) Electronic Thesis or Dissertation Industrial and Systems Engineering thesis, Ph.D.
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 CVaRregression were developed for this purpose. Next, we consider assumptionfree algorithm for pricing European Options in incomplete markets. A nonselffinancing 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. ( en )
 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.
 Thesis:
 Thesis (Ph.D.)University of Florida, 2007.
 Local:
 Adviser: Uryasev, Stanislav.
 Statement of Responsibility:
 by Sergey Vlad Sarykalin.
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 UFRGP
 Rights Management:
 Copyright Sarykalin, Sergey Vlad. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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 LD1780 2007 ( lcc )

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Full Text 
Table 31. Prices of options on the stock following the geometric Brownian motion:
calculated versus BlackScholes prices.
Strike Calc. BS 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
BS.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 riskfree 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 MonteCarlo simulation.
Strike($) option strike price, Calc. obtained option price (relative), BS BlackScholes
option price (relative), Err=(Found BS)/BS, Calc.Vol. obtained option price in
volatility form, BS.Vol.( ) BlackScholes volatility,
Vol.Err(,) (Calc.Vol. BS. Vol.)/BS. Vol.
12 00%
10.00oo
 Day 1 7r Day 2
8.00% .
oDay 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 23. Daily volume distributions
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.
Proof: Condition (524) can be expressed as
1
Q0(w)>1 SD
SM
where SV Consider a rescaled 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(1A)+AQ
(1 A) + A argmin cov(
QEQ
(1 A) + A argmin cov(
QEQ
' (1
I Q)
A) + AQ)
S (1 A)+ AQ(, ,).
Finally, if (525) holds for D, it holds for AD as well, since
(1 A) + AQv()
AQv(Lw)
QVM(L)
> 1
> 1
> AA
> A
>
S3
(525)
Table 23. 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 24. 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 25. 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
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 CVaRregression were developed for
this purpose.
Next, we consider assumptionfree algorithm for pricing European Options in
incomplete markets. A nonselffinancing 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.
Comparison at step 1. C'! .... ; the model may entail modelling error. For
example, stocks are approximately follow the geometric Brownian motion. However, the
BlackScholes 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, nonselffinancing hedging strategy, and possibility to introduce transaction costs
(this feature is not directly presented in the paper).
Calibration of processspecific 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 processspecific
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 processspecific methods would contain a
significant modelling error.
Comparison at step 3. To perform backtesting, the hedging strategy, implied by a
pricing method, is implemented on historical price paths. The backtesting hedging error is
a measure of practical usefulness of the algorithm.
The major advantage of our algorithm is that the errors of backtesting in our
case can be much lower than the errors of processspecific methods. The reason being,
the minimization of the backtesting 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
processspecific methods possess this property.
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.
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 BlackScholes
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 (37).
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 (324)(327) for put options.
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 noncoherent 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 mixedCVaR deviation of gains and losses. The risk identifier of
CVaR and mixedCVaR deviations of losses are derived and coherence of these deviation
measures is examined.
The algorithm uses the hedging portfolio to approximate the price of the option. We
aimed at making the hedging strategy close to reallife trading. The actual trading occurs
at discrete times and is not selffinancing at rebalancing points. The shortage of money
should be covered at any discrete point. Large shortages are undesirable at any time
moment, even if selffinancing 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 riskfree bond and use its value as an approximation to the
option price. We aimed at making the hedging strategy close to reallife trading. The
actual trading occurs at discrete times and is not selffinancing at rebalancing points.
The shortage of money should be covered at any discrete point. Large shortages
are undesirable at any time moment, even if selffinancing is present. We consider
nonselffinancing hedging strategies. External financing of the portfolio or withdrawal
is allowed at any rebalancing 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 rebalancing 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 "selffinancing on avi I, We incorporated monotonic,
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 ahiv
outperforms portfolio A. This follows from nonarbitrage 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 (er.(T) r(Tu))
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.
Formula (510) is the riskadjusted pricing form of generalized CAPM relations (52)
(compare with (510)), where the riskadjusted 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 (58) and (510) 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 (512) holds.
Eri ro
If cov(ri, QD) < 0, then Er, < ro, therefore < 0, and (512) 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
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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 (531) 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(), (532)
where p(r) f,, p(r', r', ..., r, f)dr[...dr'dr is the actual marginal distribution of the
master fund.
Relationship (529) 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 riskneutral 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 riskneutral
distribution, q(r') = q(r', r ...., r', r)dr...dr'dr. Applying formula (530) 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.
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 riskfree 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) (51)
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 (51) has three different
types of solution depending on the magnitude of the riskfree 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 CAPMrelations
cov(rs,, Qj)
Eri ro = D(r) [E I ro], i = ..., n, (52)
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 worstcase deviation and semideviations, see
Rockafellar et al. (2006)
Under assumptions Al and A2, all quantities in generalized CAPM relations are fixed
and welldefined, 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 riskneutral world,
the asset will be priced as follows.
= E[ (53)
1 + ro
where ro is a riskfree rate of return. The price of an asset is the discounted expected
value of its future p ivoff. The asset with random p ioff ( 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 ioff (, such that
E[(] = The formula (53) 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), (54)
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 ioff of a riskfree asset having the same price as the risky asset with
1p 'ioff (.
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.
(44)
The constraint zfi , x 1 allows to rewrite the set of constraints
bl< xj < b, m= ...,M, (45)
J Jr,
li < Xi < Ui, i = 1, (46)
in the following form
I I
b x s< xj ,< b zx, m ..., M, (47)
i= 1 jEJ i 1
I I
Sj xi < Xi < uiy xi, i = 1, ..., 1. (48)
i 1 i 1
For any x satisfying (47)(48), Ax for A > 0 will also satisfy (47)(48). Therefore,
constraints (47)(48) are special case of the constraints of type (43).
According to Theorem 1, the problem (44) can be reduced to the following problem.
For the case of scenarios (29), the optimization problem is
is
min DMAD S Ys cX cX d (212)
2.3.2 CVaRobjective
The objective we used in the second regression model will be referred to as CVaR
objective. Meanabsolute deviation equally penalizes all outcomes of the approximation
error (210), however our intention penalize the largest (by the absolute value) outcomes
of the error. To give a more formal definition of the CVaRobjective 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).
CVaRobjective consists of two CVaRdeviations (Rockafellar et al. (2005a)) and
penalizes the ahighest and the alowest outcomes of the estimation error (210) for
a specified confidence level a (a is usually expressed in percentages). We will use a
combination of CVaRdeviations as an objective:
DCVaR(c) = CVaRf(c) + CVaR(e) = (213)
= 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 (28) and the minimization (213)
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 (28)
is nonbiased.
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 21 demonstrates the two representations of the volume
distribution. Note, that WN is al i equal to 1. There is a onetoone correspondence
between representations (vi, ..., Nv) and (w1, ..., wN); the transitions between them are
given by formulas
wi "t, k= 2,..., N (26)
1 ii 1
and
k1
Vi1 =W1, vk (1 t), k 2,...,N. (27)
i=1
The last equations follow from the fact that
(1 (1 0i'_ ) ..." (1 *t _. ) t= ... .m = 1,...,i 1.
Vim +  + 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
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 MeanAbsolute Error ................... .... 18
2.3.2 CVaRobjective ................... ..... 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
to be nonselffinancing and minimize cumulative hedging error over all sample paths.
The constraints on the hedging strategy, incorporated into the optimization problem,
reflect assumptionfree properties of the option price, positions in the stock and in the
riskfree 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 nonlinear 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.
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 nonarbitrage 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 AmericanI 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.
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
Tables 31, 33, and 34 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 BlackScholes formula.
3.5.1 Pricing European options on the stock following the geometric brown
ian motion
We used a MonteCarlo 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 BlackScholes
prices for European call options are presented in Table 31.
Table 1 shows quite reasonable performance of the algorithm: the errors in the price
(Err( .), Table 31) 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 31). The volatility error slightly increases for outofthemoney
puts and inthemoney 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 32. 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 samplepaths. Nonoverlapping 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 atthemoney strike is priced correctly. This set of
paths with the adjusted volatility was used to price options with the remaining strikes.
Table 33 dipl'i1 the results of pricing using 100 historical samplepaths. The pricing
error (see Err( .), Table 33) is around 1.0' for all call and put options and increases
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 riskneutral densities, we have to assume
that no arbitrage opportunities exist in the prices of options from the crosssection. 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 riskneutral
density (T(rM) = (1 + ro)m (rM)p(rM) can be negative, and the hypothesis that the true
riskneutral densities qt(rM(u)) > 0 (estimated from options crosssection) 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 CVaRDeviation
One of the flexible parameterizations of a deviation measure is mixed CVaRdeviation
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 CVaRdeviation of gains
and losses
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 15
and a nonarbitrage 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 BlackScholes 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 BlackScholes formula and
calculating the areas of horizontal monotonicities for the options used in the case study.
The BlackScholes 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 riskfree rate, a is the volatility,
1 fy z2
N(y) =e 2 dZ, (329)
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
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 (39) for pricing
European put options.
Constraints on value of Put options.
Immediate (::; i, . constraints.
P > Xe 'T sk (319)
Option price sensitivity constraints.
pk
(320)
j 0,...,N 1, k = 1,...,K 1.
Monotonicity constraints.
0. Vertical monotonicity.
P j 1 j 0,...,N; k 1,..., K (321)
0. Horizontal monotonicity.
Pl < Pk + X(er(Ttj+l) e(Ttj)), = 0,...,N 1; k =1,...,K. (322)
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 (323)
j = 0,...,N; k = ,...,K 2.
preferences of investors holding this index. Derivatives on assets have nonlinear p lioffs
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 soobtained 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
indexassociated 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
riskfree 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 indexassociated 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
Table 33. 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~ riskfree 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.
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 (58) and (510) 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
CVaRdeviation 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 ), (513)
where the lefthand 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 (514) 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 riskfree 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 lioff 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
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}er")2 (39)
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 (36),
constraints (310)(318) (defined below) for call options,
or constraints (319)(327) (defined below) for put options,
free variables: U k, j = 0,..., N, k= 1,..., K.
The objective function in (39) is the average squared error on the set of paths (35). 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 (36). The total number of free variables in the problem is determined
by the size of the grid and is independent of the number of samplepaths. 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
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 (523)
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 ) (524)
Note that the lefthand side of condition (524) contains a random variable, while the
righthand 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 lefthand side. We show next that it does not change meaning of the condition (524).
Lemma 3. Condition (524) is invariant with respect to re. Al1.:, deviation measure D.
CAPM relations by pricing assets from the indexassociated 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 ioff 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 indexassociated 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
5.3.2 Derivation of Discount Factor for Generalized CAPM Relations
We begin by rewriting CAMPlike 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 (517)
i = E[m' (i], i 0,1,...,n. (522)
The discount factor corresponding to the deviation measure D is given by (521).
Pricing formulas (522) 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 (522) to determine a unique price of
a new asset. An investor has risk related to imperfect replication of the lp ioff of a new
asset, and specifies his risk preferences by choosing a deviation measure in pricing formula
(522).
ErrM ro EQ,
+1) 1 ,
(520)
(521)
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 Omegaoptimal portfolio with respect to a fixed benchmark with
linear constraints on portfolio weights leads to a nonlinear optimization problem. Several
approaches to solving this problem has been proposed, among which are the global
optimization approach in AvouyiGovi 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 nonlinear methods are s, i.. 1. 1 otherwise.
This paper investigates reduction of the Omegabased 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 Omegamaximizing 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 nonconvex constraint.
The second way is to modify the discounting coefficient:
7 = E[(], rra(asset) 1. (55)
1 + rra(asset)
where rra(asset) is the riskadjusted 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. (56)
1+ro o /
RB1: dliusted form of CAPM:
S= E []. (57)
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 riskreturn 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 (52), we get
C  rrrrr 
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 33.
nr nnl
3 .UU
S20.00
. 15.00
o
10.00
C 5.00
0 00
Implied volatility vs. strike: Call options in BlackScholes setting priced using
200 sample paths. Based on prices in columns Calc. Vol(..) and BS. Vol(..) of
Table 31.
Calculated Vol(.) = implied volatility of calculated options prices (200
samplepaths), Actual Vol( .) = flat volatility implied by BlackScholes
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 llir.l'.., sifted to tarlt from zero)
IActualVol  Calculated Vol
Figure 34.
Implied volatility vs. strike: Put options in BlackScholes setting priced using
200 sample paths. Based on prices in columns Calc. Vol(.. ) and BS. Vol(.. ) of
Table 31.
Calculated Vol(.) = implied volatility of calculated options prices (200
samplepaths), Actual Vol( .) = flat volatility implied by BlackScholes
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
0.05
0.10
strike (relative, shifted to start from zero)
IActual Vol  Calculated Vol
Figure 31.
35.00
30.00
25.00
20.00
15.00
10.00
5.00
nn
000
0.00
Figure 32.
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 33.
Calculated Vol(.) = implied volatility of calculated options prices (100
samplepaths), 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)
IActual VolA 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 33.
Calculated Vol(.) = implied volatility of calculated options prices (100
samplepaths), 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~~
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 putcall parity C(S, T, X) P(S, T, X) + X eT = 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 putcall 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 Putcall 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 BlackScholes formula. Second, we price
European options on S&P 500 index (ticker SPX) and compare the results with actual
market prices.
Table 21. 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 22. 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
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 (21)
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 (22)
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.
(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. AitSahalia 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 twodimensional grid and impose
constraints on the grid variables. These constraints follow under some general assumptions
from nonarbitrage considerations. Some of these constraints are taken from Merton
(1973).
MonteCarlo 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). Regressionbased
approaches in the framework of MonteCarlo 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 MonteCarlo simulation. Glasserman (2004),
showed that regressionbased approaches are special cases of the stochastic mesh method.
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 nonarbitrage assumptions we have C(SI, T, X) <
C(S1, T, X1). Applying scaling property to the righthand 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, nonarbitrage 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).
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
realization of this density at each point in time; direct estimation of the density is not
possible.
However, the formula (532) 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 (535). Formula
(533) 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 pvalue of an
appropriate statistic.
This hypothesis is further transformed as follows. Using the true riskneutral
distributions qt(rM), the actual distributions pt(rM) are estimated using (533),
4t(rM)
<(rM) (1 + ro) (FM)'
t 1, ..., T. We then test the null hypothesis that riskneutral distributions pf(rM),
t = 1,..., T, equal to the true riskneutral 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.
approximation rules (36). The price of the option is the initial value of the hedging
portfolio, calculated as uoSo + ,,
The following constraints (310)(318) for call options or (319)(327) 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 nonarbitrage 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 BlackScholes 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
riskfree 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 (310)
2 European options do not have the feature of immediate exercise. However, the right
part of constraint (310) coincides with the immediate exercise value of an American
option having the current stock price S and the strike price Xei(T).
where 7 and ( are the price and the p ,,off of an asset. Letting
qD(o) = (1 + ro)mn'()p(w), (529)
we get
t= j ()D(w)dw. (530)
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 (529) 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
(531)
Let
(~) q( ,ri,.. ,r)d...drIddrr.
If ma was strictly positive, qD( 7) would be a riskneutral marginal distribution of the
master fund. To simplify the righthand side of (531), 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 nonzero coefficient, we can represent w
as Lc (r, r,.., r,, ..., r, r ). After including r2,..., r, to f, we get the representation
= Iwk
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 35. BlackScholes call option: distribution of the total external financing on
sample paths.
Initial price= '.," strike= ~. time to expiration 70, riskfree rate 10' ,
volatility 21 I'
Stock price is modelled with 200 MonteCarlo sample paths.
3000
S2500
2000
3 1500
a 1000
u. 500
Dollars
Figure 36. BlackScholes call option: distribution of discounted inflows/outflows at
rebalancing points.
Initial price='., strike= ~. time to expiration 70, riskfree rate 10'.
volatility 21 I'
Stock price is modelled with 200 MonteCarlo sample paths.
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 (239)
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 21, 22 show the results for the meanabsolute 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 (221) 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 CVaRobjective and mixed objective (Tables 23, 24), different values
of L, S and Sbest yielded a similar order of superiority as in the case of the meanabsolute
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 nonsimilar 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.
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 (43) 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 Pgq1.
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
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 (232). 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 (221) 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 Plos (238)
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.
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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) (58)
1 + ro ((m
Pricing formula (58) the certainty equivalent pricing form of generalized CAPM
relations (52) (compare it to (54)), where the certainty equivalent
Er D ro
Q((,) E( + (r) cov(, QM) (59)
is the p ioff of a riskfree asset having the same price 7rj.
We could rearrange the formula (52) 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 = (510)
1 + ro + D(r,) cov( r, Q)
ErD ro
when D(rM) cov(ri, Q) / 1 + r0.
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(mX) + E)(] E[proj(mX) ].
It should be mentioned that the existence of socalled I:1lneutral" measure is
justified by the existence of a strictly positive discount factor. Indeed, we can rewrite
(517) as follows.
S= r[ii] /n m()((u)dP(u) = (()dQ(u), (518)
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 riskneutral pricing form of (517) is
1
1 + ro
where EQ[.] denotes expectation with respect to the riskneutral measure.
If one picks a discount factor m, which is not strictly positive, the transformation
(518) 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 (517) to the riskfree rate gives 1 = E[m(1 + ro)].
4.2.2 General Problem
This paper deals with solving the following nonlinear 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.
CHAPTER 4
METHODS OF REDUCING MAXIMIZATION OF OMEGA FUNCTION TO LINEAR
PROGRAMMING
4.1 Introduction
The classical meanvariance 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 riskfree 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, Sharpeoptimal portfolios can produce highly
nonoptimal returns. Critique of the classical approach to the portfolio management is
based on the fact that the mean and the variance of a nonnormal 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 higherorder
moments into portfolio analysis leads to more accurate solutions. One of the areas in
which the meanvariance 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 higherorder moments of return distributions in
order to adequately represent hedge fund risk.
One of the alternatives to the meanvariance 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,
* 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. (312)
0. Horizontal monotonicity. The price of an option is a decreasing function of
time.
C +1 j 0, j ..., N 1; k= 1,...,K. (313)
* 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. (315)
* 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. (316)
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 nonarbitrage 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 (39) is 2PN
and the number of constraints is O(NK), regardless of the number of sample paths. Table
36 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, MonteCarlo simulation, and other methods are usually used for
pricing.
Step 3: Backtesting. 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 (BlackScholes model, stochastic
volatility model, jumpdiffusion model, etc). We will refer to these methods as processspecific
methods. In order to judge the advantages of the proposed algorithm against the
processspecific methods, we should compare them step by step.
Table 34. 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~ riskfree 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.
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)) (526)
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().
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. (325)
Horizontal monotonicity
UT < Uk, if k > uk > U if k < k (326)
Convexity constraints
(1 3+1)U+2 + 3+1k < Uk+1, if k > k,
(1 /1)uk2 +2 1 > U1, if k < ,
(327)
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 (39) we used the following
constraints holding for options in quite a general case. We assume nonarbitrage and make
technical assumptions 15 (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 (310)(314) for call options. Some of the considered properties of option
stated as follows
I
min /DMAD + YA DVa3
i= (218)
subject to constraints in (217),
where / E [0, 1], i 1,..., I, 3 + Ei / 1.
In our experiments, we used convex combinations of two CVaRobjectives, one with
the confidence level 50'.
min 3. D R + (1 ) D
(219)
subject to constraints in (217),
and of the meanabsolute error function and the CVaRdeviation:
min 3. DMAD + (1 /) Dvan
(220)
subject to constraints in (217) without the first one,
where the balance coefficient 3 E [0, 1]. For comparison, different types of deviations are
presented on Figure 22.
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 k1),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 ) (21)
= 1
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 riskfree bond. The stock price is modelled by
a set of samplepaths generated by a MonteCarlo or historical bootstrap simulation.
We consider a nonselffinancing portfolio dynamics and minimize the sum of squared
additions/subtractions of money to/from the hedging portfolio at every rebalancing
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 riskfree bond, can incorporate various market frictions, such
as transaction costs and trading restrictions. For incomplete markets, replicationbased
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. AsiaP i..: I. Financial Markets, 11(3), 301333.
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 (221).
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(k1), ",k1),, I 1,..., L, we define the "dIi ii' between the
current dv and the scenario s in the following way:
D= P (p P 1)8,)2. (228)
i=l 1l=1
After calculating distances to all S scenarios, we choose Sbest closest scenarios corresponding
to lowest values of Di in (228). 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 5minute 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
3.4.3 Justification of Constraints on Stock Position
This subsection proves/validates inequalities (315)(318) and (324)(327) on
the stock position. Stock position bounds and vertical monotonicity are proven in the
general case (i.e. under assumptions 15 and the nonarbitrage 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 (315)(318) 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 STmax{0, ST
X} > 0 for any ST and X > 0. Nonarbitrage assumption implies that S > C(S, t, X).
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
(221).
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}. (222)
Denote the linear combination
L
1 k1),s ++ Pk)s) (223)
l 1
as wf, the collection of 7 _k as '.
In our study we consider the following optimization problems:
Pl: MAD
min Ewk kw (224)
P2: CVaR
mint CVaR (wk wk) + CVaR (wC w)
(225)
s.t. E[wk] E[,k]
P3: MAD+CVaR
mmn SEw k + (1 3) (CVaR(k k) + CVaR(k w k)) (226)
P4: Mixed CVaR
mint 3 (CVaRso(0k wk) + CVaRso%(WC wk)) +
+(1 3) (CVaR (wk k) + CVaR (wk wk)) (227)
s.t. E[wk] =E[w],
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 (34)
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} (35)
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)
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 distributionfree, 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 timecontinuous 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 distributionfree and is based
on realistic assumptions, such as discrete trading and nonselffinancing hedging strategy.
Another advantage of our algorithm is low backtesting errors. Timecontinuous
models do not account for errors of implementation on historical paths. The objective in
our algorithm is to minimize the backtesting errors on historical paths. Therefore, the
algorithm has a very attractive backtesting performance. This feature is not shared by
any of timecontinuous models.
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 CAPMlike 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 CAPMlike relations and consider pricing
forms of generalized CAPM relations. We propose a method of calibrating deviation
measures from market data and discuss viv 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 aalgebra 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
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 (310)(314) for call options and (319)(323) for
put options follow under quite general assumptions from the nonarbitrage considerations.
The type of the underlying stock price process pl .1' no role in the approach: the set
of sample paths (35) specifies the behavior of the underlying stock. For this reason,
the approach is distributionfree 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 (39) 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 (39) 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 (316)(318) 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 (325)(327) 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
LIST OF TABLES
Table page
21 Performance of tracking models: stock vs. stock+index, full history regression .28
22 Performance of tracking models: stock vs. stock+index, best sample regression 28
23 Performance of tracking models: mixed objective, changing size of history and
best sam ple .................. ................... .. 29
24 Performance of tracking models: CVaR deviation, changing size of history and
best sample . ............... ..... .... 29
25 Performance of tracking models: mixed objective ... . . 29
31 Prices of options on the stock following the geometric Brownian motion: calculated
versus BlackScholes prices. .................. .. ...... 68
32 S&P 500 options data set. .................. .. ........ 69
33 Pricing options on S&P 500 index: 100 paths .................. 70
34 Pricing options on S&P 500 index: 20 paths ................ 71
35 Summary of cashflow distributions for obtained hedging strategies presented on
Figures 3.6, 3.6, 3.6, and 3.6. .................. ..... 72
36 Calculation times of the pricing algorithm. .................. .... 72
37 Numerical values of inflexion points of the stock position as a function of the
stock price for some options. .................. .. ...... 72
41 Optimal allocation .................. . . .. 88
values are not included in the constraints of the optimization problem (39), they are
used in proofs of some of constraints (310)(314). 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 (319)(323) 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
(310)(314) and (319)(323).
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 (310)(314).
1. "Immediate (::; i, constraints. Merton (1973)
C > [St X erTt +.
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 VWAPbenchmark 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.
X() = 0 otherwise. Then,
Q(X) =argminE[XQ] argminE[ls Q] argminE[ls Q] C Q, (541)
QEQ QEQ QEQs
which contradicts with the condition Q > 0 for all Q E Q(X), as required by (540). This
concludes the proof.
Risk identifiers (539) implies that the deviation measure Q3(X) = CVaR3(X +
EX) is coherent if 2 01 > 0, which is equivalent to having f > 1/2.
Now consider the mixedCVaR 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).
(542)
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
(534)
= Sq(S)dS K q(S)dS.
K K
Differentiating (534) with respect to K, we get
r = K(K) (S)dS + Kq(K) q(S)dS.
OK JK JK
Differentiating (534) twice with respect to K, we arrive at the formula for estimating
riskneutral 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 (535)
OK2 K=S
Formula (535) allows to estimate the function q( ,) when the crosssection 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.
Properties (Q1) and (Q3) of Q follow immediately from properties of Q. To prove
property (Q2) we need to show that for each nonconstant X there exists Q E Q so
that E[XQ] < EX. Indeed, fix a nonconstant 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 (538), 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 (538), the risk envelope for the deviation measure D(X) = CVaRg(X + EX)
Q= Q 2a1
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) = Q1,
Q e Q(X) < 0 Q(w ) a1,
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).
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 ,ioff 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
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 (31), 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 (35) as an average of squared
errors Sp over all sample paths (35)
P N
S ^ i>ri, (38)
p=1 j=1
The matrices [Utf] and [Vjk] and the approximation rule (36) 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 (36) 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.
To my parents.
convex, and some other properties of option prices following from the definition of an
option, a nonarbitrage assumption, and some other general assumptions about the
market. We do not make assumptions about the stock process which makes the algorithm
distributionfree. 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 MonteCarlo
samplepaths. Calculated option prices are compared with the known prices given by the
BlackScholes 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 samplepaths (considered cases
include 100 and 20 samplepaths). 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 BlackScholes setting. The implied volatility for S&P 500 index
options (priced with 100 samplepaths) 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
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.
Therefore Q E Q(X) is equivalent to having
Q(w) = a, when X(w) < VaR,(X)
<0 < Q(w) < a1, when X() VaR,(X)
Q(w) = 0, when X(w) > VaR,(X),
or
Q(w) = a, when X(w) > VaR,(X)
0 < Q(L) < a1, 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 31 when X(w) > VaR3(X)
Q Q(X) 2 j < 2 3 < Q( ) < 2, when X(w) VaR,(X) (539)
Q(w) = 2, when X(w) < VaR,(X).
Next, we examine coherence of CVaR and mixedCVaR 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 nonnegativity 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. (540)
Proof: If D is coherent, then (540) holds since Q(X) E Q for any X.
To prove the converse statement, we need to show that (540) 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
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.
for outofthemoney options. Errors of implied volatility follow similar patterns: errors
are of the order of 1 for all options except for deep outofthemoney options. For deep
inthemoney 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 samplepaths
is available. (Table 34 shows that inthemoney S&P 500 index options can be priced
quite accurately with 20 samplepaths.) At the same time, the method is flexible enough
to take advantage of specific features of historical samplepaths. 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
BlackScholes 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 samplepaths.
This observation is in accordance with the prior known fact that the nonnormality 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 erj) on sample paths and distributions of discounted money inflows/outflows
(aperj) at rebalancing points for BlackScholes and SPX call options. We summarize
statistical properties of these distributions in Table (35).
Figures 3.6, 3.6, 3.6, and 3.6 also show that the obtained prices satisfy the nonarbitrage
condition. With respect to pricing a single option, the nonarbitrage 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 nonarbitrage condition in this sense. The value of external
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
nonarbitrage 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 arbitragefree 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).
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 (52) 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 (52) 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 CVaRdeviation with confidence level a; C = Ei for mean absolute
deviation and semideviations. Since asset prices in generalized CAPM (52) depend on the
risk identifier Q' though (ri, Q'), assumption A2 ,.. that Prob{r = C} = 0.
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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 (28)
of indicator variables Xi, ..., X,. In our study the variables are modelled by a set of
scenarios
{(Y"; X, ...,X) S =l,...,S} (29)
For a scenario s the approximation error is
e Y" ciX' ... cX d. (210)
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 MeanAbsolute Error
In the first regression model, the minimized objective is the meanabsolute error of
the approximation (28)
DMAD(C) = EBec. (211)
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 riskfree 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 riskfree bond. The hedging portfolio is
rebalanced at times tj, j 1, ..., N 1. Suppose that at the time tj_ the hedging portfolio
consists of uj1 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)vj1. 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 nonselffinancing portfolio dynamics by allowing the
difference
aj = ujSj + vj (ujSj + (1 + r)vj1) (31)
to be nonzero. 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 [tj1, 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.
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 nonarbitrage 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, (328)
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 )
In Table 25 we changed the form of the mixed objective, that is, differed 3 and a in
(217). 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.
Thus, the regression problem takes the form:
min,d CVaR [Y Y] + CVaR [F Y]
s.t. E [Y] E[Y] (214)
Y = ciXi + d.
Since
a 1
CVaR(1_) [X] = CVaR, [X]  E[X], (215)
optimization program (214) becomes:
minc,d aCVaRa [Y Y] + (1 a)CVaRi [Y Y]
s.t. E [Y] E[Y] (216)
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 (29) the optimization problem (216) 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
XIa > Ia + zS_ 1Z Eas (217)
> Y" (E c~x2 + d)
z a> vs (YE cXr + d) a1
Variables: c, d ER for i =1, ..., n; Xa, Xia 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 meanabsolute error function and CVaRobjectives with different confidence levels a.
Denote the objective in (214) by DvjaR, then the problem with the mixed objective is
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 atthemoney options on S&P500 Index; (c)
closely matches the market volatility smile; (d) is able to price options using 2050 sample
paths. We use replication idea to find option price, however we allow the hedging strategy
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. (517)
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 (.
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 Dq1 = {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.
Harrison, J. M. and Kreps, D. M. (1979) Martingales and Arbitrage in Multiperiod
Securities Markets. Journal of Economic Theory 20, 381408.
Jackwerth, J. C. (2000) Recovering Risk Aversion From Option Prices and Realized
Returns. The Review of Financial Studies 13(2), 433451
Joy, C., Boyle, P., and Tan, K.S.(1996) Quasi Monte Carlo Methods in Numerical
Finance, .i.ri,, ,, .ul. Science 42, 926936.
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), 3351.
Konishi, H. (2002) Optimal Slice of a VWAP Trade. Jourmal of Financial Markets
5, 197221.
Levy, H. (1985) Upper and Lower Bounds of Put and Call Option Value: Stochastic
Dominance Approach, Journal of Finance 40, 11971217.
Longstaff, F., Schwartz, E. (2001) Valuing American Options by Simulation: A
Simple LeastSquares Approach, A Review of Financial Studies 14(1), 113147.
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), 141184.
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. 8590.
Perrakis, S., Ryan, P. J. (1984) Option Pricing Bounds in Discrete Time, Journal of
Finance 39, 519525.
Ritchken, P. H. (1985) On Option Pricing Bounds, Journal of Finance 40,
12191233.
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

Full Text 
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IwanttothankmyadvisorProf.StanUryasevforhisguidancesupport,andenthusiasm.Ilearnedalotfromhisdeterminationandexperience.IwanttothankmycommitteemembersProf.JasonKarseski,Prof.FaridAitSahlia,andProf.R.TyrrellRockafellarfortheirconcernandinspiration.IwanttothankmycollaboratorsVladBugeraandValeriyRyabchekno,whowerealwaysgreatpleasuretoworkwith.Iwouldliketoexpressmydeepestappreciationtomyfamilyandfriendsfortheirconstantsupport. 4
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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 11 2TRACKINGVOLUMEWEIGHTEDAVERAGEPRICE ............ 13 2.1Introduction ................................... 13 2.2BackgroundandPreliminaryRemarks .................... 15 2.3GeneralDescriptionofRegressionModel ................... 18 2.3.1MeanAbsoluteError .......................... 18 2.3.2CVaRobjective ............................. 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
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.................................. 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 5.5CoherenceofMixedCVaRDeviation ..................... 116 5.6Conclusions ................................... 121 REFERENCES ....................................... 122 BIOGRAPHICALSKETCH ................................ 126 6
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Table page 21Performanceoftrackingmodels:stockvs.stock+index,fullhistoryregression 28 22Performanceoftrackingmodels:stockvs.stock+index,bestsampleregression 28 23Performanceoftrackingmodels:mixedobjective,changingsizeofhistoryandbestsample ...................................... 29 24Performanceoftrackingmodels:CVaRdeviation,changingsizeofhistoryandbestsample ...................................... 29 25Performanceoftrackingmodels:mixedobjective ................. 29 31PricesofoptionsonthestockfollowingthegeometricBrownianmotion:calculatedversusBlackScholesprices. ............................. 68 32S&P500ptionsdataset. ............................... 69 33PricingoptionsonS&P500index:100paths ................... 70 34PricingoptionsonS&P500index:20paths .................... 71 35SummaryofcashowdistributionsforobitainedhedgingstrategiespresentedonFigures 3.6 3.6 3.6 ,and 3.6 ............................ 72 36Calculationtimesofthepricingalgorithm. ..................... 72 37Numericalvaluesofinexionpointsofthestockpositionasafunctionofthestockpriceforsomeoptions. ............................. 72 41Optimalallocation .................................. 88 7
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Figure page 21Percentagesofremainingvolumevs.percentagesoftotalvolume ........ 30 22MAD,CVaR,andmixeddeviations ........................ 30 23Dailyvolumedistributions .............................. 31 31Impliedvolatilityvs.strike:CalloptionsonS&P500indexpricedusing100samplepaths ..................................... 64 32Impliedvolatilityvs.strike:PutoptionsonS&P500indexpricedusing100samplepaths ..................................... 64 33Impliedvolatilityvs.strike:CalloptionsinBlackScholessettingpricedusing200samplepaths ................................... 65 34Impliedvolatilityvs.strike:PutoptionsinBlackScholessettingpricedusing200samplepaths ................................... 65 35BlackScholescalloption:distributionofthetotalexternalnancingonsamplepaths .......................................... 66 36BlackScholescalloption:distributionofdiscountedinows/outowsatrebalancingpoints ......................................... 66 37SPXcalloption:distributionofthetotalexternalnancingonsamplepaths .. 67 38SPXcalloption:distributionofdiscountedinows/outowsatrebalancingpoints 67 8
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Ourstudydevelopednovelapproachestosolvingandanalyzingchallengingproblemsofnancialengineeringincludingoptionspricing,marketforecasting,andportfoliooptimization.Wealsomakeconnectionsoftheportfoliotheorywithgeneraldeviationmeasurestoclassicalportfolioandassetpricingtheories. WeconsideraproblemfacedbytraderswhoseperformanceisevaluatedusingtheVWAPbenchmark.Ecienttradingmarketordersincludepredictingfuturevolumedistributions.SeveralforecastingalgorithmsbasedonCVaRregressionweredevelopedforthispurpose. Next,weconsiderassumptionfreealgorithmforpricingEuropeanOptionsinincompletemarkets.Anonselfnancingoptionreplicationstrategywasmodelledonadiscretegridinthespaceoftimeandthestockprice.Thealgorithmwaspopulatedbyhistoricalsamplepathsadjustedtocurrentvolatility.Hedgingerroroverthelifetimeoftheoptionwasminimizedsubjecttoconstraintsonthehedgingstrategy.Theoutputofthealgorithmconsistsoftheoptionpriceandthehedgingstrategydenedbythegridvariables. AnotherconsideredproblemwasoptimizationoftheOmegafunction.HedgefundsoftenusetheOmegafunctiontorankportfolios.WeshowthatmaximizingOmegafunctionofaportfoliounderpositivelyhomogeneousconstraintscanbereducedtolinearprogramming. 9
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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%pricingerrorforEuropeaninandatthemoneyoptionsonS&P500Index;(c)closelymatchesthemarketvolatilitysmile;(d)isabletopriceoptionsusing2050samplepaths.Weusereplicationideatondoptionprice,howeverweallowthehedgingstrategy 11
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Chapter4provestworeductiontheoremsfortheOmegafunctionmaximizationproblem.Omegafunctionisacommoncriterionforrankingportfolios.Itisequaltotheratioofexpectedoverperformanceofaportfoliowithrespecttoabenchmark(hurdlerate)toexpectedunderperformanceofaportfoliowithrespecttothesamebenchmark.TheOmegafunctionisanonlinearfunctionofaportfolioreturn;however,itispositivelyhomogeneouswithrespecttoinstrumentexposuresinaportfolio.ThispropertyallowstransformationoftheOmegamaximizationproblemwithpositivelyhomogeneousconstraintsintoalinearprogrammingprobleminthecasewhentheOmegafunctionisgreaterthanoneatoptimality. Chapter5looksattheportfoliotheorywithgeneraldeviationmeasuresfromtheperspectiveoftheclassicalassetpricingtheory.Inparticular,weanalyzethegeneralizedCAPMrelations,whichcomeoutasanecessaryandsucientconditionsforoptimalityinthegeneralportfoliotheory.WederivepricingformsofthegeneralizedCAPMrelationsandshowhowthestochasticdiscountfactoremergesinthegeneralizedportfoliotheory.Wedevelopmethodsofcalibratingdeviationmeasuresfrommarketdataanddiscussapplicabilityofthesemethodstoestimationofriskpreferencesofmarketparticipants. 12
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ThereareseveraltypesofbenchmarkssimilartoVWAP.VWAP,asitisdenedabove,isreasonableforevaluationofrelativelysmallordersofliquidstocks.VWAPexcludingowntransactionsisappropriatewhenthetotalvolumeoftransactionsconstitutesasignicantportionofthemarket'sdailyvolume.Forhighlyvolatilestocks,valueweightedaveragepriceisalsoused,wherepricesoftransactionsareweightedbydollarvaluesofthistransactions.VWAPbenchmarksarewidespreadmostlyoutsideUSA,forexample,inJapan. ThepurposeoftheVWAPtradingistoobtainthevolumeweightedpriceoftransactionsasclosetothemarketVWAPaspossible.AninvestormayactdierentlywhenseekingforVWAPexecutionofhisorder.HecanmakeacontractwithabrokerwhoguaranteessellingorbuyingordersatthedailyWVAP.SincethebrokerassumesalltheriskoffailingtoachievetheaveragepricebetterthanVWAPandisusuallyriskaverse,commissionsarequitelarge. ThischapterisbasedonjointworkwithVladimirBugeraandStanUryasev. 13
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Aninvestorwithdirectaccesstothemarketmaytradehisorderdirectly.ButsinceVWAPevaluationmotivatestodistributetheorderoverthetradingperiodandtradebysmallportions,thisalternativeisnotpreferableduetointensityoftradingandthepresenceoftransactioncosts. ThemostrecentapproachtoVWAPtradingisparticipatinginVWAPautomatedtrading,whereatradingperiodisbrokenupintosmallintervalsandtheorderisdistributedascloselyaspossibletothemarket'sdailyvolumedistribution,thatistradedwiththeminimalmarketimpact.Thisstrategyprovidesagoodapproximationtomarket'sVWAP,althoughitgenerallyfailstoreachthebenchmark.Moreintelligentsystemsperformcarefulprojectionsofthemarketvolumedistributionandexpectedpricemovementsandusethisinformationintrading.AmoredetailedsurveyofVWAPtradingcanbefoundinMadgavan(2002). AlthoughVWAPbenchmarkhasgainedpopularity,veryfewstudiesconcerningVWAPstrategiesareavailable.Severalstudies,BertsimasandLo(1998),KonishiandMakimoto(2001)havebeendoneaboutblocktradingwhereoptimalsplittingoftheorderinordertooptimizetheexpectedexecutioncostisconsidered.Inthesetupofblocktrading,onlypricesareuncertain,whereasthepurposeofVWAPtradingistoachieveaclosematchofthemarketVWAP,whichimpliesdealingwithstochasticvolumesaswell.Konishi(2002)developsastaticVWAPtradingstrategythatminimizestheexpectedexecutionerrorwithrespecttothemarketrealizationofVWAP.Astaticstrategyisdeterminedforthewholetradingperiodanddoesnotchangeasnewinformationarrives. 14
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InthischapterwedevelopdynamicVWAPstrategies.Weconsiderliquidstocksandsmallorders,thatmakenegligibleimpactonpricesandvolumesofthemarket.Theforecastofvolumedistributionisthetarget;thestrategyconsistsintradingtheorderproportionallytoprojectedmarketdailyvolumedistribution.Wesplitatradingdayintosmallintervalsandestimatethemarketvolumeconsecutivelyforeachintervalusinglinearregressiontechniques. IfatradingdayissplitintoNequalintervalsf(tn1;tn]jn=1;::;Ng,tn=(n=N)T,whereTisthelengthoftheday,thenthecorrespondingexpressionforthedailyVWAPisgivenby where isthevolumetradedduringtimeperiod(tn1;tn], canbethoughtofasanaveragemarketpriceduringthenthinterval. 15
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Valuesofxnareassumedtobenonnegative(i.e.thetraderisnotallowedtobuystocks). Weconstructthedynamictradingstrategybyforecastingthevolumesofstocktradedinthemarketduringeachintervalofatradingday.Weassumethatduringasmallinterval(about5min)wecanperformtransactionsattheaveragemarketpriceduringthisinterval.Then,from( 2{5 )itfollowsthatapossiblewaytomeetthemarketVWAPistotradetheorderproportionallytothemarketvolumeduringeachinterval,yieldingthesamedailydistributionofthetradedvolumeasthemarket'sone.Foreachintervalofadaywemakeaforecastofthemarketvolumethatwillbetradedduringthisintervalandthentradeaccordingtothisforecast.Attheendofthedayweobtaintheforecastofthefulldailyvolumedistribution;theorderistradedaccordingtothisdistribution. Thewayofdynamiccomputingofthedistributionshouldbediscussedrst.Directestimationsofproportionsofthemarketvolumev1;:::;vNdoesnotguaranteethatthe 16
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21 demonstratesthetworepresentationsofthevolumedistribution.Note,thatwNisalwaysequalto1.Thereisaonetoonecorrespondencebetweenrepresentations(v1;:::;vN)and(w1;:::;wN);thetransitionsbetweenthemaregivenbyformulas and Thelastequationsfollowfromthefactthatwi(1wi1):::(1wim)=Vi 17
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ConsiderthegeneralregressionsettingwherearandomvariableYisapproximatedbyalinearcombination ofindicatorvariablesX1;:::;Xn.Inourstudythevariablesaremodelledbyasetofscenarios Forascenariostheapproximationerroris Weconsiderourregressionmodelasanoptimizationproblemofminimizingtheaggregatedapproximationerror.Belowwedescribepenaltyfunctionsweuseastheobjective. 2{8 ) 18
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2{9 ),theoptimizationproblemis minDMAD=1 2{10 ),howeverourintentionpenalizethelargest(bytheabsolutevalue)outcomesoftheerror.TogiveamoreformaldenitionoftheCVaRobjectiveandshowtherelevanceofusingitinregressionproblems,weeneedtorefertothenewlydevelopedtheoryofdeviationmeasuresandgeneralizedlinearregression,seeRockafellaretal.(2002b). CVaRobjectiveconsistsoftwoCVaRdeviations(Rockafellaretal.(2005a))andpenalizesthehighestandthelowestoutcomesoftheestimationerror( 2{10 )foraspeciedcondencelevel(isusuallyexpressedinpercentages).WewilluseacombinationofCVaRdeviationsasanobjective: (2{13) =CVaR()+CVaR(): 2{8 )andtheminimization( 2{13 )determinestheoptimalvaluesofvariablesc1;:::;cnonly.Theoptimalvalueofthetermdcanbefoundfromdierentconsiderations;weusetheconditionthattheestimator( 2{8 )isnonbiased. 19
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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
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minDMAD+IXi=1iDiCVaRsubjecttoconstraintsin( 2{17 ); wherei2[0;1];i=1;:::;I,+PIi=1i=1. Inourexperiments,weusedconvexcombinationsoftwoCVaRobjectives,onewiththecondencelevel50%: minD50%CVaR+(1)DCVaRsubjecttoconstraintsin( 2{17 ); andofthemeanabsoluteerrorfunctionandtheCVaRdeviation: minDMAD+(1)DCVaRsubjecttoconstraintsin( 2{17 )withouttherstone; wherethebalancecoecient2[0;1].Forcomparison,dierenttypesofdeviationsarepresentedonFigure 22 2.4.1ModelDesign Weconsiderthefollowingregressionmodel 21
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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
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Bysolvingtheseproblems,theoptimalvalueof~isobtained.Theforecastofw0kisthenmadebytheexpression( 2{21 ). AftercalculatingdistancestoallSscenarios,wechooseSbestclosestscenarioscorrespondingtolowestvaluesofDiin( 2{28 ).Bydoingso,weeliminate"outliers"withunusual,withrespecttothecurrentday,behaviorofthemarketwhichfavorstheaccuracyofforecasting. 23
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theforecastedvolumesare thentheestimationerroris Wealsocalculatedanothererror Asabenchmarkmeasuringtherelativeaccuracyofthemodel"averagedailyvolumes"(ADV)strategywasused.ThisverysimplestrategyprovidesagoodapproximationtoVWAP.Supposeasetofhistoricalvolumesofthemarket: Denote Vn=SXs=1Vns;Vtotal=NXn=1Vn:(2{34) Thentheaveragevolumedistributionis (v1;:::;vN);vn=Vn AnexampleofaveragevolumedistributionversustheactualvolumeevolutionispresentedinFigure 23 .Itcanbeseenthatdailyvolumeexhibitsthe"Ushape"andthattheaveragedistributionprovidesagoodapproximationtothedailyvolumeevolution. 24
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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
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lnV;lnPclose Theideaofusingindexinformationcomesfromthefactthatevolutionsofindexandstockarecorrelatedandthattheratiosofreturnsandpricesofstockandindexmayalsocontainusefulinformation. Tables 21 22 showtheresultsforthemeanabsolutedeviationusedasanobjectiveanddierentvaluesofL;SandSbest.ThesetablesshowthatincludingINDEXdatadoesnotimprovetheaccuracyofprediction.Also,asonecannotice,thereisabalancebetweenthenumberoftermsNterm=LPinthelinearcombination( 2{21 )andthenumberofscenarios(Sbest)usedintheregressionmodel.AsNtermincreases,themodelbecomesmoreexibleandmorescenariosareneededtoachievethesamelevelofaccuracy.Forexample,thebesttwomodelsthatusestockdata(P=2),havevaluesofSbestandNtermequal450and4,200and2,respectively.Also,whentheindexdataisused,thenumberofparametersPdoubles,andthenumberofscenariosinthebestmodelsincreasesto700800forthesameregressionlengthL. InthecaseofCVaRobjectiveandmixedobjective(Tables 23 24 ),dierentvaluesofL;SandSbestyieldedasimilarorderofsuperiorityasinthecaseofthemeanabsolutedeviation. Twomorefactscanbeseenfromtheresults.First,thatthemostsuccessfulmodelsuseinformationonlyfromthelastoneortwointervals,whichmeansthattheinformationaboutthefuturevolumeisconcentratedinthepastfewminutes.Second,theideaofchoosingtheclosestscenariosfromtheprecedinghistorydoeswork,especiallywhenasmallportionofnonsimilardays(50or100outof500or800potentialscenarios)isexcluded.Thisagreeswiththeobservationthatmostofthedaysare"regular"enoughtobeusedfortheestimationofthefuture. 26
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25 wechangedtheformofthemixedobjective,thatis,dieredandin( 2{17 ).WefoundthatthebestmodelshaveallweightputontheCVaRobjectiveandforaxedbalancethemodelswithsmallvaluesofaresuperior. ThemostaccuratemodelturnedouttobetheonewithCVaRobjectivehavingtherelativegain4%. 27
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Performanceoftrackingmodels:stockvs.stock+index,fullhistoryregression SSbestLMAD,%SD,%GMAD,%GSD,% STOCK500500234.041.13.53.2500500134.141.43.42.5500500334.241.33.02.8800800234.342.12.70.9800800134.442.52.60.1800800334.442.22.40.5STOCK+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 Table22. 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
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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 Table24. 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 Table25. 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
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Percentagesofremainingvolumevs.percentagesoftotalvolume MAD,CVaR,andmixeddeviations 30
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Dailyvolumedistributions 31
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Below,werefertooptionpricingmethodsdirectlyrelatedtoouralgorithm.AlthoughthispaperconsidersEuropeanoptions,somerelatedpapersconsiderAmericanoptions. Replicationoftheoptionpricebyaportfolioofsimplerassets,usuallyoftheunderlyingstockandariskfreebond,canincorporatevariousmarketfrictions,suchastransactioncostsandtradingrestrictions.Forincompletemarkets,replicationbasedmodelsarereducedtolinear,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.AsiaPacicFinancialMarkets,11(3),301333. 32
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Analyticalapproachestominimizationofquadraticriskareusedtocalculateanoptionpriceinanincompletemarket,seeDueandRichardson(1991),FollmerandSchied(2002),FollmerandSchweizer(1989),Schweizer(1991,1995,2001). Anothergroupofmethods,whicharebasedonasignicantlydierentprinciple,incorporatesknownpropertiesoftheshapeoftheoptionpriceintothestatisticalanalysisofmarketdata.AitSahaliaandDuarte(2003)incorporatemonotonicandconvexpropertiesofEuropeanoptionpricewithrespecttothestrikepriceintoapolynomialregressionofoptionprices.Inouralgorithmwelimitthesetoffeasiblehedgingstrategies,imposingconstraintsonthehedgingportfoliovalueandthestockposition.ThepropertiesoftheoptionpriceandthestockpositionandboundsontheoptionpricehasbeenstudiedboththeoreticallyandempiricallybyMerton(1973),PerrakisandRyan(1984),Ritchken(1985),BertsimasandPopescu(1999),GotohandKonno(2002),andLevi(85).Inthispaper,wemodelstockandbondpositionsonatwodimensionalgridandimposeconstraintsonthegridvariables.Theseconstraintsfollowundersomegeneralassumptionsfromnonarbitrageconsiderations.SomeoftheseconstraintsaretakenfromMerton(1973). MonteCarlomethodsforpricingoptionsarepioneeredbyBoyle(1977).Theyarewidelyusedinoptionspricing:Joyetal.(1996),BroadieandGlasserman(2004),LongstaandSchwartz(2001),Carriere(1996),TsitsiklisandVanRoy(2001).ForasurveyofliteratureinthisareaseeBoyle(1997)andGlasserman(2004).RegressionbasedapproachesintheframeworkofMonteCarlosimulationwereconsideredforpricingAmericanoptionsbyCarriere(1996),LongstaandSchwartz(2001),TsitsiklisandVanRoy(1999,2001).BroadieandGlasserman(2004)proposedstochasticmeshmethodwhichcombinedmodellingonadiscretemeshwithMonteCarlosimulation.Glasserman(2004),showedthatregressionbasedapproachesarespecialcasesofthestochasticmeshmethod. 33
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Thepricingalgorithmdescribedinthispapercombinesthefeaturesoftheaboveapproachesinthefollowingway.Weconstructahedgingportfolioconsistingoftheunderlyingstockandariskfreebondanduseitsvalueasanapproximationtotheoptionprice.Weaimedatmakingthehedgingstrategyclosetoreallifetrading.Theactualtradingoccursatdiscretetimesandisnotselfnancingatrebalancingpoints.Theshortageofmoneyshouldbecoveredatanydiscretepoint.Largeshortagesareundesirableatanytimemoment,evenifselfnancingispresent.Weconsidernonselfnancinghedgingstrategies.Externalnancingoftheportfolioorwithdrawalisallowedatanyrebalancingpoint.Weuseasetofsamplepathstomodeltheunderlyingstockbehavior.Thepositioninthestockandtheamountofmoneyinvestedinthebond(hedgingvariables)aremodelledonnodesofadiscretegridintimeandthestockprice.Twomatricesdeningstockandbondpositionsongridnodescompletelydeterminethehedgingportfolioonanypricepathoftheunderlyingstock.Also,theydetermineamountsofmoneyaddedto/takenfromtheportfolioatrebalancingpoints.Thesumofsquaresofsuchadditions/subtractionsofmoneyonapathisreferredtoasthesquarederroronapath. Thepricingproblemisreducedtoquadraticminimizationwithconstraints.Theobjectiveistheaveragedquadraticerroroverallsamplepaths;thefreevariablesarestockandbondpositionsdenedineverynodeofthegrid.Theconstraints,limitingthefeasiblesetofhedgingstrategies,restricttheportfoliovaluesestimatingtheoptionpriceandstockpositions.Werequiredthattheaverageoftotalexternalnancingoverallpathsequalstozero.Thismakesthestrategy"selfnancingonaverage".Weincorporatedmonotonic, 34
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Weperformedtwonumericaltestsofthealgorithm.First,wepricedoptionsonthestockfollowingthegeometricBrownianmotion.StockpriceismodelledbyMonteCarlosamplepaths.CalculatedoptionpricesarecomparedwiththeknownpricesgivenbytheBlackScholesformula.Second,wepricedoptionsonS&P500Indexandcomparedtheresultswithactualmarketprices.Bothnumericaltestsdemonstratedreasonableaccuracyofthepricingalgorithmwitharelativelysmallnumberofsamplepaths(consideredcasesinclude100and20samplepaths).Wecalculatedoptionpricesbothindollarsandintheimpliedvolatilityformat.TheimpliedvolatilitymatchesreasonablywelltheconstantvolatilityforoptionsintheBlackScholessetting.TheimpliedvolatilityforS&P500indexoptions(pricedwith100samplepaths)trackstheactualmarketvolatilitysmile. Theadvantageofusingthesquarederrorasanobjectivecanbeseenfromthepracticalperspective.Althoughweallowsomeexternalnancingoftheportfolioalongthepath,theminimizationofthesquarederrorensuresthatlargeshortagesofmoneywillnotoccuratanypointoftimeiftheobtainedhedgingstrategyispracticallyimplemented. Anotheradvantageofusingthesquarederroristhatthealgorithmproducesahedgingstrategysuchthatthesumofmoneyaddedto/takenfromthehedgingportfolioonanypathisclosetozero.Also,theobtainedhedgingstrategyrequireszeroaverageexternalnancingoverallpaths.Thisjustiesconsideringtheinitialvalueofthehedgingportfolioasapriceofanoption.Weusethenotionof"apriceofanoptioninthepracticalsetting"whichisthepriceatraderagreestobuy/selltheoption.Intheexample 35
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Weassumeanincompletemarketinthispaper.Weusetheportfoliooftwoinstrumentstheunderlyingstockandabondtoapproximatetheoptionpriceandconsidermanysamplepathstomodelthestockpriceprocess.Asaconsequence,thevalueofthehedgingportfoliomaynotbeequaltotheoptionpayoatexpirationonsomesamplepaths.Also,thealgorithmisdistributionfree,whichmakesitapplicabletoawiderangeofunderlyingstockprocesses.Therefore,thealgorithmcanbeusedintheframeworkofanincompletemarket. Usefulnessofouralgorithmshouldbeviewedfromtheperspectiveofpracticaloptionspricing.Commonlyusedmethodsofoptionspricingaretimecontinuousmodelsassumingspecictypeoftheunderlyingstockprocess.Iftheprocessisknown,thesemethodsprovideaccuratepricing.Ifthestockprocesscannotbeclearlyidentied,thechoiceofthestockprocessandcalibrationoftheprocesstotmarketdatamayentailsignicantmodellingerror.Ouralgorithmissuperiorinthiscase.Itisdistributionfreeandisbasedonrealisticassumptions,suchasdiscretetradingandnonselfnancinghedgingstrategy. Anotheradvantageofouralgorithmislowbacktestingerrors.Timecontinuousmodelsdonotaccountforerrorsofimplementationonhistoricalpaths.Theobjectiveinouralgorithmistominimizethebacktestingerrorsonhistoricalpaths.Therefore,thealgorithmhasaveryattractivebacktestingperformance.Thisfeatureisnotsharedbyanyoftimecontinuousmodels. 36
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3.2.1PortfolioDynamicsandSquaredError Thepriceoftheoptionattimetjisapproximatedbythepricecjofahedgingportfolioconsistingoftheunderlyingstockandariskfreebond.Thehedgingportfolioisrebalancedattimestj,j=1;:::;N1.Supposethatatthetimetj1thehedgingportfolioconsistsofuj1sharesofthestockandvk1dollarsinvestedinthebond tobenonzero.Thevalueajistheexcess/shortfallofthemoneyinthehedgingportfolioduringtheinterval[tj1;tj].Inotherwords,ajistheamountofmoneyaddedto(ifaj0)orsubtractedfrom(ifaj<0)theportfolioduringtheinterval[tj1;tj].Thus,theinow/outowofmoneyto/fromthehedgingportfolioisallowed. 37
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Thenonselfnancingportfoliodynamicsisgivenby wheretheportfoliovalueattimetjiscj=ujSj+vj;j=0;:::;N. Thedegreetowhichaportfoliodynamicsdiersfromaselfnancingoneisanimportantcharacteristic,essentialtoourapproach.Inthispaper,wedeneasquarederroronapath, tomeasurethedegreeof\nonselfnancity".Thereasonsforchoosingthisparticularmeasurewillbedescribedlateron. 38
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[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
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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
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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 ).Thetotalnumberoffreevariablesintheproblemisdeterminedbythesizeofthegridandisindependentofthenumberofsamplepaths.Aftersolvingtheoptimizationproblem,theoptionvalueattimetjforthestockpriceSjisdenedbyujSj+vj,whereujandvjarefoundfrommatrices[Ukj]and[Vkj],respectively,using 41
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3{6 ).Thepriceoftheoptionistheinitialvalueofthehedgingportfolio,calculatedasu0S0+v0. Thefollowingconstraints( 3{10 )( 3{18 )forcalloptionsor( 3{19 )( 3{27 )forputoptionsimposerestrictionsontheshapeoftheoptionvaluefunctionandonthepositioninthestock.Theserestrictionsreducethefeasiblesetofhedgingstrategies.Subsection3.3discussesthebenetsofinclusionoftheseconstraintsintheoptimizationproblem. Below,weconsidertheconstraintsforEuropeancalloptions.Theconstraintsforputoptionsaregiveninthenextsection,togetherwithproofsoftheconstraints.Mostoftheconstraintsarejustiedinaquitegeneralsetting.Weassumenonarbitrageandmake5additionalassumptions.Proofsoftwoconstraintsonthestockposition(horizontalmonotinicityandconvexity)inthegeneralsettingwillbeaddressedinsubsequentpapers.InthispaperwevalidatetheseinequalitiesintheBlackScholescase. ThenotationCkjstandsfortheoptionvalueinthenode(j;k)ofthegrid,Ckj=Ukj~Skj+Vkj: 3{10 )coincideswiththeimmediateexercisevalueofanAmericanoptionhavingthecurrentstockprice~SkjandthestrikepriceXer(Ttj):
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Thisconstraintsboundsensitivityofanoptionpricetochangesinthestockprice. 0. Verticalmonotonicity.Foranyxedtime,thepriceofanoptionisanincreasingfunctionofthestockprice. ~Skj 0. Horizontalmonotonicity.Thepriceofanoptionisadecreasingfunctionoftime. 0Ukj1;j=0;:::;N;k=1;:::;K:(3{15) 43
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(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\nonselfnancity"ofthehedgingstrategy.Thepricingalgorithmseeksastrategyascloseaspossibletoaselfnancingone,satisfyingtheimposedconstraints.Ontheotherhand,fromatrader'sviewpoint,theshortageofmoneyatanyportfoliorebalancingpointcausestheriskassociatedwiththehedgingstrategy.Theaveragesquarederrorcanbeviewedasanestimatorofthisriskonthesetofpathsconsideredintheproblem. Thereareotherwaystomeasuretheriskassociatedwithahedgingstrategy.Forexample,Bertsimasetal.(2001)considersaselfnancingdynamicsofahedgingportfolioandminimizesthesquaredreplicationerroratexpiration.Inthecontextofourframework,dierentestimatorsofriskcanbeusedasobjectivefunctionsintheoptimizationproblem( 3{9 )and,therefore,producedierentresults.However,consideringotherobjectivesisbeyondthescopeofthispaper. 44
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3{10 )( 3{14 )forcalloptionsand( 3{19 )( 3{23 )forputoptionsfollowunderquitegeneralassumptionsfromthenonarbitrageconsiderations.Thetypeoftheunderlyingstockpriceprocessplaysnoroleintheapproach:thesetofsamplepaths( 3{5 )speciesthebehavioroftheunderlyingstock.Forthisreason,theapproachisdistributionfreeandcanbeappliedtopricinganyEuropeanoptionindependentlyofthepropertiesoftheunderlyingstockpriceprocess.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
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3.4.1ConstraintsforPutOptions 3{9 )forpricingEuropeanputoptions. 0. Verticalmonotonicity. 0. Horizontalmonotonicity. 46
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3{9 )weusedthefollowingconstraintsholdingforoptionsinquiteageneralcase.Weassumenonarbitrageandmaketechnicalassumptions15(usedbyMerton(1973)forderivingpropertiesofcallandputoptionvalues.SomeoftheconsideredpropertiesofoptionvaluesareprovedbyMerton(1973).Otherinequalitiesareprovedbytheauthors. Therestofthesectionisorganizedasfollows.First,weformulateandproveinequalities( 3{10 )( 3{14 )forcalloptions.Someoftheconsideredpropertiesofoption 47
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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)]+:
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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
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ForanyX1,X2suchthat0X1X2,thefollowinginequalityholdsC(St;T;X1)C(St;T;X2)+(X2X1)er(Tt):
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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
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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;
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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
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Underassumptions1,2,and3,foranyinitialtimestandu,t
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3{15 )( 3{18 )and( 3{24 )( 3{27 )onthestockposition.Stockpositionboundsandverticalmonotonicityareproveninthegeneralcase(i.e.underassumptions15andthenonarbitrageassumption);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
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3)Horizontalmonotonicity(Calloptions) 2dZ;(3{29) andd1andd2aregivenbyexpressionsd1=1 2p 2p
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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): 37 ). TheError(%)columncontainserrorsofapproximatinginexionpointsbystrikeprices.Theseerrorsdonotexceed3%forabroadrangeofparameters.Weconcludethatinexionpointscanbeapproximatedbystrikepricesforoptionsconsideredinthecasestudy. 3{24 )( 3{27 )forputoptions. 57
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58
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31 33 ,and 34 report\relative"valuesofstrikesandoptionprices,i.e.strikesandpricesdividedbytheinitialstockprice.Pricesofoptionsarealsogivenintheimpliedvolatilityformat,i.e.,foractualandcalculatedpriceswefoundthevolatilityimpliedbytheBlackScholesformula. 31 Table1showsquitereasonableperformanceofthealgorithm:theerrorsintheprice(Err(%),Table 31 )arelessthan2%formostofcalculatedputandcalloptions. Also,itcanbeseenthatthevolatilityisquiteatforbothcallandputoptions.Theerrorofimpliedvolatilitydoesnotexceed2%formostcallandputoptions(Vol.Err(%),Table 31 ).Thevolatilityerrorslightlyincreasesforoutofthemoneyputsandinthemoneycalls. 32 .Theactualmarketpriceofanoptionisassumedtobetheaverageofitsbidandaskprices.ThepriceoftheS&P500indexwasmodelledbyhistoricalsamplepaths.NonoverlappingpathsoftheindexweretakenfromthehistoricaldatasetandnormalizedsuchthatallpathshavethesameinitialpriceS0.Then,thesetofpathswas\massaged"tochangethespreadofpathsuntiltheoptionwiththeclosesttoatthemoneystrikeispricedcorrectly.Thissetofpathswiththeadjustedvolatilitywasusedtopriceoptionswiththeremainingstrikes. Table 33 displaystheresultsofpricingusing100historicalsamplepaths.Thepricingerror(seeErr(%),Table 33 )isaround1:0%forallcallandputoptionsandincreases 59
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34 showsthatinthemoneyS&P500indexoptionscanbepricedquiteaccuratelywith20samplepaths.)Atthesametime,themethodisexibleenoughtotakeadvantageofspecicfeaturesofhistoricalsamplepaths.WhenappliedtoS&P500indexoptions,thealgorithmwasabletomatchthevolatilitysmilereasonablywell(Figures 3.6 3.6 ).Atthesametime,theimpliedvolatilityofoptionscalculatedintheBlackScholessettingisreasonablyat(Figures 3.6 3.6 ).Therefore,onecanconcludethattheinformationcausingthevolatilitysmileiscontainedinthehistoricalsamplepaths.Thisobservationisinaccordancewiththepriorknownfactthatthenonnormalityofassetpricedistributionisoneofcausesofthevolatilitysmile. Figures 3.6 3.6 3.6 ,and 3.6 presentdistributionsoftotalexternalnancing(PNj=1apjerj)onsamplepathsanddistributionsofdiscountedmoneyinows/outows(apjerj)atrebalancingpointsforBlackScholesandSPXcalloptions.WesummarizestatisticalpropertiesofthesedistributionsinTable( 35 ). Figures 3.6 3.6 3.6 ,and 3.6 alsoshowthattheobtainedpricessatisfythenonarbitragecondition.Withrespecttopricingasingleoption,thenonarbitrageconditionisunderstoodinthefollowingsense.Iftheinitialvalueofthehedgingportfolioisconsideredasapriceoftheoption,thenatexpirationthecorrespondinghedgingstrategyshouldoutperformtheoptionpayoonsomesamplepaths,andunderperformtheoptionpayoonsomeothersamplepaths.Otherwise,thefreemoneycanbeobtainedbyshortingtheoptionandbuyingthehedgingportfolioorviseversa.Thealgorithmproducesthepriceoftheoptionsatisfyingthenonarbitrageconditioninthissense.Thevalueofexternal 60
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Thepricingproblemisreducedtoquadraticprogramming,whichisquiteecientfromthecomputationalstandpoint.ForthegridconsistingofProws(thestockpriceaxis)andNcolumns(thetimeaxis),thenumberofvariablesintheproblem( 3{9 )is2PNandthenumberofconstraintsisO(NK),regardlessofthenumberofsamplepaths.Table 36 presentscalculationtimesfordierentsizesofthegridwithCPLEX9.0quadraticprogrammingsolveronPentium4,1.7GHz,1GBRAMcomputer. Inordertocompareouralgorithmwithexistingpricingmethods,weneedtoconsideroptionspricingfromthepracticalperspective.Pricingofactuallytradedoptionsincludesthreesteps. Mostcommonlyusedapproachforpracticalpricingofoptionsistimecontinuousmethodswithaspecicunderlyingstockprocess(BlackScholesmodel,stochasticvolatilitymodel,jumpdiusionmodel,etc).Wewillrefertothesemethodsasprocessspecicmethods.Inordertojudgetheadvantagesoftheproposedalgorithmagainsttheprocessspecicmethods,weshouldcomparethemstepbystep. 61
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Ouralgorithmdoesnotrelyonsomespecicmodelanddoesnothaveerrorsrelatedtothechoiceofthespecicprocess.Also,wehaverealisticassumptions,suchasdiscretetrading,nonselfnancinghedgingstrategy,andpossibilitytointroducetransactioncosts(thisfeatureisnotdirectlypresentedinthepaper). Calibrationofprocessspecicmethodsusuallyrequireasmallamountofmarketdata.Ouralgorithmcompeteswellinthisrespect.Weimposeconstraintsreducingfeasiblesetofhedgingstrategies,whichallowspricingwithverysmallnumberofsamplepaths. Themajoradvantageofouralgorithmisthattheerrorsofbacktestinginourcasecanbemuchlowerthantheerrorsofprocessspecicmethods.Thereasonbeing,theminimizationofthebacktestingerroronhistoricalpathsistheobjectiveinouralgorithm.Minimizationofthesquarederroronhistoricalpathsensuresthattheneedofadditionalnancingtopracticallyhedgetheoptionisthelowestpossible.Noneoftheprocessspecicmethodspossessthisproperty. 62
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Thispaperistherstintheseriesofpapersdevotedtoimplementationofthedevelopedalgorithmtovarioustypesofoptions.OurtargetispricingAmericanstyleandexoticoptionsandtreatmentactualmarketconditionssuchastransactioncosts,slippageofhedgingpositions,hedgingoptionswithmultipleinstrumentsandotherissues.Inthispaperweestablishedbasicsofthemethod;thesubsequentpaperswillconcentrateonmorecomplexcases. 63
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Impliedvolatilityvs.strike:CalloptionsonS&P500indexpricedusing100samplepaths.BasedonpricesincolumnsCalc.Vol(%)andAct.Vol(%)ofTable 33 .CalculatedVol(%)=impliedvolatilityofcalculatedoptionsprices(100samplepaths),ActualVol(%)=impliedvolatilityofmarketoptionsprices,strikepriceisshiftedleftbythevalueoftheloweststrike. Figure32. Impliedvolatilityvs.strike:PutoptionsonS&P500indexpricedusing100samplepaths.BasedonpricesincolumnsCalc.Vol(%)andAct.Vol(%)ofTable 33 .CalculatedVol(%)=impliedvolatilityofcalculatedoptionsprices(100samplepaths),ActualVol(%)=impliedvolatilityofmarketoptionsprices,strikepriceisshiftedleftbythevalueoftheloweststrike. 64
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Impliedvolatilityvs.strike:CalloptionsinBlackScholessettingpricedusing200samplepaths.BasedonpricesincolumnsCalc.Vol(%)andBS.Vol(%)ofTable 31 .CalculatedVol(%)=impliedvolatilityofcalculatedoptionsprices(200samplepaths),ActualVol(%)=atvolatilityimpliedbyBlackScholesformula,strikepriceisshiftedleftbythevalueoftheloweststrike. Figure34. Impliedvolatilityvs.strike:PutoptionsinBlackScholessettingpricedusing200samplepaths.BasedonpricesincolumnsCalc.Vol(%)andBS.Vol(%)ofTable 31 .CalculatedVol(%)=impliedvolatilityofcalculatedoptionsprices(200samplepaths),ActualVol(%)=atvolatilityimpliedbyBlackScholesformula,strikepriceisshiftedleftbythevalueoftheloweststrike. 65
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BlackScholescalloption:distributionofthetotalexternalnancingonsamplepaths.Initialprice=$62,strike=$62timetoexpiration=70,riskfreerate=10%,volatility=20%.Stockpriceismodelledwith200MonteCarlosamplepaths. Figure36. BlackScholescalloption:distributionofdiscountedinows/outowsatrebalancingpoints.Initialprice=$62,strike=$62timetoexpiration=70,riskfreerate=10%,volatility=20%.Stockpriceismodelledwith200MonteCarlosamplepaths. 66
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SPXcalloption:distributionofthetotalexternalnancingonsamplepaths.Initialprice=$1183:77,strikeprice=$1190timetoexpiration=49days,riskfreerate=2:3%.Stockpriceismodelledwith100samplepaths. Figure38. SPXcalloption:distributionofdiscountedinows/outowsatrebalancingpoints.Initialprice=$1183:77,strikeprice=$1190timetoexpiration=49days,riskfreerate=2:3%.Stockpriceismodelledwith100samplepaths. 67
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PricesofoptionsonthestockfollowingthegeometricBrownianmotion:calculatedversusBlackScholesprices. StrikeCalc.BSErr(%)Calc.Vol.(%)BS.Vol.(%)Vol.Err(%) Calloptions1.1450.00370.00383.7819.6320.001.861.1130.00750.00741.3519.9120.000.461.0810.01340.01330.6519.8720.000.651.0480.02260.02270.0419.7920.001.041.0160.03640.03610.8019.9420.000.281.0000.04460.04450.1919.8220.000.920.9680.06510.06480.4719.9420.000.310.9350.08910.08920.0819.5920.002.070.9030.11660.11680.1119.2920.003.560.8710.14640.14650.0718.7120.006.44Putoptions1.1450.12740.12760.1619.7320.001.361.1130.09950.09940.0420.0320.000.171.0810.07380.07380.0520.0220.000.121.0480.05140.05140.1019.9720.000.161.0160.03340.03320.7120.1420.000.681.0000.02580.02580.1520.0220.000.110.9680.01470.01441.8220.1920.000.930.9350.00700.00711.6019.8920.000.560.9030.00290.00315.7719.7120.001.450.8710.00100.001112.8819.5220.002.41 Initialprice=$62,timetoexpiration=69days,riskfreerate=10%,volatility=20%,200samplepathsgeneratedbyMonteCarlosimulation. Strike($)=optionstrikeprice,Calc.=obtainedoptionprice(relative),BS=BlackScholesoptionprice(relative),Err=(FoundBS)=BS,Calc.Vol.=obtainedoptionpriceinvolatilityform,BS.Vol.(%)=BlackScholesvolatility,Vol.Err(%)=(Calc:Vol:BS.Vol.)=BS.Vol.
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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
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PricingoptionsonS&P500index:100paths StrikeCalc.ActualErr(%)Calc.Vol.(%)Act.Vol.(%)Vol.Err(%) Calloptions1.1190.00020.000340.0013.1714.146.821.0980.00050.00055.2812.8012.920.901.0770.00130.001211.5712.7012.402.421.0560.00350.00335.7013.0312.801.781.0350.00790.00773.1513.3813.181.521.0220.01170.01180.7513.4313.490.481.0140.01560.01541.3213.9113.771.031.0050.01950.01950.0114.0714.060.010.9930.02690.02690.1814.6314.600.230.9710.04160.04140.5015.5715.401.090.9500.05890.05821.1216.8116.134.250.9290.07750.07700.6218.0417.353.940.4220.57890.57710.3369.39N/AN/APutoptions1.2670.26330.26380.2022.5029.0222.441.0980.09560.09600.4713.8815.148.351.0770.07560.07590.3613.7114.183.321.0350.04060.04050.3314.2214.110.771.0220.03190.03200.2514.2914.350.401.0140.02740.02701.2614.7514.511.621.0050.02290.02290.0114.8914.900.010.9930.01760.01741.3815.4715.301.100.9710.01110.01120.5216.4316.470.280.9500.00700.00721.9517.5817.720.790.9290.00450.00463.4218.8419.051.090.9080.00280.003110.0020.0220.572.680.8870.00150.002232.2720.4622.247.990.8660.00110.001526.0022.4623.785.54 Initialprice=$1183:77,timetoexpiration=49days,riskfreerate=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:
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PricingoptionsonS&P500index:20paths StrikeCalc.ActualErr(%)Calc.Vol.(%)Act.Vol.(%)Vol.Err(%) Calloptions1.1190.00050.000345.0014.9514.145.781.0980.00100.000588.8014.4812.9212.091.0770.00200.001266.8613.9512.4012.501.0560.00470.003341.8014.3912.8012.381.0350.00920.007719.8414.4313.189.421.0220.01320.011811.4114.4713.497.261.0140.01600.01544.0314.2013.773.131.0050.01950.01950.0014.0614.060.000.9930.02640.02691.6614.2814.602.150.9710.03930.04145.0113.6715.4011.230.9500.05480.05825.7612.0116.1325.520.9290.07370.07704.358.3917.3551.650.4220.57900.57710.34N/AN/AN/APutoptions1.2670.26330.26380.1923.4529.0219.161.0980.09590.09600.1314.8215.142.111.0770.07620.07590.4014.6714.183.451.0350.04150.04052.4914.9214.115.721.0220.03320.03203.6915.2014.355.931.0140.02780.02702.7415.0314.513.541.0050.02290.02290.0114.9014.900.010.9930.01680.01743.3114.9015.302.630.9710.00890.011220.7214.5816.4711.480.9500.00300.007258.7312.9917.7226.730.9290.00000.0046100.004.3819.0577.000.9080.00000.0031100.006.0720.5770.500.8870.00000.0022100.007.6822.2465.480.8660.00000.0015100.008.9823.7862.21 Initialprice=$1183:77,timetoexpiration=49days,riskfreerate=2:3%.Stockpriceismodelledwith20samplepaths.Griddimensions:P=15,N=49. Strike=optionstrikeprice(relative),Calc.=calculatedoptionprice(relative),Actual=actualoptionprice(relative),Err=(Calc:Actual)=Actual,Calc.Vol.=calculatedoptionpriceinvolatilityform,Act.Vol.(%)=actualoptionpriceinvolatilityterms,Vol.Err(%)=(Calc:Vol:Act:Vol:)=Act:Vol:
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SummaryofcashowdistributionsforobitainedhedgingstrategiespresentedonFigures 3.6 3.6 3.6 ,and 3.6 TotalnancingRebal.cashowTotalnancingRebal.cashow BlackScholesCallSPXCallmean0.00.00.00.0st.dev.0.62740.044916.15491.2730median0.07700.00080.26950.0314 Totalnancing($)=thesumofdiscountedinows/outowsofmoneyonapath;Rebal.cashow($)=discountedinow/outowofmoneyonrebalancingpoints. BlackScholesCall:Initialprice=$62,strike=$62,timetoexpiration=70,riskfreerate=10%,volatility=20%.Stockpriceismodelledwith200MonteCarlosamplepaths. SPXCall:Initialprice=$1183:77,strikeprice=$1190,timetoexpiration=49days,riskfreerate=2:3%.Stockpriceismodelledwith100samplepaths. Table36. Calculationtimesofthepricingalgorithm. #ofpathsPNBuildingtime(sec)CPLEXtime(sec)Totaltime(sec) 2020490.88.29.010025491.612.614.220025705.531.737.2 CalculationsaredoneusingCPLEX9.0onPentium4,1.7GHz,1GBRAM. #ofpaths=numberofsamplepaths,P=verticalsizeofthegrid,N=horizontalsizeofthegrid,Buildingtime=timeofbuildingthemodel(preprocessingtime),CPLEXtime=timeofsolvingoptimizationproblem,Totaltime=totaltimeofpricingoneoption. Table37. Numericalvaluesofinexionpointsofthestockpositionasafunctionofthestockpriceforsomeoptions. Expir.(days)Strike($)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(%)=(StrikeInexion)/Strike. 72
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OneofthealternativestothemeanvarianceapproachistheOmegafunction,recentlyintroducedinShadwickandKeating(2002).Omegafunctionr(rh)istheratiooftheupperandthelowerpartialmomentsofanassetrateofreturnragainstthebenchmarkrateofreturnrh.Theupperpartialmomentistheexpectedoutperformanceofanassetoverabenchmark;lowerpartialmomentistheexpectedunderperformanceofanassetwithrespecttothebenchmark.TheOmegafunctionhasseveralattractivefeatureswhichmadeitapopulartoolinriskmeasurement.First,ittakesthewholedistributionintoaccount.Asinglevaluer(rh)containstheimpactofallmomentsofthedistribution.Acollectionofr(rh)forallpossiblerhfullydescribesthereturndistribution.Second, 73
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ThechoiceoftheOmegaoptimalportfoliowithrespecttoaxedbenchmarkwithlinearconstraintsonportfolioweightsleadstoanonlinearoptimizationproblem.Severalapproachestosolvingthisproblemhasbeenproposed,amongwhicharetheglobaloptimizationapproachinAvouyiGovietal.(2004)andparametricapproachemployingthefamilyofJohnsondistributionsinPassow(2005).Mausseretal.(2006)proposesreductionoftheOmegamaximizationproblemtolinearproblemusingchangeofvariables.ThesuggestedreductionispossibleiftheOmegafunctionisgreaterthan1atoptimality,severalnonlinearmethodsaresuggestedotherwise. ThispaperinvestigatesreductionoftheOmegabasedportfoliooptimizationproblemwithxedbenchmarktolinearprogramming.WeconsideramoregeneralproblemthanMausseretal.(2006)byallowingshortpositionsinportfolioinstrumentsandconsideringconstraintsofthetypeh(x)0withthepositivelyhomogeneousfunctionh(),insteadoflinearconstraintsinMausseretal.(2006).WeprovethattheOmegamaximizingproblemcanbereducedtotwodierentproblems.Therstproblemhastheexpectedgainasanobjective,andhasaconstraintonthelowpartialmoment.Secondproblemhasthelowpartialmomentasanobjectiveandaconstraintontheexpectedgain.IftheOmegafunctionisgreaterthan1atoptimality,theOmegamaximizationproblemcanbereducedlinearprogrammingproblem.IftheOmegafunctionislowerthan1atoptimality,theproposedreductionmethodsleadtotheproblemeitherofmaximizingaconvexfunction,orwithlinearobjectiveandanonconvexconstraint. 74
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4.2.1DenitionofOmegaFunction Letxibetheexposureininstrumentiintheportfolio;thecorrespondingweightsarewi=xi=PNi=1xi,i=1;:::;N. ThelossfunctionmeasuringunderperformanceoftheportfoliowithrespecttothehurdlerateattimetisdenedbyL(t;x)=NXi=1(rhrti)xi:
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TheOmegafunctionistheratioofthetwopartialmoments(x)=(x) 76
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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)
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Conversely,supposexistheoptimalsolutiontoP00.ThentheobjectivefunctioninP0areboundedfromaboveby(x).Take=(PIi=1xi)1,thenxisfeasiblepointinP0,and(x)=(x),thereforew=xistheoptimalsolutiontoP0. EquivalenceofproblemsP1andP2isdenotedbyP1()P2. 78
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and(P1)maxKTD1q(x): and(Pq1)maxKTDq1(x): 79
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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
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Alternatively,theproblemPq1canbeattempted.IfDq+TKw6=?,thenthesolutiontoPq1afternormalizinggivesthesolutiontoP0.IfDq+TKw=?,theproblemPq1isinfeasible,duetotheconstraintq(x)1. IfKwTD=0=?,feasiblesetsinbothproblemsP0andP1areboundedandclosed,andobjectivefunctionarecontinuous,thereforebothproblemshavenite 81
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(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)$maxDqTD=1(x)(200),maxDqTD=1q(x)(300),maxDqTD1q(x)=P1: 82
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1 .ConsiderproblemPq1.Take^x2D=0TKw,thenq(^x)>0,(^x)=0.Taking^=1=q(^x),wehaveq(^^x)=1,(^^x)=0.Since(x)0forallx,theoptimalobjectivevalueinproblemPq1iszero. IfDq+TKw6=?andD=0TKw=?,thenproblemP0canbereducedtoproblemP00,aswasshownintheproofofTheorem 1 .ThefollowingsequenceofreductionstransformstheproblemP00intoPq1.P00=maxK(x)(10),maxDq+(x)(20)$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
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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
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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
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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
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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 41 87
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Optimalallocation StrategyAllocation(%) Manager110.00Manager220.00Manager37.50Manager42.22Manager50.00Manager67.50Manager70.00Manager812.78Manager920.00Manager1020.00 88
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ThispapermakesaconnectionbetweenthegeneralportfoliotheoryandtheclassicalassetpricingtheorybyexaminationofgeneralizedCAPMrelations.Inparticular,wederivediscountfactors,correspondingtotheCAPMlikerelationsandconsiderpricingformsofgeneralizedCAPMrelations.Weproposeamethodofcalibratingdeviationmeasuresfrommarketdataanddiscusswaysofidentifyingriskpreferencesofinvestorsinthemarketwithintheframeworkofthegeneralportfoliotheory. 89
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Investorssolvethefollowingportfoliooptimizationproblem. minD(x0r0+x1r1+:::+xnrn) (5{1) s.t.E(x0r0+x1r1+:::+xnrn)r0+x0+x1+:::+xn=1xi2R;i=0;:::;n: 5{1 )hasthreedierenttypesofsolutiondependingonthemagnitudeoftheriskfreerate,correspondingtocasesofthemasterfundofpositivetype,themasterfundofnegativetype,andthemasterfundofthresholdtype.Masterfundofpositivetypeistheonemostcommonlyobservedinthemarket,whenreturnofthemarketportfolioisgreaterthantheriskfreerate,andinvestorswouldtakelongpositionsinthemasterfundwhenformingtheirportfolios. Inthispaper,weconsiderthecaseofmasterfundsofpositivetypeandthecorrespondingCAPMrelations 90
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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)forCVaRdeviationwithcondencelevel;C=ErDMformeanabsolutedeviationandsemideviations.SinceassetpricesingeneralizedCAPM( 5{2 )dependontheriskidentierQDMthough(ri;QDM),assumptionA2suggeststhatProbfrDM=Cg=0. 91
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AssumptionA2cannotbesatisedforworstcasedeviationandsemideviations,seeRockafellaretal.(2006) UnderassumptionsA1andA2,allquantitiesingeneralizedCAPMrelationsarexedandwelldened,andtherelationsrepresentpricingequilibrium.InfurtherchapterswewillcloselyexaminegeneralizedCAPMrelationsundertheseassumptions. 5.2.1TwoWaystoAccountForRisk 1+r0E[];(5{3) wherer0isariskfreerateofreturn.Thepriceofanassetisthediscountedexpectedvalueofitsfuturepayo.Theassetwithrandompayowouldhavethesamepriceasanassetwithpays^=E[]withprobability1inthefuture. Iftheriskispresent,thepriceofanassetpaying^withcertaintyinfuturewould,generallyspeaking,dierfromthepriceoftheassethavingrandompayo,suchthatE[]=^.Theformula( 5{3 )needstobecorrectedforrisk.Therearetwowaystodoit. Therstwayistomodifythediscountedquantity: 1+r0(asset);(5{4) where(asset)iscalledthecertaintyequivalent.Itisafunctionofassetparametersandisequaltothepayoofariskfreeassethavingthesamepriceastheriskyassetwithpayo. 92
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1+rra(asset)E[];rra(asset)6=1:(5{5) whererra(asset)istheriskadjustedrateofreturn. PricingformsoftheclassicalCAPM(see,forexample,Luenberger(1998))areasfollows. CertaintyequivalentformofCAPM: 1+r0E[]cov(;rM)(ErMr0) RiskadjustedformofCAPM: 1+r0+(ErMr0)E[]:(5{7) Hereassetbeta=cov(r;rM) Relevanttofurtherdiscussion,thereisameasureofassetqualityknownastheShapreRatioS=E[r]r0 5{2 ),weget 93
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1+r00B@Ei+ErDMr0 Pricingformula( 5{8 )thecertaintyequivalentpricingformofgeneralizedCAPMrelations( 5{2 )(compareitto( 5{4 )),wherethecertaintyequivalent (i)=Ei+ErDMr0 isthepayoofariskfreeassethavingthesamepricei. Wecouldrearrangetheformula( 5{2 )inadierentway,namely whenErDMr0
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5{10 )istheriskadjustedpricingformofgeneralizedCAPMrelations( 5{2 )(comparewith( 5{10 )),wheretheriskadjustedrateofreturnis 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;
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5{8 )and( 5{10 )implythattheriskadjustmentisdeterminedbythecorrelationoftheassetrateofreturnwiththeriskidentierofthemasterfund. Togainabetterintuitionaboutthemeaningofthisformofriskadjustment,wecomparetheclassicalCAPMformulawiththegeneralizedCAPMrelationsfortheCVaRdeviationD(X)=CVaR(X)CVaR(XEX). Firstnote,thatmorevaluableassetsarethosewithlowerreturns.Whenpricingtwoassetswiththesameexpectedreturn,investorswillpayhigherpriceforamorevaluableasset,thereforeitsreturnwillbelowerthanthatofthelessvaluableasset. WebeginbyanalyzingtheclassicalCAPMformulawrittenintheform wherethelefthandsideoftheequationistheassetreturn.Thereturnisgovernedbythecorrelationoftheassetrateofreturnwiththemarketportfoliorateofreturn,i.e.bythequantitycov(ri;rM).Assetswithhigherreturncorrelationwiththemarketportfoliohavehigherexpectedreturns,andviceversa.Formula( 5{14 )impliesthatassetswithlowercorrelationwiththemarketaremorevaluable.Thereisthefollowingintuitionbehindthisresult.Investorsholdthemarketportfolioandtheriskfreeasset;theproportionsofholdingsdependonthetargetexpectedportfolioreturn.Theonlysourceofriskofsuchinvestmentsisintroducedbytheperformanceofthemarketportfolio.Themostundesirablestatesoffuturearethosewheremarketportfolioreturnsarelow.Theassetswithhigherpayoinsuchstateswouldbemorevalued,sincetheyserveasinsuranceagainstpoorperformanceofthemarketportfolio.Therefore,thelowerthecorrelationof 96
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NowconsiderthecaseofCVaRdeviation,D(X)=CVaR(XEX).Investorsmeasuringuncertaintyoftheportfolioperformancebythisdeviationmeasureareconcernedaboutthevalueoftheaverageofthe%worstreturnsrelativetothemeanofthereturndistribution. WeconsidergeneralizedCAPMrelationsfortheCVaRdeviationinthecaseofthemasterfundofpositivetype. wherei=cov(ri;QDM) IfprobfrDM=VaR(rDM)g=0,then Forfurtherdiscussion,assume=10%.Thenthenumeratorof( 5{16 )istheexpectedunderperformanceoftheassetrateofreturnwithrespecttoitsaveragerateofreturn,conditionalonthemasterfundbeinginits10%lowestvalues.Thedenominatorof( 5{16 )isthethesamequantityforthemasterfund.Aninvestorholdsthemasterfundandtheriskfreeassetinhisportfolio.Theportfolioriskisintroducedbytheperformanceofthemasterfund.Formula( 5{16 )suggeststhatassetsarevaluedbasedontheirrelativeperformanceversusthemasterfundperformanceinthosefuturestateswherethemaster 97
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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
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Underassumption(A1),thepayospaceX Fromtheperspectiveofdiscountfactors,acompletemarketischaracterizedbyauniquediscountfactor;inanincompletemarketthereexistsaninnitenumberofdiscountfactorsandeachdiscountfactorproducesthesamepricesofallassetswithpayosinX 5{17 ).Moredetailsonpricingassetsincompeteandincompletemarketswillbeprovidedlateron.Importantimplicationsofthesetheoremsareasfollows. 99
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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 )totheriskfreerategives1=E[m(1+r0)]. 100
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Inacompletemarket,thepayoofanynewassetliesinX Inanincompletemarket,twocasesarepossible.(1)ThepayoofanewassetbelongstoX new])ofthisassetcanbedenedasfollows. where=fmjm(!)>0withprobability1g:Includingonlystrictlypositivediscountfactorstothesetleadstoarbitragefreepricesgivenbyformula=E[m]. 101
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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
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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 Notethatthelefthandsideofcondition( 5{24 )containsarandomvariable,whiletherighthandsideisaconstant,andtheinequalitybetweenthemshouldbesatisedwithprobabilityone.ScalingthedeviationmeasureDbysome>0willchangethevalueofthelefthandside.Weshownextthatitdoesnotchangemeaningofthecondition( 5{24 ). 5{24 )isinvariantwithrespecttorescalingdeviationmeasureD.
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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
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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
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5.4.1IdenticationofRiskPreferencesofMarketParticipants providethemostaccurateassetpricing.WecouldeithertakeasetofassetreturnsrifromthemarketandestimatethemasterfundrDM,ortreatthemasterfundasgivenbythemarketandestimateexpectedreturnsEri.Theobtainedquantities,themasterfundreturnorexpectedreturnsoftheassets,willdependonthedeviationmeasureD,whichcanbecalibratedbycomparingestimatedquantitiestotheirmarketvalues. Welimitourconsiderationtothecaseofknownmasterfund;themethodbasedonestimationofamasterfundgivenasetofassetsismorecomputationallydicult,becausethegeneralizedportfolioproblemshouldbesolvedforeachchoiceofD. Assumptionthatamasterfundcanbeobtainedfromthemarketisjustiedbytheexistenceofindices,suchasS&P500,DowJonesIndustrialAverageandNasdaq100,whichrepresentthestateofsomelargepartofthemarket;moreover,investingintheseindicescanbethoughtofasinvestinginthemarket. Anybroadbasedmarketindexisassociatedwithcertainselectionofassets;theindexsummarizesthebehaviorofthemarketoftheseassets.Wecouldcalibratethegeneralized 106
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Therstcalibrationmethodisbasedonpricingassetsfromtheindexpool.Theindexserveasamasterfundinageneralizedportfolioproblemposedforassetsfromthepool.Givenaxedselectionofassets,dierentdeviationmeasureswouldproducedierentmasterfunds.Theexistenceofaparticularmasterfundfortheseassetsinthemarketcan,therefore,beusedasabasisforestimationofadeviationmeasure.The\best"deviationmeasureistheonewhichyieldsthebestmatchbetweentheexpectedreturnsofassetsfromthepoolandtheindexreturnthroughthegeneralizedCAPMrelations. Thesecondcalibrationmethodisbasedonpricingassetslyingoutsideoftheindexpool.Aswediscussedearlier,whenpricinganewassetwhosepayodoesnotbelongtotheinitiallyconsideredpayospace,thepriceinvestorswouldpaydependsontheirriskpreferences,denedbythedeviationmeasure.Thesecondmethod,therefore,usespricesof\new"assetswithrespecttotheindexpoolasthebasisforestimationofriskpreferences.Itshouldbenotedthatinthesetupofthegeneralportfoliotheorytheselectionofassetsisxed,andthemasterfunddependsonthedeviationmeasure.Inthepresentmethodweassumethatthemasterfundisxedandchangethedeviationmeasuretoobtainthebestmatchbetweenthemasterfundreturnandexpectedreturnsofnewassets.Bydoingso,weimplythatthechoiceoftheindexassociatedpoolofassetsdependsonthedeviationmeasure. Wejustifytheassumptionofaxedmasterfundbytheobservationthatmasterfunds,expectedreturnsofassets,andtheirgeneralizedbetascanbedeterminedfromthemarketdataquiteeasily,whiletheselectionofassetscorrespondingtoanindexcanbedeterminedmuchmoreapproximately.Anindexusuallyrepresentsbehaviorofapartofthemarketconsistingofmuchmoreinstrumentsthattheindexiscomprisedof.Withmuchcertainty,though,wecouldassumethatassetsconstitutingtheindexbelongtothe 107
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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
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Forthepurposesofcalibration,weassumeaparametrizationofadeviationmeasureD=D,where=(1;:::;l)isavectorofparameters. 109
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AssumethattheprobabilitymeasurePinthemarkethasadensityfunctionp(!).WeconsiderthegeneralizedCAPMrelationsintheform( 5{22 )andtransformthemasfollows(denotesthecompletesetoffutureevents!). 1+r0Z(!)(1+r0)mD(!)p(!)d!;(5{28) 110
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weget 1+r0Z(!)qD(!)d!:(5{30) Aswediscussedabove,ifthediscountfactormD(!)isstrictlypositive,thefunctionqD(!)couldbecalledthe\riskneutral"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
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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)denotethetrueriskneutraldistributionofthemasterfund.Bothfunctions~q(rDM)and~p(rDM)canbeestimatedfrommarketdata;theerrorinestimationof~q(rDM)by~qDrDM)in( 5{33 )isminimizedwithrespecttoD. First,weconsiderestimationof~q(rDM).Letq(!)bethetruemarketriskneutraldistribution,~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
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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,wearriveattheformulaforestimatingriskneutraldensityqfromcrosssectionofoptionpriceserT@2C @K2=@ @KZ+1K~q(S)dS=~q(K); ~q(S)=erT@2C @K2K=S:(5{35) Formula( 5{35 )allowstoestimatethefunction~q(rDM)whenthecrosssectionofpricesofoptionswrittenonthemasterfundisavailable.Itisworthmentioningthatthismethodestimates~q(rDM)atagivenpointintime;itisbasedonoptionspricesatthistime. Nowconsiderestimationofthemarginalprobabilitydensity~p(rDM).Themostcommonwaytoestimatethisdensityistousekerneldensityestimationbasedoncertainperiodofhistoricaldata.However,thismethodassumesthatthedensitydoesnotchangeovertime.Whentimedependenceistakenintoaccount,weareleftwithonlyone 113
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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)andmaximizingthepvalueofanappropriatestatistic. Thishypothesisisfurthertransformedasfollows.Usingthetrueriskneutraldistributions~qt(rM),theactualdistributions~pt(rM)areestimatedusing( 5{33 ),~pt(rM)=~qt(rM) (1+r0)mt(rM);t=1;:::;T.Wethentestthenullhypothesisthatriskneutraldistributions~pt(rM),t=1;:::;T,equaltothetrueriskneutraldistributions~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
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ImplementationIrequirescalculationexpectedreturnsofassetsandestimationofactualdistributionofthemasterfund.Thesequantitiescanbefoundfromthemarketdataquiteeasilyandaccurately.However,theresultsofthisimplementationdependonaparticularchoiceoftheobjectivefunctionDist(;).ItcanbearguedthatthechoiceoftheobjectiveshoulddependonatheparametrizationDofthedeviationmeasurebeingcalibrated.Forexample,ifthedeviationmeasureiscalibratedintheformofthemixedCVaRdeviation,thenDist(;)shouldbebasedontheCVaRdeviation,ratherthanonthestandarddeviation.AnotherdrawbackofimplementationIisthatthenancialliteraturedidnotusesimilaralgorithmsforcalibrationofutilityfunctions.Whenriskpreferencesareestimatedusingthisimplementationforthegeneralportfoliotheoryarecomparedwithriskpreferencesestimatedinnancialliteraturefortheutilitytheory,theresultsmaydierjustduetodierencesinnumericalprocedures,underlyingthetwoestimations. ImplementationIIiswidelyusedinnancialliteratureforestimationofriskaversioncoecientsofutilityfunctions.However,thisimplementationsuersfromsomenumericalchallengesrelatedtoevaluationoftheactualdensityintheformula~pt(rM)=~qt(rM) (1+r0)mt(rM): 115
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Thereisonemoredrawbackofthisimplementationwhenitisinthegeneralportfoliotheory.Whenusingnumericalestimationofriskneutraldensities,wehavetoassumethatnoarbitrageopportunitiesexistinthepricesofoptionsfromthecrosssection.ThisimpliesthatonlystrictlypositivediscountfactorsmD(rM)shouldbeusedforcalibration.Indeed,ifmD(rM(!))<0withpositiveprobability,thenestimatesoftheriskneutraldensity~qt(rM)=(1+r0)m(rM)~p(rM)canbenegative,andthehypothesisthatthetrueriskneutraldensities~qt(rM(!))>0(estimatedfromoptionscrosssection)areequaltotheestimateddensities~qt(rM)doesnotmakesense.However,itisnotclearatthispoint,whichdeviationmeasureshavethepropertythatmD(!)>0withprobability1. Finally,thereisanissuerelevanttoestimationofreturndistributionsofassetsbasedontheirhistoricalreturns.Historicaldatamaycontainoutliersoreectsofrarehistoricalevents.Aftersuch\cleaning",historicaldatamayprovidemorereliableconclusions.However,lteringhistoricaldatafromhistoricaleectsisanopenquestion. 116
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(5{36) WewillrefertodeviationmeasureCVaR(XEX)asCVaRdeviationofgains,todeviationmeasureCVaR(X+EX)asCVaRdeviationoflosses. Riskidentierforaconvexcombinationofdeviationmeasuresisaconvexcombinationoftheirriskidentiers.RiskidentierforCVaRdeviationoflosseswasderivedinRockafellaretal.(2006).Below,wederivetheriskidentierforCVaRdeviationoflossesandmixedCVaRdeviationoflossesandexaminecoherenceofthesedeviationmeasures. ThefollowinglemmawillhelptondriskidentierforD(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):
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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
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~Q2~Q(X)()8>>>>>><>>>>>>:~Q(!)=21;whenX(!)>VaR(X)21~Q(!)2;whenX(!)=VaR(X)~Q(!)=2;whenX(!)
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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
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2CVaR45%(X+EX)+1 2CVaR(X+EX); 121
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SergeySarykalinwasbornin1982,inVoronezh,Russia.In1999,hecompletedhishighschooleducationinHighSchool#15inVoronezh.Hereceivedhisbachelor'sdegreeinappliedmathematicsandphysicsfromMoscowInstituteofPhysicsandTechnologyinMoscow,Russia,in2003.InAugust2003,hebeganhisdoctoralstudiesintheIndustrialandSystemsEngineeringDepartmentattheUniversityofFlorida.HenishedhisPh.D.inindustrialandsystemsengineeringinDecember2007. 126

