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Path-Dependent Option Pricing

Permanent Link: http://ufdc.ufl.edu/UFE0042120/00001

Material Information

Title: Path-Dependent Option Pricing Efficient Methods for Levy Models
Physical Description: 1 online resource (61 p.)
Language: english
Creator: Gylfadottir, Gudbjort
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: asian, hilbert, levy, lookback, option, options
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This dissertation is concerned with the pricing of path-dependent options where the underlying asset is modeled as a continuous-time exponential Levy process and is monitored at discrete dates. These options enable their users to tailor random payoff outcomes to their particular risk profiles and are widely used by hedgers such as large multinational corporations and speculators alike. The use of continuous time models since the breakthrough paper of Black and Scholes has been greatly facilitated by advances in stochastic calculus and the mathematical elegance it provides. The recent financial crisis started in 2008 has highlighted the importance of models that incorporate the possibility of sudden, large jumps as well as the higher likelihood of adverse outcomes as compared with the classical Black-Scholes model. Increasingly, exponential Levy processes have become preferred alternatives, thanks in particular to the explicit Levy Khinchin representation of their characteristic functions. On the other hand, the restriction of monitoring dates to a discrete set increases the mathematical and computational complexity for the pricing of path dependent options even in the classical Black-Scholes model. This dissertation develops new techniques based on recent advances in the fast evaluation and inversion of Fourier and Hilbert transforms as well as classical results in fluctuation theory, particularly those involving random walk duality and ladder epochs.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Gudbjort Gylfadottir.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: AitSahlia, Farid.
Local: Co-adviser: Rao, Murali.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042120:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042120/00001

Material Information

Title: Path-Dependent Option Pricing Efficient Methods for Levy Models
Physical Description: 1 online resource (61 p.)
Language: english
Creator: Gylfadottir, Gudbjort
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: asian, hilbert, levy, lookback, option, options
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This dissertation is concerned with the pricing of path-dependent options where the underlying asset is modeled as a continuous-time exponential Levy process and is monitored at discrete dates. These options enable their users to tailor random payoff outcomes to their particular risk profiles and are widely used by hedgers such as large multinational corporations and speculators alike. The use of continuous time models since the breakthrough paper of Black and Scholes has been greatly facilitated by advances in stochastic calculus and the mathematical elegance it provides. The recent financial crisis started in 2008 has highlighted the importance of models that incorporate the possibility of sudden, large jumps as well as the higher likelihood of adverse outcomes as compared with the classical Black-Scholes model. Increasingly, exponential Levy processes have become preferred alternatives, thanks in particular to the explicit Levy Khinchin representation of their characteristic functions. On the other hand, the restriction of monitoring dates to a discrete set increases the mathematical and computational complexity for the pricing of path dependent options even in the classical Black-Scholes model. This dissertation develops new techniques based on recent advances in the fast evaluation and inversion of Fourier and Hilbert transforms as well as classical results in fluctuation theory, particularly those involving random walk duality and ladder epochs.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Gudbjort Gylfadottir.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: AitSahlia, Farid.
Local: Co-adviser: Rao, Murali.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042120:00001


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PATH-DEPENDENT OPTION PRICING:
EFFICIENT METHODS FOR LEVY MODELS


















By

GUDBJORT GYLFADOTTIR


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


2010




























2010 Gudbjort Gylfadottir































To my three favorite guys: My husband, Arni; my dad, Gylfi; and my brother, Thr6stur









ACKNOWLEDGMENTS

None of this would have been possible without my advisor and friend Farid. His

enthusiasm and ambition inspired me and I am thankful for all the knowledge he shared

with me during countless hours. Murali, of my committee, taught me a great deal of

math, yet I am most appreciative for getting to know him as a friend. I would like to

thank both Murali and Farid for all our inspirational conversations. Also, I would like

to thank my committee members, Dr. Pardalos and Dr. Nimalendran for their support.

Thanks go out to my family for their love, all the phone calls and for their lovely visits:

My dad Gylfi, my brother Thr6stur, my sister-in-law Una, my nephews Thorri and Fr6di

and my parents-in-law Erna and J6n. Also, thanks go out to my friends: Alex, Mireia,

Ehsan, Kelly, Vera, Altannar, Ashwin, Shantih, Emily, May, Soheil, Behnam, Renee, Clay,

Filip, Unnur, Helga Bj6rk, Helga Bj6rk, Svanhvit, Elin, Anna Gyda, Ragnheidur, Lara

and Jacki. Florida's nature with all its magical wonders made being here an amazing

experience. And lastly, my deepest gratitude goes to Arni, who is the most loving

husband I could wish for and has been here for me all of this time. I am really grateful

that we got to share this experience.









TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ... ............. ................. 4

LIST OFTABLES ... ............... ..................... 7

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

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

CHAPTER

1 PATH-DEPENDENT OPTIONS ..... .... ........ ....... ... 10

1.1 Introduction ................... ................ 10
1.2 Asian Options ... .......... ... .... ............ 12
1.3 Lookback Options ................... ........... 16
1.4 Overview ................... ................. 19

2 LEVY PROCESSES ................... ............ 20

2.1 Motivation for Levy Pricing Models ................ ...... 20
2.2 Using Levy Pricing Models .......................... 21
2.3 The Fast Hilbert Transform .......................... 25

3 QUANTILE APPROXIMATIONS FOR ASIAN OPTIONS .... 29

3.1 Introduction ..... ..... 29
3.2 Q uantile O options . .. 29
3.3 Distributions for Discrete Quantile Processes ..... 32
3.4 Quantile Approximations for Fixed Strike Asian Options ... 33
3.5 Pricing in the Black-Scholes Model .. .. 36
3.6 Hedging Parameters .............................. 39
3.7 Num erical Evaluation .. .. .. .. .. .. .. .. 41
3.8 C conclusion . .. 41

4 PRICING OF LOOKBACK OPTIONS USING LEVY PROCESSES ....... .43

4.1 Lookback Options ..................... ...... .... 43
4.2 Duality and Extrema of Random Walks ... 44
4.3 Fixed-Strike Lookback Options ... 48
4.4 Floating-Strike Lookback Options ... 49
4.5 Extensions ................. ............. .. 51
4.6 Sum m ary .. ... .. .. .. .. .. ... .. 52

5 CO NC LUSIO N . . 53

R EFER EN C ES . . 56









BIOGRAPHICAL SKETCH .................... ........... 61









LIST OF TABLES


Table page

3-1 Fixed Strike Asian call option with parameters So = 100, r = 0.1, n = 50, and
T = 1. Benchmark values result from Monte Carlo simulations with 100,000
paths (standard error in parentheses). Prices using quantile approximations
(with 3 = 3) are given in the last column. ... 42

3-2 Fixed Strike Asian call option with parameters So = 100, r = 0.1, n = 50, and
T = 1. Approximation of option's delta with 3 = 3. Benchmark values result
from Monte Carlo simulations with 100,000 paths (standard error in parentheses).
. . 4 2

4-1 Loookback option prices at time to = 0 . ... .. 43









LIST OF FIGURES
Figure page

4-1 Sample path of a log-price process for a lookback option .. 44









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

PATH-DEPENDENT OPTION PRICING:
EFFICIENT METHODS FOR LEVY MODELS

By

Gudbjort Gylfadottir

August 2010

Chair: Farid AitSahlia
Cochair: Murali Rao
Major: Industrial and Systems Engineering

This dissertation is concerned with the pricing of path-dependent options where

the underlying asset is modeled as a continuous-time exponential Levy process and is

monitored at discrete dates. These options enable their users to tailor random payoff

outcomes to their particular risk profiles and are widely used by hedgers such as

large multinational corporations and speculators alike. The use of continuous-time

models since the breakthrough paper of Black and Scholes has been greatly facilitated

by advances in stochastic calculus and the mathematical elegance it provides. The

recent financial crisis started in 2008 has highlighted the importance of models that

incorporate the possibility of sudden, large jumps as well as the higher likelihood of

adverse outcomes as compared with the classical Black-Scholes model. Increasingly,

exponential Levy processes have become preferred alternatives, thanks in particular to

the explicit Levy-Khinchin representation of their characteristic functions. On the other

hand, the restriction of monitoring dates to a discrete set increases the mathematical

and computational complexity for the pricing of path-dependent options even in the

classical Black-Scholes model. This dissertation develops new techniques based on

recent advances in the fast evaluation and inversion of Fourier and Hilbert transforms

as well as classical results in fluctuation theory, particularly those involving random walk

duality and ladder epochs.









CHAPTER 1
PATH-DEPENDENT OPTIONS

1.1 Introduction

Options are contracts in which the buyer of the option gets the right, but not the

obligation, to buy or sell the underlying asset of the contract at some date in the

future, for a predetermined strike price. If the date is pre-specified (labeled maturity

or expiration date) then the option is of European exercise-style. Otherwise, it is of

American exercise-style and can be exercised any time up to maturity. A call option

is a contract that gives the right to buy the underlying asset, and a put option is a

contract that gives the right to sell the underlying asset. The seller of the option collects

a fee upfront in order to give this right to the option holder. The determination of the

fair price of this fee for different kinds of options has been of interest for academics

and practitioners alike. It has become an area of major intellectual and commercial

development since 1973, when Black and Scholes published a breakthrough article that

allowed for the pricing of so-called standard (or vanilla) options (Black & Scholes,

1973), by only using readily available parameters, namely the prevailing riskless

rate in the market and the volatility (standard deviation of returns) of the underlying

asset upon which the option is written. Vanilla options depend only on the price of

the underlying security on the exercise date, whereas path-dependent options have

an exercise payoff that depends on the price path of the underlying security from the

beginning of the contract until the exercise date. An Asian option is an example of a

path dependent option. The payoff of a European exercise-style Asian call option is

max(AT K, 0) = (AT K) where K is the strike price of the Asian option and AT

is the average of the security over the life of the contract. In contrast, the corresponding

payoff of a standard (vanilla) call option is max(ST K, 0) = (ST K)+, where ST is the

price of the underlying security at maturity.









The use of derivatives has become popular in recent years because investment

banks have been able to hold them without having to put them on their balance sheets.

Since options allow for leveraged transactions, this has allowed banks and investors

to make highly leveraged transactions without them ever showing up on their balance

sheets. (In simple terms, leverage refers to borrowing.) An argument made in support

of this state of affair is described in the J.P. Morgan guide to credit derivatives (Morgan,

1999). After the crash of financial markets in late 2008, many became worried that

unregulated use of derivatives was dangerous to financial markets. Path-dependent

options are also called exotic options and are mostly traded between private parties,

in so-called over-the-counter-trade (OTC), not in open markets. They have therefore

been hard for the legislator to oversee. The U.S. House of Representatives and the

U.S. Senate drafted a bill that was to limit OTC trading of exotic derivatives to respond

to concerns that their opacity can be a source of instability (Gibson, 2010). In addition,

the bill proposed that some uncovered (or 'naked') derivatives trading be banned.

However, the bill came across hard opposition from a group of investors, politicians and

academics and has had some alleviating amendments added to it, including the drop of

the proposal to ban naked derivatives trading and the drop of most limits to OTC trading

of options. Many suggested that even if the use of exotic options would be limited in the

U.S. this would only spur life into foreign OTC trading since it would not be likely that

people would stop using these investment vehicles since they have become so common.

At the end of the last decade, (Boyle & Boyle, 2001) noted that growth in option

trading had increased significantly for the past 30 years and that in the first quarter

of 2000 the estimated value underlying option contracts around the world was $102

trillion. In fact, this was only the estimated value underlying exchange traded contracts,

the estimated value underlying over the counter (OTC) option contracts was estimated

to be $88 trillion (BIS, 2000) so the total value underlying option contracts was $190

trillion in the beginning of 2000. For the last quarter of 2009, the estimated value









underlying exchange traded options around the world was $444 trillion (BIS, 2010),

or roughly fourfold the value from 10 years earlier, even when it was down from $690

trillion in the beginning of 2008. However, OTC seems to have become the preferred

method of trading options, with $605 trillion in underlying value for OTC contracts in

June 2009.(BIS, 2009) In comparison, the GDP of the USA was $10 trillion in 2000

and $14 trillion in 2009 (BEA, 2010), so at the beginning of the decade, the total value

underlying option contracts in the world was roughly 19 times the GDP, and at the end of

the decade it was 75 times the GDP.

1.2 Asian Options

The first paper written on Asian options, by Boyle and Emanuel in 1980, was

rejected by the Journal of Finance, since this kind of option was not traded at that

time(Georgios Foufas and Mats G. Larson, 2008). The paper is still a working paper

(Boyle & Emanuel, 1980). Boyle and Emanuel called this new option type, averaging

options, but they were dubbed Asian options by Bankers Trust because the firms

that bought the options from Bankers Trust, were Japanese. These firms' annual

reports were based on average exchange rates over the year, so average rate options

were appropriate for them to hedge their risk(Vorst, 1996). In practice today, Asian

options are mostly traded on oil products, agricultural commodities such as corn and

soybeans and on currencies. As far back as in 1998, Microsoft was already taking

advantage of the elimination of downside risk that Asian options offer, along with the

potential of an upside gain by hedging their foreign currency exposure by using Asian

put options(William Falloon, 1998). Microsoft's treasurer at that time, Mr. Heitz, said in

an interview with Risk magazine that Microsoft had 10-12 counterparties from which

it could buy the put options. Today, Asian options are still most commonly traded over

the counter. Asian options are particularly useful in thinly traded markets or to protect

against large price variations. Investors who have an obligation due on a certain date

will want an insurance against the counterpart being able to move prices against them.









Since it will be much harder to move the average price than the price on a specific date,

Asian options have become common use in thinly traded stocks/currencies.

Nowadays, on the Chicago Mercantile Exchange (CME), average options are

constructed in the following way: The option has a swap (i.e.; a contract to exchange

an interest or currency rate for another) as the underlying security, and a fixed strike

price. The final price on the swap is used to calculate the payoff of the average option.

The final price on the swap is calculated by taking the arithmetic average of daily prices

from each day for which a price for the underlying security for the swap is determined

for the previous month. The daily price is found by taking the average of the high and

low quotations on each day for the underlying security for the swap. The payoff for an

average call option will be the final price on the swap minus the strike price, and the

payoff for an average put option will be the strike price minus the final price on the swap.

Even though this structure is intricate, the average price option payoff is simply the

difference between the arithmetic average price of the security itself over the previous

month minus a fixed strike price, so our pricing model for Asian options given in this

paper is applicable to the average price options traded on the CME. On the CME, all the

19 average options available in early 2010, had an oil product as the underlying security,

and they were all traded on CME's over the counter clearing service. The oil products

included e.g. gasoline, jet kerosene, fuel oil, propane, butane, heating oil, gasoil, ethane

and crude oil (CMEGroup, 2010). Through these examples, it is clear that Asian options

are widely used.

Asian options are less likely than vanilla options to be manipulated because it is

not possible to manipulate the price over such a long time as opposed to vanilla (or

regular) options. A recent example from the drop in the Dow Jones by almost 10% within

a few minutes (Mattich, 2010) shows that whether it is by mistake or manipulation, it

is possible for the market to be affected severely from other factors than efficiency in

just a matter of minutes. In the case of a vanilla call option, had the closing price of









the underlying asset been 10% lower than on the previous day, the option might have

expired worthless on that day, but would have expired in the money on the previous day.

In the case of an Asian option, this 10% lower price only affects the average by moving 1

out of n prices that are part of the average and therefore cannot affect the Asian option

price as much. As a result, Asian options are perceived to be cheaper and therefore

reduce the risk management costs of their bona fide users.

In the first published paper on Asian options, (Kemna & Vorst, 1990) used Monte

Carlo methods to determine the price of the arithmetic Asian option. By using the

geometric Asian option as a control variate, where the geometric average is given by


AT (i= 1


they were able to price the Asian option faster than with plain Monte Carlo. Monte Carlo

simulation works well but can be computationally expensive without the enhancement

of variance reduction techniques. One must account for the inherent discretization

bias resulting from the approximation of continuous-time processes through discrete

sampling as shown by (Broadie et al., 1999). As previously noted, the arithmetic Asian

option, where the arithmetic average is given by

I n
n
i= 1

is the one that is used in practice. However, it is not possible to find the exact analytical

price for the arithmetic Asian option. The geometric Asian option on the other hand

is lognormally distributed when the underlying price process is assumed to follow

a geometric Brownian motion. So with that assumption it is possible to derive the

exact analytical price for the geometric Asian option. (Turnbull & Wakeman, 1991)

proposed using an approximation of the density function of the arithmetic Asian option

by using an Edgeworth expansion. Among the first to derive analytic results,(Geman

& Yor, 1993) computed the Laplace transform of the price of a continuously sampled









Asian option computed as fT Stdt. Its numerical inversion remains problematic for

low volatility and/or short maturity as shown by (Fu et al., 1998). On the other hand,

in practice, sampling is performed over a discrete set of dates (daily, weekly, etc.)

In this case, no analytic results are available even in the Black-Scholes framework,

where the main source of the problem stems from the lack of an explicit distribution

for the sum of correlated log-normal random variables. As a result, a significant

number of approximations that produce closed-form expressions have appeared.

For example,(Thompson, 1998) provides tight analytical bounds and (Linetsky,

2004) derived a new integral formula for the price of a continuously sampled Asian

option, which is again slowly convergent for low volatility cases. In general, the price

of an Asian option can be found by solving a partial differential equation (PDE) in

two-dimensional spaces (see (Ingersoll, 1987) ), which is prone to oscillatory solutions.

Ingersoll also observed that the two-dimensional PDE for a floating strike Asian option

can be reduced to a one-dimensional PDE. (Rogers & Shi, 1995) simpler formulated

a one-dimensional PDE that can model both floating and fixed strike Asian options.

However this one-dimensional PDE is difficult to solve numerically since the diffusion

term is very small for values of interest on the finite difference grid. Several articles

contain attempts to improve the numerical performance of this PDE. (Andreasen,

1998) applies the reduction of Rogers and Shi to discretely sampled Asian option.

Independent efforts in recent years have attempted to unify pricing techniques for

different types of options and relate these methods to pricing Asian option. Using again

Rogers and Shi's reduction, (Lipton, 1999) noticed similarities in pricing equations

for the passport, lookback, and Asian options (Shreve & Ve6ef, 2000) developed

techniques for pricing options on a traded account, which include all options that could

be replicated by self-financing trading in the underlying asset. They include European,

passport, vacation, as well as Asian options. Numerical techniques for pricing contracts

of this type are described in (Vecer, 2001). (Hoogland & Neumann, 2001) developed









an alternative framework for pricing various types of options using scale invariance

methods and derived more general semi-analytic solutions for prices of continuously

sampled Asian options. A major shortcoming of these approaches is their inability to

help determine hedging parameters, which are crucial to the option writer. (Fusai &

Meucci, 2008) derive pricing methods for both arithmetic and geometric Asian options

under discrete monitoring and for a general Levy process. They work on the Fourier

space with a recursive pricing formula like we do. However, for their re-centering

technique they require finite moments, which we don't. In addition, it is unclear how their

method can lead to computing hedge parameters, while ours will be shown to produce

them with minimal additional computations.

1.3 Lookback Options

The payoff for lookback options depends on the extremum price observed over the

contract period. For floating lookback options, the holder of a call option gets the right

to buy at the lowest price over the contract period and sell at the price on the expiration

date, T and the holder of a put option gets the right to sell at the highest price over the

contract period and buy at the price on the expiration date. For fixed lookback options,

the holder of a call option gets the right to buy the security at a fixed strike price K, but

the selling price is the highest price over the contract period. The holder of a fixed put

lookback option gets the right to sell at the lowest price over the contract period and buy

at a fixed strike price K. There are also other variations, where for example the holder of

the option gets the right to buy or sell for a percentage of the extremum price observed.

Compared to other options, lookback options provide the biggest payoff potential

because the investor can choose the exercise date in retrospect, that is by looking

back over the life of the option. The reported uses for lookback options are mainly

speculative. It is obvious that lookback options will be more expensive than vanilla or

Asian options because the holder is getting the biggest potential payoff over the whole

life of the option.









Some analytical solutions have been proposed when the monitoring of the price

process is continuous and/or when the underlying price process follows geometric

Brownian motion, see e.g. (Heynen & Kat, 1995), (Conze & Viswanathan, 1991)

and (Goldman et al., 1979). In practice, monitoring occurs at discrete dates and

the monitoring dates t, ..., tN = T are predetermined. (Kou, 2008a) says that it

is practical that monitoring is discrete, and that if monitoring were continuous, there

would be arbitrage opportunities for barrier options, e.g. if a barrier is reached. Those

could represent themselves while markets are open in only some parts of the world.

Similarly, if the highest/lowest price for a lookback option during its contract period

so far is reached at a time when not all exchanges are open simultaneously, it would

be unfair to the traders in parts of the world where the markets are closed since they

are unable to trade upon that information immediately while others in open markets

would reap the profits. All traded lookback options have discrete monitoring, so even

if a higher/lower price is observed outside of the monitoring dates, it is not taken

into account for determining the extrema of prices over the contract period. As a

consequence of the discrete monitoring, pricing is mathematically and computationally

challenging. Substantial mis-pricing occurs when a discretely monitored contract is

priced approximately by a continuous-monitoring formula (cf. (Broadie et al., 1999),

(Heynen & Kat, 1995).) (Broadie et al., 1999) introduce correction terms so that the

continuous-monitoring formulas can be used as approximations for the discretely

monitored options. Their method also improves convergence by means of lattice

methods.(Babbs, 2000) uses a binomial model to price continuously monitored

floating-strike lookback options. Using discrete monitoring and pricing for both fixed

strikes and floating strikes, (Cheuk & Vorst, 1997) also use a binomial model to price

lookback options, improving upon Babbs. (Boyle & Tian, 1999) used a trinomial method

to value the non-Gaussian CEV process and found the price for lookback and barrier

options when the price follows the CEV process. Later they found that it was inaccurate









for lookback options and proposed a correction using Monte Carlo methods (Boyle

et al., 1999). (Davydov & Linetsky, 2001) also found pricing formulas for the lookback

option when the underlying follows the CEV process. Using Laplace transforms, their

method is faster than that of Boyle and Tian. (AitSahlia & Lai, 1998) use the duality

property of random walks to derive recursively the distribution of conditioned extrema of

the geometric Brownian motion price process and use numerical integration methods

to price lookback options. (Tse et al., 2001) use a tridiagonal procedure that takes

advantage of the properties of the geometric Brownian motion price process and

price the lookback options numerically and achieve more efficiency than previous

methods. (Andricopoulos et al., 2003) develop a quadrature method that can be used

to numerically price a wide range of options, including lookback options.(Broadie &

Yamamoto, 2003) develop a fast Gauss transform for non-path-dependent option

valuation under geometric Brownian motion and the Merton model.(Broadie &

Yamamoto, 2005) extend their previous results and derive a double-exponential

Gaussian model that can be applied to lookback options and other path-dependent

options. (Petrella & Kou, 2004) find Laplace transforms of discrete lookback options

using a recursion formula. These involve Spitzer's formula. They invert the Laplace

transforms numerically to get the lookback option price and hedging parameters for

several Levy price models. For the geometric Brownian motion price process and

discrete monitoring, (Atkinson & Fusai, 2007) find the distribution of the extrema of

prices in closed form and are thus able to find the lookback price for fixed and floating

options. The latest work on lookback options is by (Feng & Linetsky, 2009). They do

a forward recursion on the prices of the lookback option, utilizing Hilbert transforms

and Fourier transforms. Their method is efficient and accurate but is restricted by some

conditions making it inapplicable to the important pure-jump processes. In contrast,

our method, to be described in detail later, is more generally applicable and has the









same computational complexity as it also uses their fast algorithm for the evaluation and

inversion of Hilbert and Fourier transforms.

1.4 Overview

Briefly, this dissertation is broadly organized as follows: Chapter 2 reviews Levy

processes and their use in finance as well as recent advances in Fourier-based

techniques. In particular, we review those making use of Hilbert transforms due to (Feng

& Linetsky, 2008) and (Feng & Linetsky, 2009), which enable us to make additional

contributions to efficiently and accurately price lookback options as described in detail

in Chapter 4. Chapter 3 deals with the pricing of Asian options and provides a detailed

description of our approximation approach based on yet another type of path-dependent

options, namely quantile options, that are not traded but which provide mathematical

expediency. The contributions of this thesis consist of new techniques to price discretely

monitored Asian and lookback options. Their distinguishing feature lies in working on the

characteristic function of the option price distribution rather than on the characteristic

function of the price itself, as is done in (Feng & Linetsky, 2008) and (Feng & Linetsky,

2009), the most competitive approach up-to-date. Ours has the significant advantage

of enabling a direct computation of hedging parameters, the "Greeks", in contrast to the

unstable numerical derivatives and the computationally complex Malliavian calculus

required by all the other alternatives. In addition, the (Feng & Linetsky, 2009) pricing

method for lookback options is slower than ours and excludes an important class of Levy

models in finance, the popular variance gamma specification (Madan & Seneta, 1990).









CHAPTER 2
LEVY PROCESSES

2.1 Motivation for Levy Pricing Models

Up until recently, most pricing models have assumed that the underlying process for

any security follows a geometric Brownian motion a la Black Scholes, that is

dSt = Stpdt + StdW

where St is the price of the security at time t, p is the drift rate of the security, a is

the volatility rate and W is a Wiener process. By modeling the price in this way, the

assumption is that In(St) follows a Brownian motion, that is In(St) is continuous and has

independent and normally distributed increments. There are several issues regarding

modeling the underlying price process like this. First,(Merton, 1976) noted that far too

many random jumps occur in the price process in practice to be justified by constant

volatility or a continuous path of prices. He therefore suggested an addition of a jump

term to the price process, so that

dSt = St dt + StadW + dq

where q is a Poisson process with normally distributed jumps, where both are independent

of W. These random jumps lead to an empirical distribution that has fatter tails than the

normal distribution. Other issues include the empirical observation of (log) price returns

that are not symmetric, and with peaks higher than suggested by the normal distribution

(a leptokurtotic curve). These issues are addressed in e.g. (Kou, 2002) and in (Carr

et al., 2002). Kou and Carr et al. suggest models to remedy those issues, respectively,

the Kou model which has both a diffusion component and a jump component and the

CGMY model, which only has a jump component. All of the aforementioned models,

including the Black and Scholes model, are specific cases of a general class of

processes called Levy processes. Levy processes are fairly general and allow for a









wide range of models, including the Poisson process, Brownian motion, or the pure jump

process of Carr et al.

2.2 Using Levy Pricing Models

A process (Xt)t>o is called a Levy process if it has

a) Independent increments: That is for all to, t, ..... t,, the random variables

Xto, Xt Xto,...., X Xt,_ are independent
b) Stationary increments: That is Xt X, has the same distribution as Xt-_s+ X,

and

c) Continuous paths a.e: That is limh,0 P(IXt+h Xt| > c) = 0 for any e > 0.

Every Levy process can be fully described by three parameters. The first two

parameters, a and a2, describe the continuous component of the Levy process, and the
third parameter is a function v(x), called the Levy density, which identifies the discrete

component of the Levy process. Furthermore, a is the constant drift of the continuous

component and a2 is the constant variance of the continuous component.

Using only those three parameters, the Levy-Khinchin formula:

InE[ei'x] = aitO a tO2 t (eix iOxIxl<1),)v(x)dx

where aeR, o- > 0 and fRo min{l, x2}v(x)dx < oo, allows for an easy retrieval of the

characteristic function, (O) = E[eio0x], of many Levy processes, which makes them

feasible for practical use.

Levy processes can have either finite activity, which means that over any interval,

there will be a finite amount of jumps, or they can have infinite activity, which means that

any interval will have infinite amount of jumps (Wu et al., 2008). Pure jump processes

with infinite activity, are often not distinguishable from pure diffusion processes, and

when there is infinite activity it is not necessary to have a Brownian motion component

as well. When a is 0, we have a pure jump process and when v(x) is zero we have a

pure diffusion process. The arrival rate for jumps is determined by fR/ v(x)dx = A.









If A < oo, then the mean arrival rate of jumps is finite, and when A = oo the

number of jumps over any interval will be infinite. The simplest Levy process as

previously mentioned is the Black Scholes model, for which v(x) = 0 and therefore

the characteristic function is simply


E[eex] e:-.' -

and it is easy to derive the probability distribution function (pdf), which is simply the

normal density with mean a o-2t and variance o2 t. The Merton model has Levy

density:
A (X-t)2
e 262

which describes in mathematical terms that the process will have jumps that are

normally distributed with mean p and variance 62, and that the jumps come with

frequency A. For the Merton model the characteristic function can be simplified to

E[eiox]= e:"--z ,- '+At{e- '-1}


see (Cont & Tankov, 2004), however the probability density can only be represented as

an infinite series, and is thus not available in closed form. This provides an additional

computational complexity in deriving an option value which has this price process as the

underlying asset. On one hand, this is a better model for the price process, because it

is more realistic that the price exhibit some jumps, just as it might when new information

arrives to the market that immediately changes market participants' opinion on what

the price should be, so that the price immediately adjusts. It is worth noting that a more

realistic model (than the Black Scholes model) would only be useful in practice, if it

enables us to price the derivatives of the price process. For the base case, a vanilla call

option, the price of the option at initiation using the Merton model would be:











CT(K) = e-tE[Soex K] = (Soex K)dF(XT)
J/n(K)
-(XT, -t-k,)2
(SeX K)e (tke 2(2tk62) dX
= I(-0K) k= k! 2F ( 2t+ k2)
This integral can not directly be evaluated, except in a few cases (Merton, 1976)

where the infinite sum within the integral can be simplified.(Carr & Madan, 1999) derive

a method in which it is not necessary to know the pdf of the price process to calculate

the option price. Rather than working with the direct integral above, they work with its

Fourier transform which they obtain in an explicit form, albeit not trivially because in

order for the integral to be non-singular, they have to multiply the Fourier transform with

a specific remedial function. The explicit formula involves the characteristic function

of the price process, which as mentioned above, can always be retrieved from the

Levy-Khinchin formula for all Levy processes. Once they have an explicit formula for

the Fourier transform, they take the inverse Fourier transform, then multiply again with

the inverse of the remedial function to retrieve the option price. When calculating the

inverse Fourier transform, they use the discrete Fourier transform (DFT) on the integral,

which means that they have to discretize the integral. To speed up the calculations they

then transform the integral to conform to the setup for the Fast Fourier Transform (FFT)

which is faster than calculating the DFT directly, O(Nlog(N)) vs O(N2) respectively. The

FFT will give prices of several different strikes for each calculation of the FFT. When

performing these calculations, there is a choice to be made for the FFT, if the grid for the

DFT is chosen to be wide, the strike prices will be relatively close to each other, and if

the grid for the DFT is chosen to be fine, the strike prices will be far apart. So the choice

of the grid has to be made according to what strike price range is needed. Also, the

choice of the remedial function has to be made carefully so that it ensures integrability.

(Lee, 2004) discusses these choices of parameters in more detail, and shows how the









FFT method of pricing utilizing the characteristic function of the price process can be

extended to other option classes.

(Kou, 2002) proposes a model that has jumps in addition to a diffusion process,

but the jumps have double exponential distribution instead of normal distribution, like

in the Merton model. Also, the distribution of jumps is different depending on whether

it is an upward movement or a downward movement, reflecting the trend that stock

price changes seem generally to be of different magnitude for good news and bad news

(Chen et al., 2003). For the Kou model the Levy density is:

pA+e-Xx if x > 0 and

(1- p)_ex if x < 0

and although the probability density is not available in closed form, the characteristic

function can be derived from the Levy-Khinchin formula and the corresponding pricing

of vanilla options can then be done by using the methods in Carr et al. and Lee. The

Kou model achieves the high peak and the fat tails that are typical of stock returns and

eliminates the phenomenon that is called volatility smile. A volatility smile or skewness

is seen when options are priced using the Black and Scholes model (Hull, 2006). The

standard deviation, or volatility as it is called in the finance literature, is assumed to

be fixed in the Black and Scholes model. Yet, when vanilla option market prices are

observed for different strike prices, and the Black Scholes model is solved to return the

volatility, it is different for different strike prices, typically higher the further away from at

the money the strike price is. It can also be skewed, referring to that the implied volatility

is higher for strike prices under the at the money price, and lower for strike prices that

are out of the money. Because of the jumps that the Kou model incorporates, this smile

disappears and the implied volatility becomes constant.

There are two more prominent Levy models that we will mention. First is the

Variance Gamma model (Madan et al., 1998), that also makes implied volatility constant

for vanilla options, so that no volatility smile is observed. This is done in a very different









way from the Kou model; in the Variance Gamma model, there is no diffusion part, but

instead the number of jumps over any given interval is infinite, that is, it has infinite

activity. There are three parameters in the Variance Gamma model that need to be

calibrated. The CGMY model (Carr et al., 2002) is a generalization of both the Kou

model and the Variance Gamma, and it has five parameters that need to be specified.

The Levy density of the CGMY model is

C- if x > 0 and
x
C _+x if x < 0

When Y is equal to -1, the CGMY model becomes the Kou model, and when Y equals

zero, it is the same as the Variance Gamma model. The CGMY model exhibits infinite

activity for Y between 0 and 2, and finite activity for Y less than 0. Y has to be less

than 2 in all cases, so that the characteristic function may exist. It is not obvious how

to specify the parameters of the CGMY model, so practitioners have found reasonable

parameters for it by calibration with real world data, which is typically done by seeing

which models fit historical data the best. For example, (Carr et al., 2002) specify

the 5 parameters (in addition to the four parameters in the Levy density, o- needs to

be specified) that make the CGMY model fit the S&P 500 index the best, and also

display how drastically the distribution function changes by just twisting even one of the

parameters at a time, thereby showing how sensitive the model is to parameter changes.

It should also be noted that even though Levy pricing models solve a lot of the empirical

issues that using the Black-Scholes model entails, model selection of a Levy process is

hard, mainly because there are so many parameters to estimate. The data needed to

estimate the exact parameters and models would have to be enormous to justify using

one good model rather than another.(Heyde & Kou, 2004)

2.3 The Fast Hilbert Transform

The Fourier-transform method of (Carr & Madan, 1999) can be utilized to price

vanilla options for any Levy process. (Feng & Linetsky, 2008) and (Feng & Linetsky,









2009) develop a Hilbert-transform based method to price barrier and lookback options

when the underlying asset follows an exponential Levy process. Their recourse to

Hilbert transforms in the Fourier space stems from the presence of an indicator function

multiplying the function of interest; the price. This indicator function captures the

path-dependency of the option payoff such as the barrier crossing event prior to the

option expiration, for example. Succinctly, they use the following property relating Fourier

and Hilbert transforms for a given 4 defined on R:

1 i
F (1(0, o) o ) (0 = +2(2)(

where the Fourier transform for f c L'(R) is


(0) F(f)() -= Rei'x f (x)dx.

and the Hilbert transform for f LP(R), 1 < p < oo, is

1 f(y)
-H(f)(x) = P. V. ()dy.
S j_ x-y

The fast Fourier transform algorithm was available long before Carr et al's

paper and enables us to find the discretization of the Fourier transform (DFT) with

a computational complexity of O (N log, N), where N is the number of points in the

discretization. This is an advantage over computing the DFT in the naive way, which

results in a complexity of O (N2). Feng and Linetsky proceed to make their own fast

Hilbert transform algorithm since none existed. They use Whittaker cardinal series (Sinc

expansion) to approximate R-( with

Sf(= m f (mh) 1 cos[7( mh)/h]
-(f )(0() Hh, Mf () = f (mh)
m-M ( mh)/h
m -M

where h is the discretization step size and M > 0 is the truncating integer for the

integral approximation. After this discretization step, they then use the FFT and Toeplitz

matrix-vector multiplication to compute Hh,Mf( ). The overall computational complexity








to find the Hilbert transform is O (M log2 M), or the same complexity as the FFT for the
Fourier transform. Furthermore, the error in the approximation decays exponentially as h
is taken smaller. The price of a down-and-out barrier option at time zero is given by
V(S) = e-TEs [(Sr K) (L,,)(SA) .. 1(L,,)(SNA)] ,
where St is the price of the underlying at time t, A is the monitoring interval, NA = T
and L is the barrier. All the indicator functions are within the expectation because if
the price of the underlying drops below L on any monitoring date, the option becomes
worthless. Feng and Linetsky do a backward recursion on the prices of the barrier option
to find the time zero value of the option where St = Kex, for any Levy process Xt, with
Xo = In(So/K). They define the time-zero price of the option as

V(So) = e-rTO (In(So/K))

with vo obtained recursively through:

v"(x) = K(ex 1)+l(l,o)(x),

v-1 (x) = l(i,) (x) PAv (x),j = N, N 1,..., 2,

v(x) = PA v(x),

where Paf(x) := E[f(Xt+A)IXt = x] and / := In(L/K). Then for j=N,N-1,...,2, they
perform the recursion in Fourier space:

) K(1 eit) K(1 e(1i))
i( 1 i+
--() = 2(_ ^( eil 2 (e-ill(-TI)/j(0)) (a,

where j is the characteristic function of XA and for each recursion step they utilize
the fast Hilbert transform to obtain the Fourier transform on the left. Then, finally they
retrieve vo through a final Fourier transform

vO(x)- M e0(-(()d
Z7T/









and accomplish an aggregate computational complexity of O (NM log, M) to find the
barrier option price. In their extension to lookback options, (Feng & Linetsky, 2009)

utilize the Fast Hilbert transform, by working forward in their recursive scheme, rather

than backward. In our approach, based on evaluating the option price distribution

instead, we still maintain the use of the fast Hilbert transform discretization algorithm of

(Feng & Linetsky, 2008) in a backward recursive fashion. As mentioned in Chapter 1,

the main advantage of our approach is its ability to generate hedge parameters much

more seamlessly than any other alternative.









CHAPTER 3
QUANTILE APPROXIMATIONS FOR ASIAN OPTIONS

3.1 Introduction

Chapter 3 develops a new approximation approach to price and hedge discretely
monitored Asian options when the underlying asset price follows a Levy process. The

option price is shown to be accurately approximated by a weighted sum of related

quantile options. The latter are options on quantile values of the underlying asset
process. Though they are currently not traded, our work in Chapter 3 shows how

they can be used for efficient computation of Asian option prices. Furthermore, our

method offers a way to directly approximate hedge parameters with practically negligible
additional computational effort.

Chapter 3 is organized as follows. The first section summarizes the concept of
a quantile option in both the original continuous-time setting and our discrete set-up
for discrete monitoring of path-dependent options. The second section contains our

quantile-based approximation in a general Levy process framework. The last section
presents a numerical illustration on the particular case of the Black-Scholes (Brownian)
model.

3.2 Quantile Options

First introduced by (Miura, 1992), these options are path-dependent and are
meant to generalize the concept of options on extrema (minimum or maximum). For

a (p, o)-Brownian motion {Xt, t > 0} and a e (0, 1), define the a-quantile process

{M(a, t), t > 0} by:
M(a, t) = inf x : l(x at .

Then the a-quantile option payoff is defined as


(SoeM(T)- K),









where So is initial price of underlying asset (stock, currency, ...) and K is the strike price.

The corresponding option price has been extensively studied by (Akahori, 1995) and

(Dassios, 1995) who in the process generalize the arc-sine law for Brownian motion.

More precisely, they obtain


Pr {M(a, t) e dx} = g(x; a, t)dx,


(3-1)


where


g(x; a, t)= gz(x y; at)2(y;(1 a)t) dy,
D- O


(3-2)


and gl and g2 are the probability density functions associated with supo
info

Pr( sup Xs dx
\O
Pr ( inf Xs c dx
\( These functions are explicitly derived as

1 ) )12 exp ( -2- 2 exp
gl(x;-r)= -



{0,
S; 1) 2 exp (2T I + 2p

The quantile option price at time 0 is then
The quantile option price at time 0 is then


= gl(x; at)dx,


g2(x; (1 a)t)dx.


(2)f1 P x+-) for x > 0,
2(T2 V/ \ 7 0 J
for x < 0,


for x > 0,

xp ( (1) for x < 0.
2(T2 ( (T0


E [e-T (SoeM(aT) -K)+ ,


which can be evaluated through numerical integration as the associated probability

density function g is determined through Eq. 3-1 through Eq. 3-6.

The key to the derivation of the above results begins with the equivalence between

the events {M(a, t) > x} and {fat 6(Xs < x)ds < at}, where 6(A) is the indicator of


(3-3)


(3-4)


(3-5)



(3-6)




(3-7)








whether event A has occurred, thus relating the quantile process to the occupation time.
As a consequence, one can then show the following identity (cf. (Dassios, 1995)):

M(a, t)i sup X(1)(s)+ inf X(2)(s), (3-8)
O
where X(')(t) and X(2)(t) are independent copies of the process X(t) = pt + ,B(t),
with B(t) denoting a standard Brownian motion. Furthermore, (Dassios, 1995) also
derives the joint distribution of M(a, t) and X(t):

M(ca, t) i ( supos< X1)(s) + info X(t) X(1) (at) X(2)((1 a)t)

In fact, both Eq. 3-8 and Eq. 3-9 hold when the reference X is a Levy process as
(Dassios, 1996) shows. While the derivation of the results for the Brownian case
is based, respectively for Eq. 3-8 and Eq. 3-9, on the Feynman-Kac formula and
the Girsanov theorem, the method of proof for the Levy process relies in fact on an
asymptotic discretization. The latter will turn out to be exactly what we need for the
Asian option pricing with discrete monitoring. Specifically, (Dassios, 1996) develops the
following:
Proposition. Let _1, _,..., be i.i.d. random variables. Consider the random walk

(n = C k, 0 < n, where g( = 0 w.p. 1, and let C(1) and b(2) be two independent copies
of C. Then

Mjn(() a MJaj ((1)) Mo.n-j(((2)), (3-10)

where, for integers 0 < j < n and a discrete process X = (Xo, X1, X2, ...), MJn(X) is the
(j, n)th quantile of X defined as

Mj,n(X) =inf : 6 (X, < ) >j
i .









We should note that in fact the joint distribution

Mn( A/UC+) ^) l J(j(+) MO-j ((2)) 1)
) (3-1(1)

has been known since (Wendel, 1960).
3.3 Distributions for Discrete Quantile Processes

Whereas the use of an order statistic to consistently estimate a single quantile
implies its convergence in probability, our approach here via Eq. 3-10 deals with
quantile processes. Thus we make use of corresponding collections of order statistics
with the associated mode of weak convergence. For this purpose, we shall show that we
can rely on either convergence of characteristic functions in the general Levy case, or on
random walk approximation in the case of Brownian motion. For the latter, we will show
through a numerical illustration how Bernoulli random walks results due to (Takacs,
1996) can be exploited. For the former, we exploit the Levy-Khinchine characterization
theorem for the increment of a Levy process and make use of results due to (Pollaczek,
1975) on order statistics as we show next.
Let Xi, X2, ... be a collection of i.i.d random variables. We are interested in
determining the characteristic functions of the order statistics of the random walk
samples Xi, X + X2,..., X,. Thus, we define for n > 1 and 1 < v < n,

Xn, = max() X, X + X, ... Xi (3-12)
(i 1=1

where, for real numbers a,, a2,..., an, max(")(ai, a,,..., an) represents the Vth number
taken in descending order in the collection. With this convention, we have
max(l)(ai, a2,..., an) = max(al, a2,,..., an). In other words, Xn,,, 1 < v < n represent
(an) order statistics (process) for the random walks values (X1, X + X2,... 1X).
We now adopt the approach followed by (Pollaczek, 1975) in order to determine the
moment generating functions of the characteristic functions for Xn,,. More specifically,









with q a complex number and 0 the characteristic function of a Levy increment with cdf
F, namely,

(-q) = Eexp(-qX) = e-qdF(s), (3-13)

let
oo n
G(q,x, y) = -xn- ly-l'Eexp(-qXn,),
n=l v=l 1
where x| < 1, Ixyl < ~ (-q)'n-1. (Pollaczek, 1975) then shows

G(ix,y) =) exp xl(1 ) e it+dFn(t) (3-14)
( xy)(I xy (- i)) n

for Ix < 1, Ixy| < 1, where Fn is the n-fold convolution of F with itself, so that

on(-q) = -qtdFn(t)

and
1 / e-ql
exp(-qa+) = 2 (
27r c (q 0
for a real, q such that Re(q) > 0, and where C is a parallel to the right of the imaginary
axis such that Re(q ) > 0 for e C.
For a Levy process, 0(-q) is explicitly given and thus Fn can be obtained via Fast
Fourier Transform. The characteristic function of any X(n,) is then trivially retrievable
through derivatives with respect to x and y evaluated at x = 0 and y = 0.
3.4 Quantile Approximations for Fixed Strike Asian Options

Under the risk-neutral measure, the time-0 price is

e-rTE (AT K)+, (3-15)

which can be evaluated in closed-form with geometric averaging in the standard
Black-Scholes model. In practice, averaging is arithmetic over discretely sampled
prices of the underlying. In this case, there are no known closed-form expressions
for the distribution of a sum of correlated log-normal random variables. As a result,









pricing approximations for fixed strike options (arithmetic average) have involved mostly

Monte Carlo simulation, moment matching approaches, density perturbation, PDE, and

convolution (FFT) techniques (cf. references in (Benhamou, 2002) and (Linetsky, 2004).)

In Chapter 3, we propose using quantile options, for which analytic expressions are

readily available, to approximate the price of a discretely sampled Asian option with a

fixed strike.

In this section we detail our quantile approximation. It is based on three elements:

(i) the payoff of an Asian option is a monotone transformation of the average price, (ii)

the arithmetic average of a random sample is the same as that of the associated order

statistics, and (iii) the latter are generally consistent estimators of quantiles. Ultimately,

our task is to evaluate expectations of the form E[Z], where Z = (SoeM(aT) K) and

M(a, T) is the a-quantile of the underlying process over the interval [0, T]. Note that for

now we refer to a generic quantile. However, we will later define such processes using

notation referring directly to the discrete sampling of the underlying.

With discrete monitoring, AT in Eq. 3-15 is the arithmetic average taken over a set

of prices monitored at times tl, t,,..., t := T :
n
AT=n zSt
i1=

We now define discrete-time quantile and occupation-time processes, respectively

M(a, T, n) and r(x, T, n):


M(a, T, n) = inf : 6 (Xt < x)a

1 6(Xt x)
T(x, T, n) =)
n
where X, is the ti-time value of the Levy process X such that X0o 0 and 6(A) = 1 if A

occurs and 6(A) = 0 otherwise. Here, St, = Soexi is the underlying asset price at time ti.









Theorem 3.1. For any positive integer 3, there exist A1, A2,... AX and cd, a, ..., c,3

(0, 1) such that 1, i = 1, and


AiE (SoeM(a'n) K) -- E(AT- K)+,
i=1
as /3 oo.

Proof. Let (S(1) < S(2) < 2 < S(n)) be the order statistic of the sample (St,, St, .. St).

Then for any positive integer 3, there trivially exist 0 < ca < a2 < ... < aos such that

, ai = 1. Furthermore, we have
n n
St, S(,)
i=1 i= 1
[nal] 1-1 [na+ll n
= S(i) S()
i 1 j 1 [nj+ 1 [na3]

Note that the sequence S(), S(2),..., S(n) is monotone, non-decreasing with

probability 1. As such, it may be considered as deterministic and thus the sum Y ( S,)

may be viewed as a Riemann sum (with probability 1) to the extent that one can write
b
5(,) (S(b) 5(a)) (b a)
i=a

almost surely. The quantities
[naj +

[na+ 1
have the same properties and thus one can also write

83-1 Fn"j++i /3-1
SSW Yo (S([nai) S([na,])) ([naj+, [na])
j=1 Fna]+l j 1

almost surely. As a result, we can now write the following approximation










n 13-1
St (2 [nal] [na, 1) S(nal) + (2 [na]1 [na \]) S(Faj1)
i=1 j 2
+ (2 [nai] n) S([na3) + (n [no ]) S(n) ([nal] 1) (1).

Note that by choosing cr such that [nal] = 1 (e.g. a 1/n) and a3, such that

[na8 ,] n (e.g. a 3 > 0.95) we see that the extreme statistics S(i) = min{St} and
S(n) = max{S, } can be omitted from the approximation. Recall that S([na]) is an
estimator of the oth quantile of the price process {S}. Thus, with the monotonicity of the

functions x ex and x (x K) we can write


AiE (Soe(aM(an) K) E(A K)
i= 1
as/3 oo. O

With this approximation and given the determination of the distributions of the

variables M(a,, T, n) as described in the previous section, we now have all the

ingredients to proceed with the pricing of a discretely monitored Asian option.

3.5 Pricing in the Black-Scholes Model

In an earlier discussion we mentioned that the distributions of the discrete quantile

process can also be determined through the random walk approximation route. We

proceed to do so in this section, where we focus on the Black-Scholes model, with its

underlying Brownian motion as a special Levy process. As shown above, the core of our

approximate pricing of an Asian option is now the determination of a set of expectations,

namely E (SoeM(am,"',Tn) K) for various values of aj, where m = r a2/2 is the

drift of the Brownian motion followed by the natural logarithm of the underlying asset

price, the volatility of which is a in a market where the riskless rate of return is r. With

this notation, M(oa, m, o, T, n) represents the ajh quantile of n equally spaced segments

of this Brownian motion on the interval [0, T]. Correspondingly, we also use the notation

r(x, m, a, T, n) for the occupation time as we soon shall exploit the space-time scaling








property of Brownian motion, thus justifying the explicit reference to the drift m and
volatility a. Using a basic property of expectation for non-negative random variables, we
have
E (SoeM(a'"'"Tn) K) = So P{M(a, m, a, T, n) > x}ex dx.
In(K/So)
Observe that

P{M(a, m, a, T, n) > x} = P{r(x, m, a, T, n) < a}.

Furthermore, by the space-time scaling property of Brownian motion, we can write

P{r(x, m, a, T, n)
where T(x', m', 1, 1, n), for x' = and m' = "T, is defined over the process

Xt, = m'ti + (ti) with t = 1 and standard Brownian motion ((t). Assume that the
number of monitoring dates n of the underlying asset process satisfies n > m'2. This
is generally easy to fulfill given that m' = (r/o -/2) /T, where 0 < r/o- < 1, with a
typically in the range of 0.2 to 0.60, and T < 1. Consider now a random walk (,, r > 0)
with increments ( such that P{( = 1} = p and P{( = -1} = q, where

1 m' 1 m'
p = and q =2 2- -
2 2Vn 2 Vn

for n > m'2. Forj c {0, 1, 2, ... n}, define An(j) = y: 16(( > j), which counts the number
of times the random walk is above in the time interval {0, 1..., n}. From (Takacs, 1996)
we have for x > 0 the approximation

P m 1, 1,n n -j
I ( x m

where j = [na], k = [ ], k > 0, and 0 < a < 1. Furthermore, for n> T2

1 mVTa 1 m/T
p = 2 + and q =
2 2ca1n 2 2ca1n









Note that we can extend the definition of An(/) to I < 0 by observing that An(/) has the
same the distribution as n A~(-/ 1), where A((k), k > 0, is defined in the same

manner as An(k) with the roles of p and q interchanged. Thus, for x < 0,


i n

where j = [na], k = [x- k < 0 and 0 < a < 1. Furthermore, for n >


1 m a
p = and q
2 2 v/n

Our expectation formula then becomes


1 m/T
2 2u f


E (SoeM(a',m.a, Tn)


K) + so
In(K/So)


P{An(k) > n -j}e dx,


{ i n P{A,(k) = i}
0


forj > k > 0,

forj < k,


P{A(k) > n j} =
0


nk+ P{An(-k -1) i} for0 > k >j-n,
fork

Forl < i < n- k,


P{A,(k) = i} = P{Ai


- i} [P{p(k + 1) > n i} P{p(k) > n i}],


where


p(k)= inf{r : = k, r > 0},

P{A = i} = p qPp(1) < i},


P{p(1) < i} =1 -(P{(-1 = -1}

P{(-1 < -1}


+ P{(_-1 = 0} + (1 P/q)P{(i-1 < -1})
-2
= E P{(-1 = a}
a 1-i


where


P{An(k) > n j}


and









and
Pf (- ( = 1 P (i--l a)/2 q(i+l-a)/2
{-1 [(i- 1 + a)/2])

and

P{p(k + 1) > n i} P{p(k) > n i} = P{,-i = k/ + (q/p)P{-i = k +1}

and
(1 P) P ,(n-i < -k- 1}

and
-k-2
P{n- < -k- 1} = P{(n- = a}
a i-n
There are clearly several choices available for the weights and percentile levels for the

approximation in Theorem 1. In fact, one may refer to a simple choice inspired from

Tukey's tri-mean as a starting point. Through some numerical evidence, we show that

this is amply adequate for practical purposes. In this case, we use the approximation
3
(A K) Ai (SoeeM(aT)- K)
i= 1
where a = A1 = 0.25, a2 = A2 = 0.50, a3 = 1 A3 = 0.75.
3.6 Hedging Parameters

To obtain the Greeks, one needs the version below of the Leibniz integral formula

d r" 9 d
d J g(x, )dx = g(x, ) dx g(a(), d) a(0).
< a(0) a(0 ) <

Therefore, letting g(x) = P{M(a, m, o, T, n) > x}ex, we have

Delta = dSo g(x)dx
dSO I n(K/So)

g(x)dx+So dg (x) dx
In(K/So) dSO JIn(K/So)









Now, by the Leibnitz rule, we therefore have

Delta = g(x)dx + g (In(K/So))
In(K/So)

Another crucial hedging parameter, namely Gamma, can be computed as easily:

Gamma = Delta
dSo
d d
Sg(x)dx g (In(K/So))
dSO n(K/So) So dSO
1 1 d
g (In(K/So)) g (In(K/So))
So So dx

Recalling that g(x) = P{M(a, m, a, T, n) > x}ex and letting

fM(x) = dP{M(a, m, a, T, n) < x}, we have

d gx)= (P{M(a,m,, T,n) > x}+ d P{M(a, m,, T, n)>x}" ex
dx dx

and

1 1
Gamma = g (In(K/So)) [g (In(K/So))]
So So

Then we may use the following:
odp d "
Delta =dp = So P{M(a,m, o, T,n) > x}ex dx
dSo dSO JIn(K/So)

= P{M(a, m, a, T, n) > x}ex dx
In(K/So)

-So ( ) P{M(a, m,u T, n) > x}exx -/n(K/So)


= P{M(a, m, a, T, n) > x}ex dx + KP{M(a, m, a, T, n) > In(K/So)}
In(K/So) so


Gamma = d (d d[P{M(a, m, a, T, n) > In(K/So)}]
dSo dSo So dSo
Kd 1
-= [P{M(a, m, 7, T, n) < x}]
So dx So

= M(In(K/So)
-2









where fM(x) is the probability distribution of M(a, m, o, T, n) which can be approximated

as

fM(In(K/So) P{An(k) = j}

[ In( K/So) n~
where = n- [na] and k = [ n(K /S)

Additional hedging parameters, such as rho, vega and theta, can be approximated

similarly.

3.7 Numerical Evaluation

In this section we compare the accuracy of our approximation against benchmark

values computed via significantly much-slower Monte-Carlo simulation. Though our

main theorem requires that 3 -p oc, our results as displayed in Tables 3-1 and 3-2

indicate that the approximation is in fact very well behaved even when 3 is as small as

3. From Table 3-1, observe that the accuracy of the approximation deteriorates only in a

small number of cases that have no practical interest. They are deep out-of-the money

(thus unlikely to be exercised) options with negligible prices. In all the other cases,

the deviations from the benchmark values are in fact well within the bid-ask spread for

over-the-counter option contracts. Similar observations can be made regarding the

results displayed in Table 3-2. In this case, we are able to obtain hedging parameters

that are as important for the option writer, typically a bank as counterpart to a hedge

fund, a manufacturer, or airline company. These hedging parameters have traditionally

been omitted from the option pricing literature or relegated to numerical derivation via

finite-differences, which are numerically unstable, or Monte Carlo simulation, which is

very time-consuming.

3.8 Conclusion

Chapter 3 develops an approximation technique for Asian option pricing and

hedging based on analytic expressions for quantile options when the underlying

asset follows an exponential Levy process. Our numerical results indicate that this











Table 3-1. Fixed Strike Asian call option with parameters So = 100, r = 0.1, n = 50, and
T = 1. Benchmark values result from Monte Carlo simulations with 100,000
paths (standard error in parentheses). Prices using quantile approximations


(with 3 = 3) are given in the last column.
Volatility K From Benhamou's Paper Benchmark Price (Expected
(Monte Carlo Price and Value and SE)
SE)
0.1 80 22.78 (0.00) 22.78 (0.00)
0.1 90 13.73 (0.00) 13.73 (0.00)
0.1 100 5.24 (0.00) 5.25 (0.00)
0.1 110 0.72 (0.00) 0.73 (0.00)
0.1 120 0.03 (0.00) 0.03 (0.00)
0.3 80 23.07 (0.01) 23.09 (0.01)
0.3 90 15.22 (0.01) 15.20 (0.02)
0.3 100 9.01 (0.01) 9.00 (0.02)
0.3 110 4.83 (0.01) 4.86 (0.02)
0.3 120 2.35 (0.01) 2.39 (0.01)
0.5 80 24.83 (0.03) 24.86 (0.03)
0.5 90 18.32 (0.03) 18.29 (0.04)
0.5 100 13.18 (0.03) 13.13 (0.04)
0.5 110 9.23 (0.03) 9.24 (0.04)
0.5 120 6.36 (0.03) 6.32 (0.03)


Option Price Using
Quantile Options

22.71
13.68
5.29
1.07
0.13
22.94
15.23
9.07
5.15
2.83
24.56
18.13
12.99
9.33
6.69


Table 3-2. Fixed Strike Asian call option with parameters So = 100, r = 0.1, n = 50, and
T = 1. Approximation of option's delta with 3 = 3. Benchmark values result
from Monte Carlo simulations with 100,000 paths (standard error in
parentheses).


Sigma K
0.1 80
0.1 90
0.1 100
0.1 110
0.1 120
0.3 80
0.3 90
0.3 100
0.3 110
0.3 120
0.5 80
0.5 90
0.5 100
0.5 110
0.5 120


Benchmark
0.95 (0.000)
0.95 (0.000)
0.78 (0.001)
0.22 (0.001)
0.01 (0.000)
0.91 (0.000)
0.79 (0.001)
0.61 (0.001)
0.41 (0.001)
0.24 (0.001)
0.82 (0.000)
0.71 (0.001)
0.58 (0.000)
0.46 (0.001)
0.35 (0.001)


Delta
0.95
0.94
0.72
0.20
0.03
0.86
0.72
0.55
0.38
0.23
0.75
0.61
0.52
0.43
0.28


approximation is very competitive with alternatives that are computationally more

expensive.








CHAPTER 4
PRICING OF LOOKBACK OPTIONS USING LEVY PROCESSES
4.1 Lookback Options
Chapter 4 presents an efficient method to price lookback options in the Levy
process context by extending the random walk duality results of (AitSahlia & Lai,
1998) originally developed in the Black-Scholes set-up and by exploiting the very fast
numerical scheme recently developed by (Feng & Linetsky, 2008) and (Feng & Linetsky,
2009) to compute and invert Hilbert transforms. Though (Feng & Linetsky, 2009) also

apply the Hilbert transform technology to price lookback options, their approach is
significantly more complex than ours and is about twice as long. In addition, they need
to determine the transition probability density of the Levy process and impose conditions
that exclude pure jumps processes, such as the popular Variance Gamma model (cf.
(Madan & Seneta, 1990), (Milne & Madan, 1991), and (Madan et al., 1998).) In contrast,
our approach is much simpler and makes use of only the characteristic function of
the log-increment, which is central to Levy processes. Furthermore, by focusing our
approach on determining the distribution function of the maximum of the Levy process
we can also determine hedging parameters with minimal additional computational effort.
For ease of comparison we adopt the notation in (AitSahlia & Lai, 1998) originally
developed for Brownian motion but now assume that the underlying price process {St}
follows an exponential Levy process (i.e.; that which is followed by log St.) Given N
discrete monitoring dates t1, t2,... tN, the maximum price MN = max {St,... S } and
minimum price AN = min {St, ..., 5} of the underlying asset lead to inception (time
to = 0) prices for both fixed strike and floating strike lookback options summarized in
Table 4-1.
Table 4-1. Loookback option prices at time to = 0
Fixed strike Floating strike
e-rTE(M K) e-rTE(M- St,
e- rTE (K AN) e-rTE (S AN)









The difficulty in pricing these options is essentially due to the fact that the

distributions of MN and AN are not known in analytical form even for the standard

geometric Brownian motion of the Black-Scholes model.

4.2 Duality and Extrema of Random Walks

Under the assumption that the underlying price {St} follows an exponential Levy

process and given the discrete monitoring of the maximum and minimum at dates

tl, t2, ..., tN, we can write St = Soeu, where {Un : n > 1, U0 = 0} is a random walk with

i.i.d. increments X, such that their common characteristic function "v is explicitly known

thanks to the Levy-Khinchine formula.

Define now = inf {n : Un < 0} to be the first passage of the log-price process

below zero, observed on a monitoring date, and T+ = inf {n : Un > 0} the corresponding

first passage of the log-price process above zero. r_ or 7+ are called 'ladder epochs'.

The duality property of this random walk will enable us, through -_ and 7+, to derive

recursive expressions leading to the distributions of the extrema MN and AN.

Fixed strike lookback option example







0 1-
0 2



02)
S 01-






Monitoring dates


Figure 4-1. Sample path of a log-price process for a lookback option









Looking at Figure 4-1 we see that = 2 even though the log-price has dropped

below zero before time 1. Since we observe the prices only on the discrete monitoring

dates, this does not affect T_ as the price is back above zero at time 1. Also, -+ = 1 and

MN is equal to the price on the 10th monitoring date, even though the continuous price

process has a higher price since this higher price is not observed on a monitoring date.

From (AitSahlia & Lai, 1998) we know that the distribution of the maximum log-price can

be written as


P{MN E dx}

P {U1 dx} PX2 < 0, X2+ X3 < 0,..., X2 + ''XN <_ 0}
N
+ P {U, > U, i < v; U, c dx} x
v=2
P {X 1< 0, X+ + X,2 < 0,... X,+ + XN 0}]

for x > 0. Furthermore, the duality of random walks (Feller, 1971), lets us rewrite one of

the above probabilities in terms of one of the ladder epochs


P {U,> U,, i < v; U, E dx}

= P{U,- U,_ > 0 ..., U, U1 > 0; U, e dx}

= P{U, > 0,..., U,_1 > 0; U, e dx}

= P{_ > v; U, E dx}


And another of the above probabilities can also be written in terms of one of the ladder

epochs


P {X,,+ < 0, X+,, + X.+2 < 0, ... X,+I + + XN < 0}

= P{Ul <0, U2 < 0, ..., UN-V < 0}

= P { > N v}








Putting the simplified probabilities into the original equation yields, for x > 0,

P {MN E dx} = P {U1 dx} P {+ > N- 1} (4-1)
N
+ P {T_ > v; U, e dx} P {+ > N v}
v=2
and for x = 0, it is clear that P {MN = 0} = P {r+ > N}. The advantage of writing
the above probabilities in terms of the ladder epochs 7_ and 7+ is that they can be
determined recursively.
Define now the Fourier transform or characteristic function of a distribution function
F of a real random variable X as (cf. (Chung, 1974)) as:

F(F)() = E (ei'x) = JR e'xdF(x).

Alternatively, the notation F will also be used. Furthermore, we define the Hilbert
transform for such F by the Cauchy principal value integral

R-(F)() = p.v. d(x)
7T JR X,

which reduces to the earlier definition of a Hilbert transform when F is absolutely
continuous (with respect to the Lebesgue measure) with a density f e LP (R). We can
now state the following generalization to Proposition 1 in (AitSahlia & Lai, 1998).
Proposition 1. Let J be either (0, oo) or (-oc, 0] and 7 = inf{n : Un i J}. For x e J,
let dFn(x) = P {7_ > n; Un e dx} and let V(x) be the cumulative distribution function
(cdf) of a log-increment X, and "V its characteristic function. Then the characteristic
functions Fi, 2, ..., FN can be determined recursively through the following relations:

Fi = 4 (4-2)

Fn-- = RFn-i",i -i_ ) for 2 2 2








Proof. A straightforward generalization of the recursion on density functions in
(AitSahlia & Lai, 1998), pg. 230, Eq. 10, can be expressed as

Fi(x) = v(x)

Fn(x) = 1j(x) (F_-1 Wf) (x), for 2 < n < N

We now recall the following property that relates Fourier and Hilbert transforms for a
function 0 on R (cf. (Stenger, 1993) and (Feng & Linetsky, 2008)):

F2(1(,) 2 =(H (H ,

which together with the independence of the Levy increments leads, for 2 < n < N, to:

.'(F, ) = (1j-(Fn_I* ))
I /
2 2



Remarks. First, note that the preceding applies to the distribution of the minimum of
the random walk as well. Simply replace U, by U,. Then

AN = min{U : 0 < n < max{-U : 0 < n < N}

and for x < 0,

P {AN E dx} = P {U e dx} P {_ > N 1}
N
+ P {T > v; U, e dx} P { > N v}
v=2
Second, note that the recursions in Eq. 4-2 and Eq. 4-3 fit perfectly the set-up
of (Feng & Linetsky, 2008) to apply their highly efficient algorithm to compute all
the Fourier and Hilbert transforms and invert the last (FN)for pricing purposes at a
computational cost of 0 (NM log(M)), where M is the number of quadrature points









in the integrals and N is the number of discrete observation dates, with a resulting

error O (M'/(l ) exp(-cM" (+ ")), c > 0, which decays exponentially. The ultimate

determination of FN (via its Fourier inversion) is at the root of the computation of the

option price as we show next.

4.3 Fixed-Strike Lookback Options

We are now ready to apply the main result of the last section to price a fixed strike

(a.k.a. hindsight) lookback option, which, upon exercise, grants the right to purchase

the underlying asset at the minimum price and re-sell it at the strike K, for a put, or to

buy it at the strike K and re-sell it at the maximum for a call. To enable comparisons with

earlier results involving only Brownian motion, we shall focus on the call, whose payoff is

(SoeM K) .

Proposition 2. The value of a hindsight (or fixed-strike) lookback call at inception is
N /o
e-rTE (SoeM K) =e-rTaN(So K) e- rT (Soe K) dF,(x), (4-4)
vi~

where F,(x) are obtained through the application of the numerical scheme of (Feng &

Linetsky, 2008) to the recursions in Eq. 4-2 and Eq. 4-3 for x > 0, with J = (-o, 0],

and ao, al, ..., aN defined by

ao = 1, an = G,(0) lim Gn(x) for n > 1,
X--O0

where G, defined for x < 0 by replacing Fn by Gn in Eq. 4-2 and Eq. 4-3 and using

J =(-o, 0].

Proof. By definition, we have

E (SoeMN K) = (Soex- K) P{MN c dx},

the right hand side of which can be re-expressed as


(So-K) PMN =0}+ (Soex-K) P{MNedx}.
o+









Recall that 7+ = inf{n : U, > 0} and dG,(x) = P{r+ > n; Un, dx} for x < 0 and n > 1.

Therefore

an= dG,(x)= P{r+ > n} P{U < 0,...,Un <0}. (4-5)

The latter, together with (4-1) and the decomposition above, yields
N
P{MN C dx} = aNP{U1 e dx} -aN,dF,(x) for x > 0,
v=2

which in turn concludes the proof by virtue of P{MN = 0} = P{r+ > N}.

4.4 Floating-Strike Lookback Options

We show in this section that the pricing via the recursions in Eq. 4-2 and Eq. 4-3

extends to floating-strike lookback options. These are contrasted to the fixed-strike

by making the strike set to the price of the underlying upon exercise. Thus with a

floating-strike put, its holder can purchase the underlying at its trading price upon

exercise and sell it at the maximum it has achieved over the life of the contract, resulting

in a payoff (SoeMN SM) On the other hand, a floating-strike call allows its holder

to purchase the asset at the minimum it achieved during its life and sell it at the price

it trades upon exercise. Again, to allow for comparison with the classical Brownian

process in the Black-Scholes model we illustrate the application of the approach on the

put. Incidentally, floating-strike options are sometimes labeled standard.

Proposition 3. The value at inception of floating-strike lookback put is given by
N-1
e- E (SoeM SN) = e-"TSo /3N i/,
v=0
where


N-v = \ (1 e) dGN_,(x) for 0 < v < N,

1, = jC exdF(x) for v> 1,
Io = 1, l, = exdF,(x) for _> 1,
JO/








with F, and G, obtained through the recursions in Eq. 4-2 and Eq. 4-3 as in Proposition


Proof. Since SN = SoeUN, we have (SoeMp


E (eMN eUN) =


SN) = So (eM eUN) from which


where each of the above cases corresponds to the maximum being achieved at,
respectively, to = 0, tl, or t,, 2 < v < N 1. Observe that P{U, = Uj}=0 for i / j. By
definition, -+ = inf{n : Un > 0} and = inf{n : Un < 0}, but since P{Un = 0} = 0 for all
n > 0, we have 7+ = inf{n : Un > 0} almost surely. Therefore


E (1 eU ) l{u,

E (1- eU) l{->o}

0 e)dG(x)


Furthermore, we have
E (eul eUN) l{u1>o.u>u2,... U>U}


N XN
E (eU' eUl 2X) l{Ii>oUi2-uli<....UN-UI


xP Ul e dx, X2 <0, X2... X X2 X3 -- XN <0 Xi e dx
i 2
j exP {U e dx}

x (1- e)P X2 <0, X2 X3 < 0,..., X2 X3 -... XN < 0, ei dx
id 2
SexW(x) /1 (I e)dG1(y) ,
/o 0Jo


E (1 eu ) {u1
+E (eu' euN) {u1>o.ui>u2,...,UI>UN}
N-1
SE (eu eu ) l{ou+, ... u,>u},
v 2









where we make use of the independence between U, and (X2, ..., XN) in the next to last
step above.
Finally,

3E 2 E (eU eu") l{ouu, >u..., U
N-1 oo
= I (ex ex y)P{U < U... U < U,,; U dx}
2 x=0 J y=-oo
xP{X,+i < 0,..., X,+i + + XN < 0;X,+ + + XN E dy}

= exdFj(x) 0 (1 e)dGN _,(y) 0.
v-2 0 L-J

4.5 Extensions

Further applications of the technique presented above can be made with straightforward
modifications to situations where the payoff depends on the minimum. In addition, all

these options can be valued at other times than their inceptions by conditioning on the
supreme up to the valuation time prior to expiration. Other variations on the pricing of
these lookback include the situation, for example, where the supreme are observed
over a predefined window within the life of the contract. In all these cases, the general
relations provided by (AitSahlia & Lai, 1998) also apply here, with obvious modifications
and will therefore not be repeated here.

Additionally, our approach is particularly well-suited for the computation of hedging
parameters, which are especially crucial to the option writer's risk management practice.
For example, the fixed-strike lookback price at time 0 of Proposition 2, Eq. 4-4, can be
re-written as
N oo
e- E (Soe" K) = e-a (S K)+ + e-r (Soe K) dF,,(x)

J e- rrZ1 Jlog(K/So) (Soex K))d(x) if o < K

e-rTaN (So K) e- rT N Jo (Soex -K) dF,(x) if So > K









from which the delta and gamma parameters (first and second derivatives with respect

to So, respectively) can easily be computed.

4.6 Summary

In Chapter 4 we extended a recursive algorithm that was originally developed for

lookback option pricing when the underlying asset follows a geometric Brownian motion

and is monitored at discrete dates within the life of the contract. Our extension to the

geometric Levy processes exploited the duality property of random walks through the

use of ladder epochs resulting in recursion expressions for characteristic functions of the

extrema that are perfectly tailored for a powerful algorithm for Hilbert transform akin to

the Fast Fourier Transform. In addition, our approach yields hedging parameters with

little additional computational effort. The ability to develop such results is inherently

linked to the characterization of Levy processes as consisting of continuous-time

processes with independent and identically distributed increments. Thus their discrete

monitoring is in fact very helpful as it enables us to use readily available results from

fluctuation theory.









CHAPTER 5
CONCLUSION

Derivatives such as options are essential to the functioning of a modern economy.

They provide opportunities for hedgers seeking to reduce their financial risks as well

as speculators, whose hits and misses in the marketplace can provide additional

liquidity. The pricing and hedging of these financial instruments has become increasingly

challenging as ever more complex models have emerged to account for practical

features that cannot be ignored. Over the past few years, continuous-time asset pricing

models that rely on Levy processes have gained significant prominence. Their widening

adoption is due to their ability to capture salient features such as jumps and fat tails in

asset return distributions that cannot be ignored. For example, if one were to maintain

using the classical Black-Scholes-Merton model that gave mathematical finance its

impetus in the early 1970's and which relies on the normality assumption of asset

returns, one would seriously underestimate the actual probability of significant and

unusual drops. For example, (Kou, 2008b) shows that over the period Jan 2, 1980 to

December 31, 2005, the standardized (de-meaned and scaled by standard deviation)

daily return of the critically watched S&P 500 index ranged from a minimum of -21.1550,

to a maximum of 7.9967, which both occurred during the market crash year of 1987. Yet

the probability of a standard normal distribution falling 21 units below its zero mean is

approximately 1 x 10-107. For comparison, it is estimated that the universe is about 15

billion years (or 5 x 1017 seconds) old. There is therefore clearly a need for alternative

models, and those based on Levy processes have many favorable features, including

independence of increments and their infinite-divisibility, a variety of ways to capture

large deviations, the possibility to incorporate jumps, particularly the popular pure-jump

and jump-diffusion models. Finally, from a mathematical and computational tractability

perspective, there is the remarkable Levy-Khinchin representation which makes explicit

the characteristic function of the process in terms of three parameters. In addition,









recent developments in the inversion of Fourier (otherwise known as characteristic

functions in stochastic modeling) and related Hilbert transforms have spurred great

interest in Levy models.

The focus of this dissertation is on path-dependent options in the particular context

of Levy models. With payoffs depending on the entire path followed by the asset price

of the underlying up until exercise, these options are especially useful when their

holders wish to address a specific risk issue in a fashion that cannot be achieved by

standard (or vanilla) options alone. For example, they could be concerned only if the

underlying asset moves outside a certain range of values, say of interest or currency

exchange rates, in which case they would be interested in barrier options, which come

in the knock-in and knock-out flavors. The former entitle their holder the acquisition of a

standard option only if the underlying asset price crosses a barrier. They however have

to pay for the privilege upfront, with the possibility of never acquiring the option if the

underlying does not cross the barrier before expiration. On the other hand, a knock-out

option yields the same payoff as a standard option as long as the underlying asset price

does not cross a barrier prior to expiration. Though barrier options were not explicitly

addressed in this thesis, they are in fact intimately linked with lookback options, where

the statistical distribution of the maximum (or minimum) is paramount as it is clear that

a barrier above the initial asset price can only be breached if the maximum is above

while, correspondingly, a barrier below would only be breached when the minimum is

below it. Lookback options (or options on extrema) have the most flexible payoffs, and

are thus the most expensive. They are used by either speculators or by very risk-averse

operators. The other type of path-dependent options addressed in the present work

concerns Asian (also known as average) options, which are widely used by multinational

corporations to smooth their costs as well as their revenues in the face of highly variable

raw material prices and large fluctuations in currency exchange rates.









Since the successful use of continuous-time modeling based on stochastic

calculus to derive the celebrated Black-Scholes model, mathematical finance has

developed mainly in this realm and has accomplished much. However, the monitoring

of asset prices for path-dependent options is effected on a discrete set of dates. The

resulting mathematical problem is significantly more complex than the operational use

of stochastic calculus as it involves a mix of discrete and continuous methods. This

dissertation contains new results regarding the efficient pricing of lookback options

that exploit judiciously the random walk duality inherent to discretely observed Levy

processes together with recent algorithmic advances on Hilbert transforms that afford

computational complexity comparable to the Fast Fourier Transform. The other topic in

this dissertation concerns discretely monitored Asian options, the pricing and hedging of

which we address through the use of conceptual quantile options. Though they may not

be yet traded, history has proved that options initially started as concepts, such as Asian

and lookback options, do eventually enjoy acceptance in practice. In our case, they

enable a mathematical approach which when coupled with yet another set of results

from fluctuation theory based on characteristic functions leads to efficient pricing and

hedging computational advances.









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BIOGRAPHICAL SKETCH

Gudbjort Gylfadottir was born in Sweden, to Icelandic parents Gylfi Haraldsson and

Halla Arnlj6tsd6ttir. She grew up in Laugaras, Biskupstungur, a village in Iceland with a

population around 100 people; before moving to the capital, Reykjavik, where she went

to Verzlunarsk6linn high school. After that, she received her B.S. in mathematics from

the University of Iceland in 2006. In the fall of 2006, she moved to Gainesville, FL, to

pursue her doctoral studies in the department of Industrial and Systems Engineering at

The University of Florida, with concentration in quantitative finance. She received her

M.S. in finance from the Warrington College of Business at the University of Florida

in 2008 and her Ph.D. in industrial and systems engineering from the College of

Engineering in 2010.





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NoneofthiswouldhavebeenpossiblewithoutmyadvisorandfriendFarid.HisenthusiasmandambitioninspiredmeandIamthankfulforalltheknowledgehesharedwithmeduringcountlesshours.Murali,ofmycommittee,taughtmeagreatdealofmath,yetIammostappreciativeforgettingtoknowhimasafriend.IwouldliketothankbothMuraliandFaridforallourinspirationalconversations.Also,Iwouldliketothankmycommitteemembers,Dr.PardalosandDr.Nimalendranfortheirsupport.Thanksgoouttomyfamilyfortheirlove,allthephonecallsandfortheirlovelyvisits:MydadGyl,mybrotherThrostur,mysister-in-lawUna,mynephewsThorriandFrodiandmyparents-in-lawErnaandJon.Also,thanksgoouttomyfriends:Alex,Mireia,Ehsan,Kelly,Vera,Altannar,Ashwin,Shantih,Emily,May,Soheil,Behnam,Renee,Clay,Filip,Unnur,HelgaBjork,HelgaBjork,Svanhvt,Eln,AnnaGyda,Ragnheidur,LaraandJacki.Florida'snaturewithallitsmagicalwondersmadebeinghereanamazingexperience.Andlastly,mydeepestgratitudegoestoArni,whoisthemostlovinghusbandIcouldwishforandhasbeenhereformeallofthistime.Iamreallygratefulthatwegottosharethisexperience. 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1PATH-DEPENDENTOPTIONS ........................... 10 1.1Introduction ................................... 10 1.2AsianOptions .................................. 12 1.3LookbackOptions ............................... 16 1.4Overview .................................... 19 2LEVYPROCESSES ................................. 20 2.1MotivationforLevyPricingModels ...................... 20 2.2UsingLevyPricingModels .......................... 21 2.3TheFastHilbertTransform .......................... 25 3QUANTILEAPPROXIMATIONSFORASIANOPTIONS ............. 29 3.1Introduction ................................... 29 3.2QuantileOptions ................................ 29 3.3DistributionsforDiscreteQuantileProcesses ................ 32 3.4QuantileApproximationsforFixedStrikeAsianOptions .......... 33 3.5PricingintheBlack-ScholesModel ...................... 36 3.6HedgingParameters .............................. 39 3.7NumericalEvaluation ............................. 41 3.8Conclusion ................................... 41 4PRICINGOFLOOKBACKOPTIONSUSINGLEVYPROCESSES ....... 43 4.1LookbackOptions ............................... 43 4.2DualityandExtremaofRandomWalks ................... 44 4.3Fixed-StrikeLookbackOptions ........................ 48 4.4Floating-StrikeLookbackOptions ....................... 49 4.5Extensions ................................... 51 4.6Summary .................................... 52 5CONCLUSION .................................... 53 REFERENCES ....................................... 56 5

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Table page 3-1FixedStrikeAsiancalloptionwithparametersS0=100,r=0.1,n=50,andT=1.BenchmarkvaluesresultfromMonteCarlosimulationswith100,000paths(standarderrorinparentheses).Pricesusingquantileapproximations(with=3)aregiveninthelastcolumn. ..................... 42 3-2FixedStrikeAsiancalloptionwithparametersS0=100,r=0.1,n=50,andT=1.Approximationofoption'sdeltawith=3.BenchmarkvaluesresultfromMonteCarlosimulationswith100,000paths(standarderrorinparentheses). ............................................. 42 4-1Loookbackoptionpricesattimet0=0 43 7

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Figure page 4-1Samplepathofalog-priceprocessforalookbackoption ............. 44 8

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Thisdissertationisconcernedwiththepricingofpath-dependentoptionswheretheunderlyingassetismodeledasacontinuous-timeexponentialLevyprocessandismonitoredatdiscretedates.Theseoptionsenabletheiruserstotailorrandompayoffoutcomestotheirparticularriskprolesandarewidelyusedbyhedgerssuchaslargemultinationalcorporationsandspeculatorsalike.TheuseofcontinuoustimemodelssincethebreakthroughpaperofBlackandScholeshasbeengreatlyfacilitatedbyadvancesinstochasticcalculusandthemathematicaleleganceitprovides.Therecentnancialcrisisstartedin2008hashighlightedtheimportanceofmodelsthatincorporatethepossibilityofsudden,largejumpsaswellasthehigherlikelihoodofadverseoutcomesascomparedwiththeclassicalBlack-Scholesmodel.Increasingly,exponentialLevyprocesseshavebecomepreferredalternatives,thanksinparticulartotheexplicitLevyKhinchinrepresentationoftheircharacteristicfunctions.Ontheotherhand,therestrictionofmonitoringdatestoadiscretesetincreasesthemathematicalandcomputationalcomplexityforthepricingofpathdependentoptionsevenintheclassicalBlack-Scholesmodel.ThisdissertationdevelopsnewtechniquesbasedonrecentadvancesinthefastevaluationandinversionofFourierandHilberttransformsaswellasclassicalresultsinuctuationtheory,particularlythoseinvolvingrandomwalkdualityandladderepochs. 9

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Black&Scholes 1973 ),byonlyusingreadilyavailableparameters,namelytheprevailingrisklessrateinthemarketandthevolatility(standarddeviationofreturns)oftheunderlyingassetuponwhichtheoptioniswritten.Vanillaoptionsdependonlyonthepriceoftheunderlyingsecurityontheexercisedate,whereaspath-dependentoptionshaveanexercisepayoffthatdependsonthepricepathoftheunderlyingsecurityfromthebeginningofthecontractuntiltheexercisedate.AnAsianoptionisanexampleofapathdependentoption.ThepayoffofaEuropeanexercise-styleAsiancalloptionismax(ATK,0)=(ATK)+,whereKisthestrikepriceoftheAsianoptionandATistheaverageofthesecurityoverthelifeofthecontract.Incontrast,thecorrespondingpayoffofastandard(vanilla)calloptionismax(STK,0)=(STK)+,whereSTisthepriceoftheunderlyingsecurityatmaturity. 10

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Morgan 1999 ).Afterthecrashofnancialmarketsinlate2008,manybecameworriedthatunregulateduseofderivativeswasdangeroustonancialmarkets.Path-dependentoptionsarealsocalledexoticoptionsandaremostlytradedbetweenprivateparties,inso-calledover-the-counter-trade(OTC),notinopenmarkets.Theyhavethereforebeenhardforthelegislatortooversee.TheU.S.HouseofRepresentativesandtheU.S.SenatedraftedabillthatwastolimitOTCtradingofexoticderivativestorespondtoconcernsthattheiropacitycanbeasourceofinstability( Gibson 2010 ).Inaddition,thebillproposedthatsomeuncovered(or`naked')derivativestradingbebanned.However,thebillcameacrosshardoppositionfromagroupofinvestors,politiciansandacademicsandhashadsomealleviatingamendmentsaddedtoit,includingthedropoftheproposaltobannakedderivativestradingandthedropofmostlimitstoOTCtradingofoptions.ManysuggestedthateveniftheuseofexoticoptionswouldbelimitedintheU.S.thiswouldonlyspurlifeintoforeignOTCtradingsinceitwouldnotbelikelythatpeoplewouldstopusingtheseinvestmentvehiclessincetheyhavebecomesocommon. Attheendofthelastdecade,( Boyle&Boyle 2001 )notedthatgrowthinoptiontradinghadincreasedsignicantlyforthepast30yearsandthatintherstquarterof2000theestimatedvalueunderlyingoptioncontractsaroundtheworldwas$102trillion.Infact,thiswasonlytheestimatedvalueunderlyingexchangetradedcontracts,theestimatedvalueunderlyingoverthecounter(OTC)optioncontractswasestimatedtobe$88trillion( BIS 2000 )sothetotalvalueunderlyingoptioncontractswas$190trillioninthebeginningof2000.Forthelastquarterof2009,theestimatedvalue 11

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BIS 2010 ),orroughlyfourfoldthevaluefrom10yearsearlier,evenwhenitwasdownfrom$690trillioninthebeginningof2008.However,OTCseemstohavebecomethepreferredmethodoftradingoptions,with$605trillioninunderlyingvalueforOTCcontractsinJune2009.( BIS 2009 )Incomparison,theGDPoftheUSAwas$10trillionin2000and$14trillionin2009( BEA 2010 ),soatthebeginningofthedecade,thetotalvalueunderlyingoptioncontractsintheworldwasroughly19timestheGDP,andattheendofthedecadeitwas75timestheGDP. GeorgiosFoufasandMatsG.Larson 2008 ).Thepaperisstillaworkingpaper( Boyle&Emanuel 1980 ).BoyleandEmanuelcalledthisnewoptiontype,averagingoptions,buttheyweredubbedAsianoptionsbyBankersTrustbecausethermsthatboughttheoptionsfromBankersTrust,wereJapanese.Theserms'annualreportswerebasedonaverageexchangeratesovertheyear,soaveragerateoptionswereappropriateforthemtohedgetheirrisk( Vorst 1996 ).Inpracticetoday,Asianoptionsaremostlytradedonoilproducts,agriculturalcommoditiessuchascornandsoybeansandoncurrencies.Asfarbackasin1998,MicrosoftwasalreadytakingadvantageoftheeliminationofdownsideriskthatAsianoptionsoffer,alongwiththepotentialofanupsidegainbyhedgingtheirforeigncurrencyexposurebyusingAsianputoptions( WilliamFalloon 1998 ).Microsoft'streasureratthattime,Mr.Heitz,saidinaninterviewwithRiskmagazinethatMicrosofthad10-12counterpartiesfromwhichitcouldbuytheputoptions.Today,Asianoptionsarestillmostcommonlytradedoverthecounter.Asianoptionsareparticularlyusefulinthinlytradedmarketsortoprotectagainstlargepricevariations.Investorswhohaveanobligationdueonacertaindatewillwantaninsuranceagainstthecounterpartybeingabletomovepricesagainstthem. 12

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Nowadays,ontheChicagoMercantileExchange(CME),averageoptionsareconstructedinthefollowingway:Theoptionhasaswap(i.e.;acontracttoexchangeaninterestorcurrencyrateforanother)astheunderlyingsecurity,andaxedstrikeprice.Thenalpriceontheswapisusedtocalculatethepayoffoftheaverageoption.Thenalpriceontheswapiscalculatedbytakingthearithmeticaverageofdailypricesfromeachdayforwhichapricefortheunderlyingsecurityfortheswapisdeterminedforthepreviousmonth.Thedailypriceisfoundbytakingtheaverageofthehighandlowquotationsoneachdayfortheunderlyingsecurityfortheswap.Thepayoffforanaveragecalloptionwillbethenalpriceontheswapminusthestrikeprice,andthepayoffforanaverageputoptionwillbethestrikepriceminusthenalpriceontheswap.Eventhoughthisstructureisintricate,theaveragepriceoptionpayoffissimplythedifferencebetweenthearithmeticaveragepriceofthesecurityitselfoverthepreviousmonthminusaxedstrikeprice,soourpricingmodelforAsianoptionsgiveninthispaperisapplicabletotheaveragepriceoptionstradedontheCME.OntheCME,allthe19averageoptionsavailableinearly2010,hadanoilproductastheunderlyingsecurity,andtheywerealltradedonCME'soverthecounterclearingservice.Theoilproductsincludede.g.gasoline,jetkerosene,fueloil,propane,butane,heatingoil,gasoil,ethaneandcrudeoil( CMEGroup 2010 ).Throughtheseexamples,itisclearthatAsianoptionsarewidelyused. Asianoptionsarelesslikelythanvanillaoptionstobemanipulatedbecauseitisnotpossibletomanipulatethepriceoversuchalongtimeasopposedtovanilla(orregular)options.ArecentexamplefromthedropintheDowJonesbyalmost10%withinafewminutes( Mattich 2010 )showsthatwhetheritisbymistakeormanipulation,itispossibleforthemarkettobeaffectedseverelyfromotherfactorsthanefciencyinjustamatterofminutes.Inthecaseofavanillacalloption,hadtheclosingpriceof 13

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IntherstpublishedpaperonAsianoptions,( Kemna&Vorst 1990 )usedMonteCarlomethodstodeterminethepriceofthearithmeticAsianoption.ByusingthegeometricAsianoptionasacontrolvariate,wherethegeometricaverageisgivenbyAT=nYi=1Sti!1=n Broadieetal. 1999 ).Aspreviouslynoted,thearithmeticAsianoption,wherethearithmeticaverageisgivenbyAT=1 Turnbull&Wakeman 1991 )proposedusinganapproximationofthedensityfunctionofthearithmeticAsianoptionbyusinganEdgeworthexpansion.Amongthersttoderiveanalyticresults,( Geman&Yor 1993 )computedtheLaplacetransformofthepriceofacontinuouslysampled 14

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Fuetal. 1998 ).Ontheotherhand,inpractice,samplingisperformedoveradiscretesetofdates(daily,weekly,etc.)Inthiscase,noanalyticresultsareavailableevenintheBlack-Scholesframework,wherethemainsourceoftheproblemstemsfromthelackofanexplicitdistributionforthesumofcorrelatedlog-normalrandomvariables.Asaresult,asignicantnumberofapproximationsthatproduceclosed-formexpressionshaveappeared.Forexample,( Thompson 1998 )providestightanalyticalboundsand( Linetsky 2004 )derivedanewintegralformulaforthepriceofacontinuouslysampledAsianoption,whichisagainslowlyconvergentforlowvolatilitycases.Ingeneral,thepriceofanAsianoptioncanbefoundbysolvingapartialdifferentialequation(PDE)intwo-dimensionalspaces(see( Ingersoll 1987 )),whichispronetooscillatorysolutions.Ingersollalsoobservedthatthetwo-dimensionalPDEforaoatingstrikeAsianoptioncanbereducedtoaone-dimensionalPDE.( Rogers&Shi 1995 )simplerformulatedaone-dimensionalPDEthatcanmodelbothoatingandxedstrikeAsianoptions.Howeverthisone-dimensionalPDEisdifculttosolvenumericallysincethediffusiontermisverysmallforvaluesofinterestonthenitedifferencegrid.SeveralarticlescontainattemptstoimprovethenumericalperformanceofthisPDE.( Andreasen 1998 )appliesthereductionofRogersandShitodiscretelysampledAsianoption.IndependenteffortsinrecentyearshaveattemptedtounifypricingtechniquesfordifferenttypesofoptionsandrelatethesemethodstopricingAsianoption.UsingagainRogersandShi'sreduction,( Lipton 1999 )noticedsimilaritiesinpricingequationsforthepassport,lookback,andAsianoptions.( Shreve&Vecer 2000 )developedtechniquesforpricingoptionsonatradedaccount,whichincludealloptionsthatcouldbereplicatedbyself-nancingtradingintheunderlyingasset.TheyincludeEuropean,passport,vacation,aswellasAsianoptions.Numericaltechniquesforpricingcontractsofthistypearedescribedin( Vecer 2001 ).( Hoogland&Neumann 2001 )developed 15

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Fusai&Meucci 2008 )derivepricingmethodsforbotharithmeticandgeometricAsianoptionsunderdiscretemonitoringandforageneralLevyprocess.TheyworkontheFourierspacewitharecursivepricingformulalikewedo.However,fortheirre-centeringtechniquetheyrequirenitemoments,whichwedon't.Inaddition,itisunclearhowtheirmethodcanleadtocomputinghedgeparameters,whileourswillbeshowntoproducethemwithminimaladditionalcomputations. Comparedtootheroptions,lookbackoptionsprovidethebiggestpayoffpotentialbecausetheinvestorcanchoosetheexercisedateinretrospect,thatisbylookingbackoverthelifeoftheoption.Thereportedusesforlookbackoptionsaremainlyspeculative.ItisobviousthatlookbackoptionswillbemoreexpensivethanvanillaorAsianoptionsbecausetheholderisgettingthebiggestpotentialpayoffoverthewholelifeoftheoption. 16

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Heynen&Kat 1995 ),( Conze&Viswanathan 1991 )and( Goldmanetal. 1979 ).Inpractice,monitoringoccursatdiscretedatesandthemonitoringdatest1,...,tN=Tarepredetermined.( Kou 2008a )saysthatitispracticalthatmonitoringisdiscrete,andthatifmonitoringwerecontinuous,therewouldbearbitrageopportunitiesforbarrieroptions,e.g.ifabarrierisreached.Thosecouldrepresentthemselveswhilemarketsareopeninonlysomepartsoftheworld.Similarly,ifthehighest/lowestpriceforalookbackoptionduringitscontractperiodsofarisreachedatatimewhennotallexchangesareopensimultaneously,itwouldbeunfairtothetradersinpartsoftheworldwherethemarketsareclosedsincetheyareunabletotradeuponthatinformationimmediatelywhileothersinopenmarketswouldreaptheprots.Alltradedlookbackoptionshavediscretemonitoring,soevenifahigher/lowerpriceisobservedoutsideofthemonitoringdates,itisnottakenintoaccountfordeterminingtheextremaofpricesoverthecontractperiod.Asaconsequenceofthediscretemonitoring,pricingismathematicallyandcomputationallychallenging.Substantialmis-pricingoccurswhenadiscretelymonitoredcontractispricedapproximatelybyacontinuous-monitoringformula(cf.( Broadieetal. 1999 ),( Heynen&Kat 1995 ).)( Broadieetal. 1999 )introducecorrectiontermssothatthecontinuous-monitoringformulascanbeusedasapproximationsforthediscretelymonitoredoptions.Theirmethodalsoimprovesconvergencebymeansoflatticemethods.( Babbs 2000 )usesabinomialmodeltopricecontinuouslymonitoredoatingstrikelookbackoptions.Usingdiscretemonitoringandpricingforbothxedstrikesandoatingstrikes,( Cheuk&Vorst 1997 )alsouseabinomialmodeltopricelookbackoptions,improvinguponBabbs.( Boyle&Tian 1999 )usedatrinomialmethodtovaluethenon-GaussianCEVprocessandfoundthepriceforlookbackandbarrieroptionswhenthepricefollowstheCEVprocess.Latertheyfoundthatitwasinaccurate 17

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Boyleetal. 1999 ).( Davydov&Linetsky 2001 )alsofoundpricingformulasforthelookbackoptionwhentheunderlyingfollowstheCEVprocess.UsingLaplacetransforms,theirmethodisfasterthanthatofBoyleandTian.( AitSahlia&Lai 1998 )usethedualitypropertyofrandomwalkstoderiverecursivelythedistributionofconditionedextremaofthegeometricBrownianmotionpriceprocessandusenumericalintegrationmethodstopricelookbackoptions.( Tseetal. 2001 )useatridiagonalprocedurethattakesadvantageofthepropertiesofthegeometricBrownianmotionpriceprocessandpricethelookbackoptionsnumericallyandachievemoreefciencythanpreviousmethods.( Andricopoulosetal. 2003 )developaquadraturemethodthatcanbeusedtonumericallypriceawiderangeofoptions,includinglookbackoptions.( Broadie&Yamamoto 2003 )developafastGausstransformfornon-path-dependentoptionvaluationundergeometricBrownianmotionandtheMertonmodel.( Broadie&Yamamoto 2005 )extendtheirpreviousresultsandderiveadouble-exponentialGaussianmodelthatcanbeappliedtolookbackoptionsandotherpath-dependentoptions.( Petrella&Kou 2004 )ndLaplacetransformsofdiscretelookbackoptionsusingarecursionformula.TheseinvolveSpitzer'sformula.TheyinverttheLaplacetransformsnumericallytogetthelookbackoptionpriceandhedgingparametersforseveralLevypricemodels.ForthegeometricBrownianmotionpriceprocessanddiscretemonitoring,( Atkinson&Fusai 2007 )ndthedistributionoftheextremaofpricesinclosedformandarethusabletondthelookbackpriceforxedandoatingoptions.Thelatestworkonlookbackoptionsisby( Feng&Linetsky 2009 ).Theydoaforwardrecursiononthepricesofthelookbackoption,utilizingHilberttransformsandFouriertransforms.Theirmethodisefcientandaccuratebutisrestrictedbysomeconditionsmakingitinapplicabletotheimportantpure-jumpprocesses.Incontrast,ourmethod,tobedescribedindetaillater,ismoregenerallyapplicableandhasthe 18

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Feng&Linetsky 2008 )and( Feng&Linetsky 2009 ),whichenableustomakeadditionalcontributionstoefcientlyandaccuratelypricelookbackoptionsasdescribedindetailinChapter4.Chapter3dealswiththepricingofAsianoptionsandprovidesadetaileddescriptionofourapproximationapproachbasedonyetanothertypeofpath-dependentoptions,namelyquantileoptions,thatarenottradedbutwhichprovidemathematicalexpediency.ThecontributionsofthisthesisconsistofnewtechniquestopricediscretelymonitoredAsianandlookbackoptions.Theirdistinguishingfeatureliesinworkingonthecharacteristicfunctionoftheoptionpricedistributionratherthanonthecharacteristicfunctionofthepriceitself,asisdonein( Feng&Linetsky 2008 )and( Feng&Linetsky 2009 ),themostcompetitiveapproachup-to-date.Ourshasthesignicantadvantageofenablingadirectcomputationofhedgingparameters,theGreeks,incontrasttotheunstablenumericalderivativesandthecomputationallycomplexMalliaviancalculusrequiredbyalltheotheralternatives.Inaddition,the( Feng&Linetsky 2009 )pricingmethodforlookbackoptionsisslowerthanoursandexcludesanimportantclassofLevymodelsinnance,thepopularvariancegammaspecication( Madan&Seneta 1990 ). 19

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Merton 1976 )notedthatfartoomanyrandomjumpsoccurinthepriceprocessinpracticetobejustiedbyconstantvolatilityoracontinuouspathofprices.Hethereforesuggestedanadditionofajumptermtothepriceprocess,sothatdSt=Stdt+StdW+dq Kou 2002 )andin( Carretal. 2002 ).KouandCarretal.suggestmodelstoremedythoseissues,respectively,theKoumodelwhichhasbothadiffusioncomponentandajumpcomponentandtheCGMYmodel,whichonlyhasajumpcomponent.Alloftheaforementionedmodels,includingtheBlackandScholesmodel,arespeciccasesofageneralclassofprocessescalledLevyprocesses.Levyprocessesarefairlygeneralandallowfora 20

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a)Independentincrements:Thatisforallt0,t1,....,tn,therandomvariablesXt0,Xt1Xt0,....,XtnXtn1areindependent b)Stationaryincrements:ThatisXtXshasthesamedistributionasXts+uXuand c)Continuouspathsa.e:Thatislimh!0P(jXt+hXtj)=0forany>0. EveryLevyprocesscanbefullydescribedbythreeparameters.Thersttwoparameters,aand2,describethecontinuouscomponentoftheLevyprocess,andthethirdparameterisafunction(x),calledtheLevydensity,whichidentiesthediscretecomponentoftheLevyprocess.Furthermore,aistheconstantdriftofthecontinuouscomponentand2istheconstantvarianceofthecontinuouscomponent. Usingonlythosethreeparameters,theLevy-Khinchinformula:lnE[eiXt]=ait1 22t2+tZ(eix1ixIjxj<1)(x)dx Levyprocessescanhaveeitherniteactivity,whichmeansthatoveranyinterval,therewillbeaniteamountofjumps,ortheycanhaveinniteactivity,whichmeansthatanyintervalwillhaveinniteamountofjumps( Wuetal. 2008 ).Purejumpprocesseswithinniteactivity,areoftennotdistinguishablefrompurediffusionprocesses,andwhenthereisinniteactivityitisnotnecessarytohaveaBrownianmotioncomponentaswell.Whenis0,wehaveapurejumpprocessandwhen(x)iszerowehaveapurediffusionprocess.ThearrivalrateforjumpsisdeterminedbyRR=0(x)dx=. 21

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22t2 22tandvariance2t.TheMertonmodelhasLevydensity: 22 22t2+tfe22 2+i1g, Cont&Tankov 2004 ),howevertheprobabilitydensitycanonlyberepresentedasaninniteseries,andisthusnotavailableinclosedform.Thisprovidesanadditionalcomputationalcomplexityinderivinganoptionvaluewhichhasthispriceprocessastheunderlyingasset.Ononehand,thisisabettermodelforthepriceprocess,becauseitismorerealisticthatthepriceexhibitsomejumps,justasitmightwhennewinformationarrivestothemarketthatimmediatelychangesmarketparticipants'opiniononwhatthepriceshouldbe,sothatthepriceimmediatelyadjusts.Itisworthnotingthatamorerealisticmodel(thantheBlackScholesmodel)wouldonlybeusefulinpractice,ifitenablesustopricethederivativesofthepriceprocess.Forthebasecase,avanillacalloption,thepriceoftheoptionatinitiationusingtheMertonmodelwouldbe: 22

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2(2t+k2) Merton 1976 )wheretheinnitesumwithintheintegralcanbesimplied.( Carr&Madan 1999 )deriveamethodinwhichitisnotnecessarytoknowthepdfofthepriceprocesstocalculatetheoptionprice.Ratherthanworkingwiththedirectintegralabove,theyworkwithitsFouriertransformwhichtheyobtaininanexplicitform,albeitnottriviallybecauseinorderfortheintegraltobenon-singular,theyhavetomultiplytheFouriertransformwithaspecicremedialfunction.Theexplicitformulainvolvesthecharacteristicfunctionofthepriceprocess,whichasmentionedabove,canalwaysberetrievedfromtheLevy-KhinchinformulaforallLevyprocesses.OncetheyhaveanexplicitformulafortheFouriertransform,theytaketheinverseFouriertransform,thenmultiplyagainwiththeinverseoftheremedialfunctiontoretrievetheoptionprice.WhencalculatingtheinverseFouriertransform,theyusethediscreteFouriertransform(DFT)ontheintegral,whichmeansthattheyhavetodiscretizetheintegral.TospeedupthecalculationstheythentransformtheintegraltoconformtothesetupfortheFastFourierTransform(FFT)whichisfasterthancalculatingtheDFTdirectly,O(Nlog(N))vsO(N2)respectively.TheFFTwillgivepricesofseveraldifferentstrikesforeachcalculationoftheFFT.Whenperformingthesecalculations,thereisachoicetobemadefortheFFT,ifthegridfortheDFTischosentobewide,thestrikepriceswillberelativelyclosetoeachother,andifthegridfortheDFTischosentobene,thestrikepriceswillbefarapart.Sothechoiceofthegridhastobemadeaccordingtowhatstrikepricerangeisneeded.Also,thechoiceoftheremedialfunctionhastobemadecarefullysothatitensuresintegrability.( Lee 2004 )discussesthesechoicesofparametersinmoredetail,andshowshowthe 23

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( Kou 2002 )proposesamodelthathasjumpsinadditiontoadiffusionprocess,butthejumpshavedoubleexponentialdistributioninsteadofnormaldistribution,likeintheMertonmodel.Also,thedistributionofjumpsisdifferentdependingonwhetheritisanupwardmovementoradownwardmovement,reectingthetrendthatstockpricechangesseemgenerallytobeofdifferentmagnitudeforgoodnewsandbadnews( Chenetal. 2003 ).FortheKoumodeltheLevydensityis: Hull 2006 ).Thestandarddeviation,orvolatilityasitiscalledinthenanceliterature,isassumedtobexedintheBlackandScholesmodel.Yet,whenvanillaoptionmarketpricesareobservedfordifferentstrikeprices,andtheBlackScholesmodelissolvedtoreturnthevolatility,itisdifferentfordifferentstrikeprices,typicallyhigherthefurtherawayfromatthemoneythestrikepriceis.Itcanalsobeskewed,referringtothattheimpliedvolatilityishigherforstrikepricesundertheatthemoneyprice,andlowerforstrikepricesthatareoutofthemoney.BecauseofthejumpsthattheKoumodelincorporates,thissmiledisappearsandtheimpliedvolatilitybecomesconstant. TherearetwomoreprominentLevymodelsthatwewillmention.FirstistheVarianceGammamodel( Madanetal. 1998 ),thatalsomakesimpliedvolatilityconstantforvanillaoptions,sothatnovolatilitysmileisobserved.Thisisdoneinaverydifferent 24

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Carretal. 2002 )isageneralizationofboththeKoumodelandtheVarianceGamma,andithasveparametersthatneedtobespecied.TheLevydensityoftheCGMYmodelis Carretal. 2002 )specifythe5parameters(inadditiontothefourparametersintheLevydensity,needstobespecied)thatmaketheCGMYmodelttheS&P500indexthebest,andalsodisplayhowdrasticallythedistributionfunctionchangesbyjusttwistingevenoneoftheparametersatatime,therebyshowinghowsensitivethemodelistoparameterchanges.ItshouldalsobenotedthateventhoughLevypricingmodelssolvealotoftheempiricalissuesthatusingtheBlack-Scholesmodelentails,modelselectionofaLevyprocessishard,mainlybecausetherearesomanyparameterstoestimate.Thedataneededtoestimatetheexactparametersandmodelswouldhavetobeenormoustojustifyusingonegoodmodelratherthananother.( Heyde&Kou 2004 ) Carr&Madan 1999 )canbeutilizedtopricevanillaoptionsforanyLevyprocess.( Feng&Linetsky 2008 )and( Feng&Linetsky 25

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)developaHilbert-transformbasedmethodtopricebarrierandlookbackoptionswhentheunderlyingassetfollowsanexponentialLevyprocess.TheirrecoursetoHilberttransformsintheFourierspacestemsfromthepresenceofanindicatorfunctionmultiplyingthefunctionofinterest;theprice.Thisindicatorfunctioncapturesthepath-dependencyoftheoptionpayoffsuchasthebarriercrossingeventpriortotheoptionexpiration,forexample.Succinctly,theyusethefollowingpropertyrelatingFourierandHilberttransformsforagivendenedon<:F1(0,1)()=1 2^+i 26

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1+i^vj1()=1 2^()^vj()+i 2Z1eix^()^v1()d

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Feng&Linetsky 2009 )utilizetheFastHilberttransform,byworkingforwardintheirrecursivescheme,ratherthanbackward.Inourapproach,basedonevaluatingtheoptionpricedistributioninstead,westillmaintaintheuseofthefastHilberttransformdiscretizationalgorithmof( Feng&Linetsky 2008 )inabackwardrecursivefashion.AsmentionedinChapter1,themainadvantageofourapproachisitsabilitytogeneratehedgeparametersmuchmoreseamlesslythananyotheralternative. 28

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Chapter3isorganizedasfollows.Therstsectionsummarizestheconceptofaquantileoptioninboththeoriginalcontinuous-timesettingandourdiscreteset-upfordiscretemonitoringofpath-dependentoptions.Thesecondsectioncontainsourquantile-basedapproximationinageneralLevyprocessframework.ThelastsectionpresentsanumericalillustrationontheparticularcaseoftheBlack-Scholes(Brownian)model. Miura 1992 ),theseoptionsarepath-dependentandaremeanttogeneralizetheconceptofoptionsonextrema(minimumormaximum).Fora(,)BrownianmotionfXt,t0gand2(0,1),denethequantileprocessfM(,t),t0gby:M(,t)=infx:Zt01(Xsx)ds>t.

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Akahori 1995 )and( Dassios 1995 )whointheprocessgeneralizethearc-sinelawforBrownianmotion.Moreprecisely,theyobtain where andg1andg2aretheprobabilitydensityfunctionsassociatedwithsup0stXsandinf0s(1)tXs,respectively,i.e: Thesefunctionsareexplicitlyderivedas 2exp2x p 2exp2x p Thequantileoptionpriceattime0isthen whichcanbeevaluatedthroughnumericalintegrationastheassociatedprobabilitydensityfunctiongisdeterminedthroughEq. 3 throughEq. 3 ThekeytothederivationoftheaboveresultsbeginswiththeequivalencebetweentheeventsfM(,t)>xgandfRt0(Xsx)ds
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Dassios 1995 )): whereX(1)(t)andX(2)(t)areindependentcopiesoftheprocessX(t)=t+B(t),withB(t)denotingastandardBrownianmotion.Furthermore,( Dassios 1995 )alsoderivesthejointdistributionofM(,t)andX(t): Infact,bothEq. 3 andEq. 3 holdwhenthereferenceXisaLevyprocessas( Dassios 1996 )shows.WhilethederivationoftheresultsfortheBrowniancaseisbased,respectivelyforEq. 3 andEq. 3 ,ontheFeynman-KacformulaandtheGirsanovtheorem,themethodofprooffortheLevyprocessreliesinfactonanasymptoticdiscretization.ThelatterwillturnouttobeexactlywhatweneedfortheAsianoptionpricingwithdiscretemonitoring.Specically,( Dassios 1996 )developsthefollowing: where,forintegers0jnandadiscreteprocessX=(X0,X1,X2,...),Mj,n(X)isthe(j,n)thquantileofXdenedasMj,n(X)=inf(x:nXi=0(Xix)>j).

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hasbeenknownsince( Wendel 1960 ). 3 dealswithquantileprocesses.Thuswemakeuseofcorrespondingcollectionsoforderstatisticswiththeassociatedmodeofweakconvergence.Forthispurpose,weshallshowthatwecanrelyoneitherconvergenceofcharacteristicfunctionsinthegeneralLevycase,oronrandomwalkapproximationinthecaseofBrownianmotion.Forthelatter,wewillshowthroughanumericalillustrationhowBernoullirandomwalksresultsdueto( Takacs 1996 )canbeexploited.Fortheformer,weexploittheLevy-KhinchinecharacterizationtheoremfortheincrementofaLevyprocessandmakeuseofresultsdueto( Pollaczek 1975 )onorderstatisticsasweshownext. LetX1,X2,...beacollectionofi.i.drandomvariables.WeareinterestedindeterminingthecharacteristicfunctionsoftheorderstatisticsoftherandomwalksamplesX1,X1+X2,...,Pn1Xi.Thus,wedeneforn1and1n, where,forrealnumbersa1,a2,...,an,max()(a1,a2,...,an)representsthethnumbertakenindescendingorderinthecollection.Withthisconvention,wehavemax(1)(a1,a2,...,an)max(a1,a2,...,an).Inotherwords,Xn,,1nrepresent(an)orderstatistics(process)fortherandomwalksvaluesX1,X1+X2,...,Pni=1Xi.Wenowadopttheapproachfollowedby( Pollaczek 1975 )inordertodeterminethemomentgeneratingfunctionsofthecharacteristicfunctionsforXn,.Morespecically, 32

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letG(q,x,y)=1Xn=1nX=1xn1y1Eexp(qXn,), Pollaczek 1975 )thenshowsG(i,x,y)=(i) (1xy)(1xy(i))exp"1Xn=1xn forjxj<1,jxyj<1,whereFnisthen-foldconvolutionofFwithitself,sothatn(q)=Z1eqtdFn(t) 2ZCeaq ForaLevyprocess,(q)isexplicitlygivenandthusFncanbeobtainedviaFastFourierTransform.ThecharacteristicfunctionofanyX(n,)isthentriviallyretrievablethroughderivativeswithrespecttoxandyevaluatedatx=0andy=0. whichcanbeevaluatedinclosed-formwithgeometricaveraginginthestandardBlack-Scholesmodel.Inpractice,averagingisarithmeticoverdiscretelysampledpricesoftheunderlying.Inthiscase,therearenoknownclosed-formexpressionsforthedistributionofasumofcorrelatedlog-normalrandomvariables.Asaresult, 33

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Benhamou 2002 )and( Linetsky 2004 ).)InChapter3,weproposeusingquantileoptions,forwhichanalyticexpressionsarereadilyavailable,toapproximatethepriceofadiscretelysampledAsianoptionwithaxedstrike. Inthissectionwedetailourquantileapproximation.Itisbasedonthreeelements:(i)thepayoffofanAsianoptionisamonotonetransformationoftheaverageprice,(ii)thearithmeticaverageofarandomsampleisthesameasthatoftheassociatedorderstatistics,and(iii)thelatteraregenerallyconsistentestimatorsofquantiles.Ultimately,ourtaskistoevaluateexpectationsoftheformE[Z],whereZ=S0eM(,T)K+andM(,T)isthe-quantileoftheunderlyingprocessovertheinterval[0,T].Notethatfornowwerefertoagenericquantile.However,wewilllaterdenesuchprocessesusingnotationreferringdirectlytothediscretesamplingoftheunderlying. Withdiscretemonitoring,ATinEq. 3 isthearithmeticaveragetakenoverasetofpricesmonitoredattimest1,t2,...,tn:=T:AT=1 34

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Proof. 35

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WiththisapproximationandgiventhedeterminationofthedistributionsofthevariablesM(i,T,n)asdescribedintheprevioussection,wenowhavealltheingredientstoproceedwiththepricingofadiscretelymonitoredAsianoption. 36

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2+m0 2m0 Takacs 1996 )wehaveforx>0theapproximationP(x 2+mp 2mp

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2mp 2+mp andPfn(k)njg=8><>:1Pn+k+1i=j+1Pfn(k1)=igfor0>kjn,0fork
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q)p qkPfni
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dS0Delta=d dS0Z1ln(K=S0)g(x)dx1 dS0g(ln(K=S0))=1 dxg(ln(K=S0)) dxPfM(,m,,T,n)xg+d dxPfM(,m,,T,n)>xgex dS0=d dS0S0Z1ln(K=S0)PfM(,m,,T,n)>xgexdx=Z1ln(K=S0)PfM(,m,,T,n)>xgexdxS0S0 S20PfM(,m,,T,n)>xgexjx=ln(K=S0)=Z1ln(K=S0)PfM(,m,,T,n)>xgexdx+K S0PfM(,m,,T,n)>ln(K=S0)gGamma=d dS0dp dS0=K S0d dS0[PfM(,m,,T,n)>ln(K=S0)g]=K S0d dx[PfM(,m,,T,n)
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3-1 and 3-2 indicatethattheapproximationisinfactverywellbehavedevenwhenisassmallas3.FromTable 3-1 ,observethattheaccuracyoftheapproximationdeterioratesonlyinasmallnumberofcasesthathavenopracticalinterest.Theyaredeepout-of-themoney(thusunlikelytobeexercised)optionswithnegligibleprices.Inalltheothercases,thedeviationsfromthebenchmarkvaluesareinfactwellwithinthebid-askspreadforover-the-counteroptioncontracts.SimilarobservationscanbemaderegardingtheresultsdisplayedinTable 3-2 .Inthiscase,weareabletoobtainhedgingparametersthatareasimportantfortheoptionwriter,typicallyabankascounterpartytoahedgefund,amanufacturer,orairlinecompany.Thesehedgingparametershavetraditionallybeenomittedfromtheoptionpricingliteratureorrelegatedtonumericalderivationvianite-differences,whicharenumericallyunstable,orMonteCarlosimulation,whichisverytime-consuming. 41

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FixedStrikeAsiancalloptionwithparametersS0=100,r=0.1,n=50,andT=1.BenchmarkvaluesresultfromMonteCarlosimulationswith100,000paths(standarderrorinparentheses).Pricesusingquantileapproximations(with=3)aregiveninthelastcolumn. FromBenhamou'sPaper(MonteCarloPriceandSE)BenchmarkPrice(ExpectedValueandSE)OptionPriceUsingQuantileOptions 0.18022.78(0.00)22.78(0.00)22.710.19013.73(0.00)13.73(0.00)13.680.11005.24(0.00)5.25(0.00)5.290.11100.72(0.00)0.73(0.00)1.070.11200.03(0.00)0.03(0.00)0.130.38023.07(0.01)23.09(0.01)22.940.39015.22(0.01)15.20(0.02)15.230.31009.01(0.01)9.00(0.02)9.070.31104.83(0.01)4.86(0.02)5.150.31202.35(0.01)2.39(0.01)2.830.58024.83(0.03)24.86(0.03)24.560.59018.32(0.03)18.29(0.04)18.130.510013.18(0.03)13.13(0.04)12.990.51109.23(0.03)9.24(0.04)9.330.51206.36(0.03)6.32(0.03)6.69 FixedStrikeAsiancalloptionwithparametersS0=100,r=0.1,n=50,andT=1.Approximationofoption'sdeltawith=3.BenchmarkvaluesresultfromMonteCarlosimulationswith100,000paths(standarderrorinparentheses). 0.1800.95(0.000)0.950.1900.95(0.000)0.940.11000.78(0.001)0.720.11100.22(0.001)0.200.11200.01(0.000)0.030.3800.91(0.000)0.860.3900.79(0.001)0.720.31000.61(0.001)0.550.31100.41(0.001)0.380.31200.24(0.001)0.230.5800.82(0.000)0.750.5900.71(0.001)0.610.51000.58(0.000)0.520.51100.46(0.001)0.430.51200.35(0.001)0.28 42

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AitSahlia&Lai 1998 )originallydevelopedintheBlack-Scholesset-upandbyexploitingtheveryfastnumericalschemerecentlydevelopedby( Feng&Linetsky 2008 )and( Feng&Linetsky 2009 )tocomputeandinvertHilberttransforms.Though( Feng&Linetsky 2009 )alsoapplytheHilberttransformtechnologytopricelookbackoptions,theirapproachissignicantlymorecomplexthanoursandisabouttwiceaslong.Inaddition,theyneedtodeterminethetransitionprobabilitydensityoftheLevyprocessandimposeconditionsthatexcludepurejumpsprocesses,suchasthepopularVarianceGammamodel(cf.( Madan&Seneta 1990 ),( Milne&Madan 1991 ),and( Madanetal. 1998 ).)Incontrast,ourapproachismuchsimplerandmakesuseofonlythecharacteristicfunctionofthelog-increment,whichiscentraltoLevyprocesses.Furthermore,byfocusingourapproachondeterminingthedistributionfunctionofthemaximumoftheLevyprocesswecanalsodeterminehedgingparameterswithminimaladditionalcomputationaleffort. Foreaseofcomparisonweadoptthenotationin( AitSahlia&Lai 1998 )originallydevelopedforBrownianmotionbutnowassumethattheunderlyingpriceprocessfStgfollowsanexponentialLevyprocess(i.e.;thatwhichisfollowedbylogSt.)GivenNdiscretemonitoringdatest1,t2,...,tN,themaximumprice~MN=maxfSt1,...,StNgandminimumprice~N=minfSt1,...,StNgoftheunderlyingassetleadtoinception(timet0=0)pricesforbothxedstrikeandoatingstrikelookbackoptionssummarizedinTable 4-1 Table4-1. Loookbackoptionpricesattimet0=0

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Denenow=inffn:Un0gtobetherstpassageofthelog-priceprocessbelowzero,observedonamonitoringdate,and+=inffn:Un>0gthecorrespondingrstpassageofthelog-priceprocessabovezero.or+arecalled'ladderepochs'.Thedualitypropertyofthisrandomwalkwillenableus,throughand+,toderiverecursiveexpressionsleadingtothedistributionsoftheextrema~MNand~N. Figure4-1. Samplepathofalog-priceprocessforalookbackoption 44

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4-1 weseethat=2eventhoughthelog-pricehasdroppedbelowzerobeforetime1.Sinceweobservethepricesonlyonthediscretemonitoringdates,thisdoesnotaffectasthepriceisbackabovezeroattime1.Also,+=1andMNisequaltothepriceonthe10thmonitoringdate,eventhoughthecontinuouspriceprocesshasahigherpricesincethishigherpriceisnotobservedonamonitoringdate.From( AitSahlia&Lai 1998 )weknowthatthedistributionofthemaximumlog-pricecanbewrittenasPfMN2dxg=PfU12dxgPfX20,X2+X30,...,X2+XN0g+NX=2hPfU>Ui,i<;U2dxgPfX+10,X+1+X+20,...,X+1++XN0gi Feller 1971 ),letsusrewriteoneoftheaboveprobabilitiesintermsofoneoftheladderepochsPfU>Ui,i<;U2dxg=PfUU1>0,...,UU1>0;U2dxg=PfU1>0,...,U1>0;U2dxg=Pf>;U2dxg

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DenenowtheFouriertransformorcharacteristicfunctionofadistributionfunctionFofarealrandomvariableXas(cf.( Chung 1974 ))as:F(F)()=EeiX=ZReixdF(x). AitSahlia&Lai 1998 ). 2bFn1b+i 46

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AitSahlia&Lai 1998 ),pg.230,Eq.10,canbeexpressedasF1(x)=(x)Fn(x)=1J(x)(Fn1)(x),for2nN Stenger 1993 )and( Feng&Linetsky 2008 )):F1(0,1)()=1 2^+i 2F(Fn1)+i 2bFn1b+i 4 andEq. 4 tperfectlytheset-upof( Feng&Linetsky 2008 )toapplytheirhighlyefcientalgorithmtocomputealltheFourierandHilberttransformsandinvertthelast(bFN)forpricingpurposesatacomputationalcostofO(NMlog(M)),whereMisthenumberofquadraturepoints 47

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whereF(x)areobtainedthroughtheapplicationofthenumericalschemeof( Feng&Linetsky 2008 )totherecursionsinEq. 4 andEq. 4 forx>0,withJ=(,0],and0,1,...,Ndenedby0=1,n=Gn(0)limx!Gn(x)forn1, 4 andEq. 4 andusingJ=(,0].

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Thelatter,togetherwith( 4 )andthedecompositionabove,yieldsPfMN2dxg=N1PfU12dxg+NX=2NdF(x)forx>0, 4 andEq. 4 extendstooating-strikelookbackoptions.Thesearecontrastedtothexed-strikebymakingthestrikesettothepriceoftheunderlyinguponexercise.Thuswithaoating-strikeput,itsholdercanpurchasetheunderlyingatitstradingpriceuponexerciseandsellitatthemaximumithasachievedoverthelifeofthecontract,resultinginapayoffS0eMNSM+.Ontheotherhand,aoating-strikecallallowsitsholdertopurchasetheassetattheminimumitachievedduringitslifeandsellitatthepriceittradesuponexercise.Again,toallowforcomparisonwiththeclassicalBrownianprocessintheBlack-Scholesmodelweillustratetheapplicationoftheapproachontheput.Incidentally,oating-strikeoptionsaresometimeslabeledstandard.

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4 andEq. 4 asinProposition2.

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Finally, AitSahlia&Lai 1998 )alsoapplyhere,withobviousmodicationsandwillthereforenotberepeatedhere. Additionally,ourapproachisparticularlywell-suitedforthecomputationofhedgingparameters,whichareespeciallycrucialtotheoptionwriter'sriskmanagementpractice.Forexample,thexed-strikelookbackpriceattime0ofProposition2,Eq. 4 ,canbere-writtenaserTES0eMNK+=erTN(S0K)++erTNX=1Z10(S0exK)+dF(x)=8><>:erTPN=1R1log(K=S0)(S0exK))dF(x)ifS0KerTN(S0K)+erTPN=1R10(S0exK)dF(x)ifS0>K

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Derivativessuchasoptionsareessentialtothefunctioningofamoderneconomy.Theyprovideopportunitiesforhedgersseekingtoreducetheirnancialrisksaswellasspeculators,whosehitsandmissesinthemarketplacecanprovideadditionalliquidity.Thepricingandhedgingofthesenancialinstrumentshasbecomeincreasinglychallengingasevermorecomplexmodelshaveemergedtoaccountforpracticalfeaturesthatcannotbeignored.Overthepastfewyears,continuoustimeassetpricingmodelsthatrelyonLevyprocesseshavegainedsignicantprominence.Theirwideningadoptionisduetotheirabilitytocapturesalientfeaturessuchasjumpsandfattailsinassetreturndistributionsthatcannotbeignored.Forexample,ifoneweretomaintainusingtheclassicalBlack-Scholes-Mertonmodelthatgavemathematicalnanceitsimpetusintheearly1970'sandwhichreliesonthenormalityassumptionofassetreturns,onewouldseriouslyunderestimatetheactualprobabilityofsignicantandunusualdrops.Forexample,( Kou 2008b )showsthatovertheperiodJan2,1980toDecember31,2005,thestandardized(de-meanedandscaledbystandarddeviation)dailyreturnofthecriticallywatchedS&P500indexrangedfromaminimumof-21.1550,toamaximumof7.9967,whichbothoccurredduringthemarketcrashyearof1987.Yettheprobabilityofastandardnormaldistributionfalling21unitsbelowitszeromeanisapproximately110107.Forcomparison,itisestimatedthattheuniverseisabout15billionyears(or51017seconds)old.Thereisthereforeclearlyaneedforalternativemodels,andthosebasedonLevyprocesseshavemanyfavorablefeatures,includingindependenceofincrementsandtheirinnite-divisibility,avarietyofwaystocapturelargedeviations,thepossibilitytoincorporatejumps,particularlythepopularpure-jumpandjump-diffusionmodels.Finally,fromamathematicalandcomputationaltractabilityperspective,thereistheremarkableLevy-Khinchinrepresentationwhichmakesexplicitthecharacteristicfunctionoftheprocessintermsofthreeparameters.Inaddition, 53

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Thefocusofthisdissertationisonpath-dependentoptionsintheparticularcontextofLevymodels.Withpayoffsdependingontheentirepathfollowedbytheassetpriceoftheunderlyingupuntilexercise,theseoptionsareespeciallyusefulwhentheirholderswishtoaddressaspecicriskissueinafashionthatcannotbeachievedbystandard(orvanilla)optionsalone.Forexample,theycouldbeconcernedonlyiftheunderlyingassetmovesoutsideacertainrangeofvalues,sayofinterestorcurrencyexchangerates,inwhichcasetheywouldbeinterestedinbarrieroptions,whichcomeintheknock-inandknock-outavors.Theformerentitletheirholdertheacquisitionofastandardoptiononlyiftheunderlyingassetpricecrossesabarrier.Theyhoweverhavetopayfortheprivilegeupfront,withthepossibilityofneveracquiringtheoptioniftheunderlyingdoesnotcrossthebarrierbeforeexpiration.Ontheotherhand,aknock-outoptionyieldsthesamepayoffasastandardoptionaslongastheunderlyingassetpricedoesnotcrossabarrierpriortoexpiration.Thoughbarrieroptionswerenotexplicitlyaddressedinthisthesis,theyareinfactintimatelylinkedwithlookbackoptions,wherethestatisticaldistributionofthemaximum(orminimum)isparamountasitisclearthatabarrierabovetheinitialassetpricecanonlybebreachedifthemaximumisabovewhile,correspondingly,abarrierbelowwouldonlybebreachedwhentheminimumisbelowit.Lookbackoptions(oroptionsonextrema)havethemostexiblepayoffs,andarethusthemostexpensive.Theyareusedbyeitherspeculatorsorbyveryrisk-averseoperators.Theothertypeofpath-dependentoptionsaddressedinthepresentworkconcernsAsian(alsoknownasaverage)options,whicharewidelyusedbymultinationalcorporationstosmooththeircostsaswellastheirrevenuesinthefaceofhighlyvariablerawmaterialpricesandlargeuctuationsincurrencyexchangerates. 54

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Tse,W.,Li,L.,&Ng,K.(2001).Pricingdiscretebarrierandhindsightoptionswiththetridiagonalprobabilityalgorithm.Managementscience,47(3),383. Turnbull,S.,&Wakeman,L.(1991).AquickalgorithmforpricingEuropeanaverageoptions.JournalofFinancialandQuantitativeAnalysis,26(3),377. Vecer,J.(2001).AnewPDEapproachforpricingarithmeticaverageAsianoptions.JournalofComputationalFinance,4(4),105. Vorst,T.(1996).Averagingoptions.TheHandbookofExoticOptions:Instruments,Analysis,andApplicationsbyI.Nielken,Honeywood,Il,Irwin,(pp.175). Wendel,J.(1960).Orderstatisticsofpartialsums.TheAnnalsofMathematicalStatistics,(pp.1034). WilliamFalloon(1998).Windowsonrisk.Risk,June,42. Wu,Editors:,Birge,J.,&Linetsky,V.(2008).ModelingFinancialSecurityReturnsUsingLevyProcesses.HandbooksinOperationsResearchandManagementScience,(pp.117). 60

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GudbjortGylfadottirwasborninSweden,toIcelandicparentsGylHaraldssonandHallaArnljotsdottir.ShegrewupinLaugaras,Biskupstungur,avillageinIcelandwithapopulationaround100people;beforemovingtothecapital,Reykjavk,whereshewenttoVerzlunarskolinnhighschool.Afterthat,shereceivedherB.S.inmathematicsfromtheUniversityofIcelandin2006.Inthefallof2006,shemovedtoGainesville,FL,topursueherdoctoralstudiesinthedepartmentofIndustrialandSystemsEngineeringatTheUniversityofFlorida,withconcentrationinquantitativenance.ShereceivedherM.S.innancefromtheWarringtonCollegeofBusinessattheUniversityofFloridain2008andherPh.D.inindustrialandsystemsengineeringfromtheCollegeofEngineeringin2010. 61