Zhu J. Applications of Fourier Transform to Smile Modeling: Theory and Implementation

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Jianwei Zhu

Applications of Fourier

Transform to Smile Modeling

Theory and Implementation

Dr. Jianwei Zhu

Lucht Probst Associates

Große Gallusstr. 9

60311 Frankfurt/Main

Germany

jianwei.zhu@l-p-a.com

ISBN 978-3-642-01807-7 e-ISBN 978-3-642-01808-4

DOI 10.1007/978-3-642-01808-4

Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2009937646

Mathematics Subject Classiﬁcation (2000): 42A99, 60E10, 60G05, 60H05, 60J65, 60J75, 91B28, 91B70

JEL Classiﬁcation: C02, C46, C63, E43, G12, G13, G24

 Springer-Verlag Berlin Heidelberg 2000, 2010

Originally published as volume 493 in the series Lecture Notes in Economics and Mathematical Systems

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To my wife Shiquan

Preface

This book addresses the applications of Fourier transform to smile modeling. Smile

effect is used generically by ﬁnancial engineers and risk managers to refer to the

inconsistences of quoted implied volatilities in ﬁnancial markets, or more mathe-

matically, to the leptokurtic distributions of ﬁnancial assets and indices. Therefore,

a sound modeling of smile effect is the central challenge in quantitative ﬁnance.

Since more than one decade, Fourier transform has triggered a technical revolution

in option pricing theory. Almost all new developed option pricing models, espe-

cially in connection with stochastic volatility and random jump, have extensively

applied Fourier transform and the corresponding inverse transform to express op-

tion pricing formulas. The large accommodation of the Fourier transform allows for

a very convenient modeling with a general class of stochastic processes and dis-

tributions. This book is then intended to present a comprehensive treatment of the

Fourier transform in the option valuation, covering the most stochastic factors such

as stochastic volatilities and interest rates, Poisson and L

evy jumps, including some

asset classes such as equity, FX and interest rates, and providing numerical exam-

ples and prototype programming codes. I hope that readers will beneﬁt from this

book not only by gaining an overview of the advanced theory and the vast large lit-

erature on these topics, but also by gaining a ﬁrst-hand feedback from the practice

on the applications and implementations of the theory.

This book has grown out of my early research manuscript “Modular Pricing of

Options” that was published in 2000 in the series “Lecture Notes in Economics

and Mathematical Systems”. From the point of view of a practitioner, the modular

pricing approach is the one of the best applications of Fourier transform in the option

valuation. By the modular pricing approach, each stochastic factor may be regarded

as a module in a generalized pricing framework with Fourier transform. Following

this building-block way for pricing, I address in this book stochastic volatilities,

stochastic interest rates, Poisson jumps and L

evy jumps successively, and at the

same time, expound the advanced option pricing theory step by step.

Additionally, I give detailed discussions on the numerical issues of stochastic

volatility models, for instance, how to perform an efﬁcient numerical integration of

the inverse Fourier transform in a trading environment. Since stochastic volatility

vii

viii Preface

models are the representative examples of the applications of Fourier transform,

almost all numerical methods for stochastic volatilities can be applied to other

stochastic factors, for instance, L

evy jumps. In this book, L

evy jumps are referred

to as the jumps with inﬁnite activity, and are also the driving factor of the so-

called L

evy pricing models in ﬁnancial literature. This book delivers an illustrative

and compact introduction to L

evy jumps. Exotic options and Libor Market Model

(LMM) with stochastic factors are two other focuses of my book. Valuing exotic

equity/FX option with stochastic factors, especially with stochastic volatility ana-

lytically, is generally not feasible, and therefore, requires some special treatments

and modiﬁcations. The stochastic volatility LMMs are proposed to capture the smile

effects in cap and swaption markets. Some of these extended LMMs use the Fourier

transform extensively to arrive at some tractable solutions.

I organize this book in a self-contented way, and present the basic concepts of

option pricing theory and the advanced techniques of asset modeling, especially

with respect to Fourier transform and characteristic functions. The prerequisites for

this book are some basic knowledge of stochastic theory and ﬁnancial products, al-

though the main results and theorems such as the Feynman-Kac theorem and the

Girsanov theorem are brieﬂy reviewed in the ﬁrst chapter. This book is structured

in 11 chapters. Chapter 1 summarizes the Black-Scholes model and explains the

related smile effect of implied volatilities. Capturing smile effects is the main moti-

vation of this book. At the same time, the key deﬁnitions and concepts in quantitative

ﬁnance such as It

o process, the Feynman-Kac theorem and change of measure, are

introduced. Chapter 2 addresses in details the fundamentals of a generalized pricing

framework where characteristic functions play a key role in unifying all stochas-

tic factors. Additionally, the Fourier transform as a pricing technique is expounded

from the different aspects, both economically and mathematically. Afterwards, I be-

gin to deal with various stochastic factors separately, namely, stochastic volatility

and interest rate, Poisson jump and L

evy jump. Chapter 3 considers three analyti-

cal tractable stochastic volatility models: the Heston model (1993), the Sch

obel and

Zhu model (1999) and a double square root model, where the stochastic volatility or

variance follows a mean-reverting square root process, a mean-reverting Ornstein-

Uhlenbeck process and a mean-reverting double square root process, respectively.

As an example, I derive the Heston model in details with two approaches: the PDE

approach and the expectation approach, and show that the latter approach is much

more powerful and can be applied to calculate the characteristic functions of broader

stochastic processes. Next in Chapter 4, I discuss a more important problem for the

practical applications of stochastic volatility models, namely, the numerical imple-

mentation of these models, for instance, the efﬁcient numerical integration of in-

verse Fourier transform, the complex logarithms, the calculation of Greeks as well

as quick calibration. All numerical methods and techniques for stochastic volatil-

ity are also relevant and valid for other stochastic factors. Chapter 5 focuses on the

simulation of stochastic volatility models. The simulation of square root process,

particularly the Heston model, has been a large challenge in computational ﬁnance.

Chapter 5 presents some recently developed efﬁcient simulation methods for the He-

ston model. Numerical examples show that the QE scheme of Andersen (2007) and

Preface ix

the transformed volatility scheme of Zhu (2008) are two most efﬁcient methods for

simulating mean-reverting square root process. Additionally, we will discuss how

to simulate extremes in a diffusive process and a multi-asset model with stochastic

volatilities. Chapter 6 explains how to incorporate stochastic interest rates into an

option pricing model in a short rate setup. Three representative short rate models of

CIR (1985), Vasicek (1977) and Longstaff (1989) are examined where two cases,

zero correlation and non-zero correlation, for the correlation between the stock re-

turns and the interest rates are expounded.

Chapter 7 and Chapter 8 are dedicated to jump modeling. The Fourier trans-

form is a very elegant and efﬁcient technique when incorporating discontinuous

jump events in an asset process. Chapter 7 deals with the traditional jump mod-

els where the jump mechanism is characterized by a compound Poisson process. A

compound Poisson process is a mixed process with a jump counting described by

a pure Poisson process and a random jump size described by a distribution. Depen-

dently on the speciﬁcation of jump sizes, we have different Poisson jump models,

for instance, lognormal jumps, Pareto jumps and double exponential jumps. The

afﬁne jump-diffusion model of Dufﬁe, Pan and Singleton (2000) is also reviewed

brieﬂy. Chapter 8 describes the L

evy jumps that are characterized by inﬁnite activ-

ity. After an introduction to the L

evy process, two simple and popular L

evy jump

models, the variance Gamma model and the normal inverse Gaussian model, are

ﬁrst discussed. Both of these processes belong to the class of stochastic clock mod-

els. Next I move to the so-called time-changed L

evy processes that are generated by

applying a stochastic time change to the underlying L

evy processes, or equivalently,

can be generated by a subordination. Two different cases of stochastic time-change,

the zero correlation and the non-zero correlation between the stochastic time and

the underlying L

evy process, are considered where the non-zero correlation is ref-

ereed to as the leverage effect, and can be dealt with conveniently by introducing

a complex-valued leverage-risk neutral measure. Additionally, two particular L

evy

models, the Barndorff-Nielsen and Shephard model (2001) and the alpha log-stable

model are brieﬂy examined.

After the stochastic volatility, the stochastic interest rate, the Poisson jump and

the L

evy jumps have been discussed, Chapter 9 integrates all these stochastic fac-

tors into an uniﬁed pricing framework. I use two approaches for the integration, the

modular approach and the time-change approach. While the modular approach has

a simple, clear and transparent structure, and is also designed for practical imple-

mentations, the approach based on time-change is more technical and may embrace

some nested factors. The variety of the resulting option valuation models by the

modular pricing is immense. As numerical examples, the option prices of 108 differ-

ent models are given to illustrate the large accommodation of the Fourier transform.

Finally, I provide some criterions for choosing the “best” pricing models from the

point of view of a practitioner.

Chapter 10 addresses the valuation of exotic options with stochastic factors, par-

ticularly with stochastic volatility. Generally, there are no closed-form pricing for-

mulas for exotic options with stochastic volatility. However, for some special pro-

cesses and model setups, I attempt to derive the tractable pricing solutions for some

x Preface

exotic options, for example, barrier options, correlation options and Asian options.

In Chapter 11, I present ﬁve stochastic volatility Libor market models (LMM) that

are applied to capture the smile effects in cap and swaption markets. All of these

ﬁve models apply characteristic functions or moment-generating functions for pric-

ing caps and swaptions to different extent. While characteristic functions do not ﬁnd

signiﬁcant applications in the models of Andersen and Brotherton-Ratcliffe (2001),

and Piterbarg (2003), the models of Zhang and Wu (2006), Zhu (2007) as well as

Belomestny, Matthew and Schoenmakers (2007) have extensively and consequently

used characteristic functions to arrive at the closed-form pricing formulas for caplets

and swaptions.

This book is aimed at

• ﬁnancial engineers and risk managers who are seeking for a simple and clearly

structured book on the smile modeling with Fourier transform, and want to imple-

ment some pricing models with characteristic functions efﬁciently and robustly;

• graduate students who need a compact introduction to the option pricing theory

with Fourier transform, and want to gain an overview of the vast literature on

these advanced topics;

• researchers who want to know what is the most important for the smile modeling

in practice, and want to get some feedbacks from practice on the applications of

the option pricing models, particularly stochastic volatility models.

During the writing of this book and its early version, I have beneﬁted from many

discussions with my colleagues and large supports from friends. First, I would like

to thank Professor Rainer Sch

obel and Professor Gerd Ronning for their helpful

comments and encouragement. During my research activities at the University of

ubingen, my mentor Professor Sch

obel provided me with rich freedom in research-

ing the interesting and exciting topics that are discussed in this book. The four years

of my doctoral time have been a highly spiritual experience for me to enjoy the

beauty and rigor of ﬁnancial economics. In addition, I am grateful to Juri Hinz for

some discussions and, to Professor Herbert Heyer for the lectures on stochastic cal-

culus, both from the mathematical institute at the University of T

ubingen.

It is my pleasure to express my thanks here to my colleagues at the university,

Philipp Kellerhals, Stephan Heilig and Vera Kl

ockner, my colleagues J

ugen Linde,

Andreas Gaida and Reinhard Batin at Dresdner Bank, WestLB and Sal. Oppenheim,

and my current colleagues Daniel Herzbach, Martin Kleine Kalvelage, Phillip Re-

infeld and Ger

ald Wigger, as well as Professor Wolfgang Schmidt at HfB. I am

grateful to Wolfram Boenkost, my current chef at LPA, for providing me with the

challenging tasks to implement some sophisticated models and with the opportuni-

ties to do experiment with new ideas.

Finally, I would like to express my gratitude to my family for their unbroken

and warm support. Especially, without the support of my wife, I would have never

ﬁnished writing this book.

Frankfurt am Main, Germany

August 2009 Jianwei Zhu