Equity

derivatives

Theory and Applications

Marcus Overhaus

Andrew Ferraris

Thomas Knudsen

Ross Milward

Laurent Nguyen-Ngoc

Gero Schindlmayr

John Wiley & Sons, Inc.

Equity

derivatives

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Equity

derivatives

Theory and Applications

Marcus Overhaus

Andrew Ferraris

Thomas Knudsen

Ross Milward

Laurent Nguyen-Ngoc

Gero Schindlmayr

John Wiley & Sons, Inc.

Copyright (c) 2002 by Marcus Overhaus. All rights reserved.

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Library of Congress Cataloging-in-Publication Data:

Overhaus, Marcus.

Equity derivatives: theory and applications / Marcus Overhaus.

p. cm.

Includes index.

ISBN 0-471-43646-1 (cloth : alk. paper)

1. Derivative securities. I. Title.

HG6024.A3 O94 2001

332.63’2-dc21

2001026547

Printed in the United States of America

10

9

8

7

6 5

4

3

2

1

about the authors

Marcus Overhaus is Managing Director and Global Head of Quantitative

Research at Deutsche Bank AG. He holds a Ph.D. in pure mathematics.

Andrew Ferraris is a Director in Global Quantitative Research at Deutsche

Bank AG. His work focuses on the software design of the model

library and its integration into client applications. He holds a D.Phil. in

experimental particle physics.

Thomas Knudsen is a Vice President in Global Quantitative Research at

Deutsche Bank AG. His work focuses on modeling volatility. He holds

a Ph.D. in pure mathematics.

Ross Milward is a Vice President in Global Quantitative Research at

Deutsche Bank AG. His work focuses on the architecture of analytics

services and web technologies. He holds a B.Sc. (Hons.) in computer

science.

Laurent Nguyen-Ngoc works in Global Quantitative Research at Deutsche

´

Bank AG. His work focuses on Levy

processes applied to volatility

modeling. He is completing a Ph.D. in probability theory.

Gero Schindlmayr is an Associate in Global Quantitative Research at

Deutsche Bank AG. His work focuses on ﬁnite difference techniques.

He holds a Ph.D. in pure mathematics.

v

preface

Equity derivatives and equity-linked structures—a story of success that still

continues. That is why, after publishing two books already, we decided

to publish a third book on this topic. We hope that the reader of this

book will participate and enjoy this very dynamic and proﬁtable business

and its associated complexity as much as we have done, still do, and will

continue to do.

Our approach is, as in our ﬁrst two books, to provide the reader with

a self-contained unit. Chapter 1 starts with a mathematical foundation for

all the remaining chapters. Chapter 2 is dedicated to pricing and hedging in

´

incomplete markets. In Chapter 3 we give a thorough introduction to Levy

processes and their application to ﬁnance, and we show how to push the

Heston stochastic volatility model toward a much more general framework:

the Heston Jump Diffusion model.

How to set up a general multifactor ﬁnite difference framework to

incorporate, for example, stochastic volatility, is presented in Chapter 4.

Chapter 5 gives a detailed review of current convertible bond models, and

expounds a detailed discussion of convertible bond asset swaps (CBAS) and

their advantages compared to convertible bonds.

Chapters 6, 7, and 8 deal with recent developments and new technologies in the delivery of pricing and hedging analytics over the Internet and

intranet. Beginning by outlining XML, the emerging standard for representing and transmitting data of all kinds, we then consider the technologies

available for distributed computing, focusing on SOAP and web services.

Finally, we illustrate the application of these technologies and of scripting

technologies to providing analytics to client applications, including web

browsers.

Chapter 9 describes a portfolio and hedging simulation engine and its

application to discrete hedging, to hedging in the Heston model, and to

CPPIs. We have tried to be as extensive as we could regarding the list of

references: Our only regret is that we are unlikely to have caught everything

that might have been useful to our readers.

We would like to offer our special thanks to Marc Yor for careful

reading of the manuscript and valuable comments.

The Authors

London, November 2001

vii

contents

CHAPTER 1

Mathematical Introduction

1.1

1.2

1.3

1.4

1.5

Probability Basis

1

Processes

2

Where in Time?

3

Martingales and Semimartingales

Markov Processes

7

Stochastic Calculus

8

Ito’s

9

¯ Formula

Girsanov’s Theorem

10

Financial Interpretations

11

Two Canonical Examples

11

1

4

CHAPTER 2

Incomplete Markets

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

15

Martingale Measures

15

Self-Financing Strategies, Completeness, and

No Arbitrage

17

Examples

21

Martingale Measures, Completeness, and

No Arbitrage

28

Completing the Market

30

Pricing in Incomplete Markets

37

Variance-Optimal Pricing and Hedging

43

Super Hedging and Quantile Hedging

46

CHAPTER 3

´ Processes

Financial Modeling with Levy

3.1

´ Processes

A Primer on Levy

52

First Properties

52

Measure Changes

58

Subordination

61

Levy

´ Processes with No Positive Jumps

51

66

ix

x

3.2

3.3

3.4

3.5

3.6

Contents

´ Processes

Modeling with Levy

68

Model Framework

69

The Choice of a Pricing Measure

69

European Options Pricing

70

Products and Models

72

Exotic Products

72

Some Particular Models

77

Model Calibration and Smile Replication

88

´ Processes

Numerical Methods for Levy

95

Fast Fourier Transform

95

Monte Carlo Simulation

95

Finite-Difference Methods

97

´

A Model Involving Levy Processes

98

CHAPTER 4

Finite-Difference Methods for Multifactor Models

4.1

4.2

4.3

4.4

4.5

4.6

Pricing Models and PDES

103

Multiasset Model

104

Stock-Spread Model

105

The Vasicek Model

106

The Heston Model

106

The Pricing PDE and Its Discretization

106

Explicit and Implicit Schemes

109

The ADI Scheme

110

Convergence and Performance

113

Dividend Treatment in Stochastic Volatility Models

Modeling Dividends

117

Stock Process with Dividends

117

Local Volatility Model with Dividends

122

Heston Model with Dividends

123

103

116

CHAPTER 5

Convertible Bonds and Asset Swaps

5.1

5.2

Convertible Bonds

125

Introduction

125

Deterministic Risk Premium in Convertible Bonds

Non–Black-Scholes Models for Convertible Bonds

Convertible Bond Asset Swaps

137

Introduction

137

Pricing and Analysis

140

125

127

132

xi

Contents

CHAPTER 6

Data Representation

6.1

6.2

6.3

6.4

XML

149

Tags and Elements

150

Attributes

151

Namespaces

151

Processing Instructions

152

Comments

152

Nesting

152

Parsing XML

153

Multiple Representation

154

XML Schema

154

XML Transformation

157

XML Document Transformation

158

Transformation into HTML

160

Representing Equity Derivative Market Data

147

162

CHAPTER 7

Application Connectivity

7.1

7.2

7.3

7.4

Components

166

Distributed Components

167

DCOM and CORBA

168

SOAP

168

SOAP Structure

171

SOAP Security

173

State and Scalability

175

Web Services

177

WSDL

177

UDDI

179

CHAPTER 8

Web-Based Quantitative Services

8.1

8.2

8.3

165

Web Pricing Servers

183

Thread Safety Issues in Web Servers

186

Model Integration into Risk Management and

Booking Systems

187

A Position Server

190

Web Applications and Dynamic Web Pages

191

Option Calculator Pages

193

Providing Pricing Applications to Clients

195

181

xii

Contents

CHAPTER 9

Portfolio and Hedging Simulation

9.1

9.2

9.3

9.4

9.5

9.6

Introduction

199

Algorithm and Software Design

199

Example: Discrete Hedging and Volatility

Misspeciﬁcation

201

Example: Hedging a Heston Market

205

Example: Constant Proportion Portfolio Insurance

Server Integration

209

199

206

REFERENCES

211

INDEX

219

Equity

derivatives

CHAPTER

1

Mathematical

Introduction

use of probability theory and stochastic calculus is now an established

T hestandard

in the ﬁeld of ﬁnancial derivatives. During the last 30 years, a

large amount of material has been published, in the form of books or papers,

on both the theory of stochastic processes and their applications to ﬁnance

problems. The goal of this chapter is to introduce notions on probability

theory and stochastic calculus that are used in the applications presented afterwards. The notations used here will remain identical throughout the book.

We hope that the reader who is not familiar with the theory of stochastic

processes will ﬁnd here an intuitive presentation, although rigorous enough

for our purposes, and a set of useful references about the underlying

mathematical theory. The reader acquainted with stochastic calculus will

ﬁnd here an introduction of objects and notations that are used constantly,

although maybe not very explicitly.

This chapter does not aim at giving a thorough treatment of the theory

of stochastic processes, nor does it give a detailed view of mathematical

ﬁnance theory in general. It recalls, rather, the main general facts that will

be used in the examples developed in the next chapters.

1.1

PROBABILITY BASIS

Financial models used for the evaluation of derivatives are mainly concerned

with the uncertainty of the future evolution of the stock prices. The theory

of probability and stochastic processes provides a framework with a form

of uncertainty, called randomness. A probability space ⍀ is assumed to be

given once and for all, interpreted as consisting of all the possible paths

of the prices of securities we are interested in. We will suppose that this

probability space is rich enough to carry all the random objects we wish

to construct and use. This assumption is not restrictive for our purposes,

because we could always enlarge the space ⍀ , for example, by considering

a product space. Note that ⍀ can be chosen to be a “canonical space,”

1

2

MATHEMATICAL INTRODUCTION

such as the space of continuous functions, or the space of cadlag (French

acronym for “continuous from the right, with left limits”) functions.

We endow the set ⍀ with a -ﬁeld Ᏺ which is also assumed to be ﬁxed

throughout this book, unless otherwise speciﬁed. Ᏺ represents all the events

that are or will eventually be observable.

Let ސbe a probability measure on the measurable space (⍀ , Ᏺ). The

(Lebesgue) integral with respect to ސof a random variable X (that is, a

measurable function from (⍀ , Ᏺ) to (ޒN , ᏮN ), where ᏮN is the Borel -ﬁeld

on ޒN ) is denoted by [ޅX] instead of Ύ⍀ X d ސand is called the expectation

of X. If we need to emphasize that the expectation operator ޅis relative to

ސ, we denote it by ސޅ. We assume that the reader is familiar with general

notions of probability theory such as independence, correlation, conditional

expectation, and so forth. For more details and references, we refer to [9],

[45], or [49].

The probability space (⍀ , Ᏺ, )ސis endowed with a ﬁltration (Ᏺt , t Ն 0),

that is, a family of sub- -ﬁelds of Ᏺ such that Ᏺs ʚ Ᏺt for all 0 Յ s Յ t.

The ﬁltration is said to be ސ-complete if for all t, all ސ-null sets belong to

every Ᏺt ; it is said to be right-continuous if for all t > 0,

Ᏺt ס

ʝᏲ

t⑀ם

⑀ Ͼ0

It will be implicit in the sequel that all the ﬁltrations we use have been

previously completed and made right-continuous (this is always possible).

The ﬁltration Ᏺt represents the “ﬂow of information” available; we will

often deal with the ﬁltration generated by some process (e.g., stock price

process), in which case Ᏺt represents past observations up to time t. For

detailed studies on ﬁltrations the reader can consult any book concerned

with stochastic calculus, such as [44], [63], and [103].

1.2

PROCESSES

We will be concerned with random quantities whose values depend on time.

Denote by ᐀ a subset of ;םޒ᐀ can be םޒitself, a bounded interval [0, T ], or

a discrete set ᐀ = {0, 1, . . .}. In general, given a measurable space (E, Ᏹ), a

process with values in E is an application X : ⍀ × ᐀

E that is measurable

with respect to the -ﬁelds Ᏹ and Ᏺ Ꮾ᐀ , where Ꮾ᐀ denotes the Borel

-ﬁeld on ᐀ .

In our applications we will need to consider only the case in which

E = ޒN and Ᏹ is the Borel -ﬁeld ᏮN . From now on, we make these

assumptions. A process will be denoted by X or (Xt , t ʦ ᐀ ); the (random)

value of the process at time t ʦ ᐀ will be denoted by Xt or X(t); we

may sometimes wish to emphasize the dependence on , in which case

1.2

3

Processes

we will use the notation Xt ( ) or X(t, ). The jump at time t of a process

X, is denoted by ⌬Xt and deﬁned by ⌬Xt = Xt – XtϪ , where XtϪ =

lim⑀Q0 XtϪ⑀ .

Where in Time?

Before we take on the study of processes themselves, we deﬁne a class of

random times that form a cornerstone in the theory of stochastic processes.

These are the times that are “suited” to the ﬁltration Ᏺt .

DEFINITION 1.1

A random time T , that is a random variable with values in םޒʜ {ϱ}, is

called an Ᏺt -stopping time if for all t ʦ ᐀

͕T Յ t͖ ʦ Ᏺt

This deﬁnition means that at each time t, based on the available information

Ᏺt , one is able to determine whether T is in the past or in the future.

Stopping times include constant times, as well as hitting times (i.e., random

times of the form = inf{t ʦ ᐀ : Xt ʦ B}, where B is a Borel set), among

others.

From a ﬁnancial point of view, the different quantities encountered are

constrained to depend only on the available information at the time they

are given a value. In mathematical words, we state the following:

DEFINITION 1.2

A process X is said to be adapted to the ﬁltration Ᏺt (or Ᏺt -adapted) if,

for all t ʦ ᐀ , Xt is Ᏺt -measurable.

A process used to model the price of an asset must be adapted to the ﬂow

of information available in the market. On the other hand, this information

consists mainly in the prices of different assets. Given a process X, we can

deﬁne a ﬁltration (ᏲtX ), where ᏲtX is the smallest sub- -ﬁeld of Ᏺ that makes

the variables (Xu , u Յ t) simultaneously measurable. The ﬁltration ᏲtX is

said to be generated by X, and X is clearly adapted to it. One also speaks of

X “in its own ﬁltration.”

Because we do not make the assumption that the processes we consider

have continuous paths, we need to introduce a ﬁne view of the “past.”

Continuous processes play a special role in this setting.

4

MATHEMATICAL INTRODUCTION

DEFINITION 1.3

1. The predictable -ﬁeld ᏼ is the -ﬁeld on ⍀ × ᐀ generated by

Ᏺt -adapted processes whose paths are continuous.

2. A process X is said to be predictable if it is measurable with respect

to ᏼ.

That is, ᏼ is the smallest -ﬁeld on ⍀ × ᐀ such that every process X, viewed

as a function of (, t), for which t ۋX(t) is continuous, is ᏼ-measurable.

It can be shown that ᏼ is also generated by random intervals (S, T ] where

S < T are stopping times.

A process that describes the number of shares in a trading strategy must

be predictable, because the investment decision is taken before the price has

a possible instantaneous shock.

In discrete time, the deﬁnition of a predictable process is much simpler,

since then a process (Xi , i ʦ )ގis predictable if for each n, Xn is ᏲnϪ1 measurable. However, we have the satisfactory property that if X is an

Ᏺt -adapted process, then the process of left limits (XtϪ , t Ն 0) is predictable.

For more details about predictable processes, see [27] or [63].

Let us also mention the optional -ﬁeld: It is the -ﬁeld ᏻ on ⍀ × ᐀

generated by Ᏺt -adapted processes with right-continuous paths. It will not

be, for our purposes, as crucial as the predictable -ﬁeld; see, however,

Chapter 2 for a situation where this is needed.

We end this discussion by introducing the notion of localization, which

is the key to establishing certain results in a general case.

DEFINITION 1.4

A localizing sequence (Tn ) is an increasing sequence of stopping times

ϱ as n

ϱ.

such that Tn

In this chapter, a property is said to hold locally if there exists a localizing

sequence such that the property holds on every interval [0, Tn ]. This notion

is important, because there are many interesting cases in which important

properties hold only locally (and not on a ﬁxed interval, [0, ϱ), for example).

Martingales and Semimartingales

Among the adapted processes deﬁned in the foregoing section, not all

are suitable for ﬁnancial modelling. The work of Harrison and Pliska

[60] shows that only a certain class of processes, called semimartingales

1.2

5

Processes

are good candidates. Indeed, the reader familiar with the theory of arbitrage

knows that the stock price process must be a local martingale under an

appropriate probability measure; Girsanov’s theorem then implies that

it must be a semimartingale under any (locally) equivalent probability

measure.

DEFINITION 1.5

A process X is called an Ᏺt -martingale if it is integrable (i.e., |[ޅXt| ] < ϱ

for all t), Ᏺt -adapted, and if it satisﬁes, for all 0 Յ s Յ t

[ޅXt ͉Ᏺs ] סXs

(1.1)

X is called a local martingale if there is a localizing sequence (Tn ) such

that for all n, (XtٙTn , t Ն 0) is a martingale. X is called a semimartingale

if it is Ᏺt -adapted and can be written

X t סX0 םM t םV t

(1.2)

where M is a local martingale, V has a.s. (almost surely) ﬁnite variation,

and M and V are null at time t = 0. If V can be chosen to be predictable,

X is called a special semimartingale and the decomposition with such V

is called the canonical decomposition.

If we need to emphasize the underlying probability measure ސ, we will say

that X is a ސ-(semi)martingale.

With a semimartingale X are associated two increasing processes,

called the quadratic variation and the conditional quadratic variation.

These processes are interesting because they allow us to compute the

decomposition of a semimartingale under a change of probability measure:

This is the famous Girsanov theorem (see Section 1.3). We give a brief

introduction to these processes here; for more details, see for example [27],

[44], [63], [100], [103], [104], [105].

We ﬁrst turn to the quadratic variation of semimartingale.

DEFINITION 1.6

Let X be a semimartingale such that [ޅXt2 ] < ϱ for all t. There exists an

increasing process, denoted by [X, X], and called the quadratic variation

of X, such that

[X, X]t סplim

Α (Xt Ϫ Xt

nqϱ t ʦ (n)

i

i

iϪ 1

)2

(1.3)

6

MATHEMATICAL INTRODUCTION

where for each n, (n) = (0 = t0 < t1 < иии < tpn = t) is a subdivision

of [0, t] whose mesh sup1 Յ i Յ pn (ti – tiϪ1 ) tends to 0 as n tends to ϱ.

The abbreviation “plim” stands for “limit in probability.” It can be shown

that the above deﬁnition is actually meaningful: The limit does not depend

on a particular sequence of subdivisions. Moreover, if X is a martingale,

the quadratic variation is a compensator of X2 ; that is, X2 – [X, X] is again

a martingale. More generally, given a process X, another process Y will be

called a compensator for X if X – Y is a local martingale. Because of the

properties of martingales, compensation is the key to many properties when

paths are not supposed to be continuous.

Given two semimartingales X and Y , we deﬁne the quadratic covariation

of X and Y by a polarization identity:

[X, Y ] ס

1

([X םY, X םY ] Ϫ [X, X] Ϫ [Y, Y ])

2

Let M be a martingale. It can be shown that there exist two uniquely

determined martingales Mc and Md such that: M = Mc + Md , Mc has continuous paths and Md is orthogonal to any continuous martingale; that is,

Md N is a martingale for any continuous martingale N. Mc is called the

martingale continuous part of M, while Md is called the purely discontinuous part. If X is a special semimartingale, with canonical decomposition

X = M + V , Xc denotes the martingale continuous part of M, that is

X c ϵ Mc .

Note that the jump at time t of the quadratic variation of a semimartingale X is simply given by ⌬[X, X]t = (⌬Xt )2 . We have the following

important property:

[Xc , Xc ]t [ סX, X]ct [ סX, X]t Ϫ Α ⌬[X, X]s

(1.4)

sՅt

where the last sum is actually meaningful (see [100]).

We now turn to the conditional quadratic variation.

DEFINITION 1.7

Let X be a semimartingale such that [ޅXt2 ] < ϱ for all t. If

plim

Αޅ

nqϱ t ʦ (n)

i

ͫ

(Xti Ϫ XtiϪ1 )2 ͉ᏲtiϪ1

ͬ

(1.5)

1.2

7

Processes

exists, where for each n, (n) = (0 = t0 < t1 < иии < tpn = t) is a subdivision of [0, t] whose mesh sup1 Յ i Յ pn ti – tiϪ1 tends to 0 as n tends to

ϱ, and the limit does not depend on a particular subdivision, this limit is

called the conditional quadratic variation of X and is denoted by ͗X, X͘t .

In that case, ͗X, X͘t is an increasing process.

In contrast to the quadratic variation, the limit in (1.5) may fail to exist for

some semimartingales X. However, it can be shown that the limit exists, and

that the process ͗X, X͘ is well-deﬁned, if X is a special semimartingale, in

´ process or a continuous semimartingale, for example.

particular for a Levy

Similar to the case of quadratic variation, the conditional quadratic

covariation is deﬁned as

͗X, Y ͘ ס

1

(͗X םY, X םY ͘ Ϫ ͗X, X͘ Ϫ ͗Y, Y ͘)

2

as soon as this expression makes sense.

It can also be proven that when it exists, the conditional quadratic

variation is the predictable compensator of the quadratic variation; that is,

͗X, X͘ is a predictable process and [X, X] – ͗X, X͘ is a martingale. It follows

that if X is a martingale, X2 – ͗X, X͘ is also a martingale, and the quadratic

variation is the predictable compensator of X2 . The (conditional) quadratic

variation has the following well-known properties, provided the quantities

considered exist:

Ⅲ The applications (X, Y ) [ ۋX, Y ] and (X, Y ) ͗ ۋX, Y ͘ are linear in X

and Y .

Ⅲ If X has ﬁnite variation, [X, Y ] = ͗X, Y ͘ = 0 for any semimartingale Y .

Moreover we have the following important identity (see [100]):

[Xc , Xc ] ͗ סXc , Xc ͘

so that if X has continuous paths, ͗X, X͘ is identical to [X, X]. The (conditional) quadratic variation will appear into the decomposition of F(X)

¯ formula, which lies at the heart of stochastic

for suitable F, given by Ito’s

calculus.

Markov Processes

We now introduce brieﬂy another class of processes that are memoryless at

stopping times.

8

MATHEMATICAL INTRODUCTION

DEFINITION 1.8

1. An Ᏺt -adapted process X is called a Markov process in the ﬁltration

(Ᏺt ) if for all t Ն 0, for every measurable and bounded functional F,

[ޅF(Xtםs , s Ն 0)͉Ᏺt ] [ޅ סF(Xtםs , s Ն 0)͉Xt ]

(1.6)

2. X is called a strong Markov process if (1.6) holds with t replaced by

any ﬁnite stopping time T .

In other words, for a Markov process, at each time t, the whole past is

summarized in the present value of the process Xt . For a strong Markov

process, this is true with a stopping time. In ﬁnancial words, an investment

decision is often made on the basis of the present state of the market, that

in some sense sums up its history.

A nice feature of Markov processes is the Feynman-Kac formula; this

formula links Markov processes to (integro-)partial differential equations

and makes available numerical techniques such as the ﬁnite difference

method explained in Chapter 4. We do not go further into Markov processes

and go on with stochastic calculus. Some relationships between Markov

processes and semimartingales are discussed in [28].

1.3

STOCHASTIC CALCULUS

With the processes deﬁned in the previous section (semimartingales), a

theory of (stochastic) integral calculus can be built and used to model

ﬁnancial time series. Accordingly, this section contains the two results of

¯ formula and the

probability theory that are most useful in ﬁnance: Ito’s

Girsanov theorem, both in a quite general form.

The construction and properties of the stochastic integral are well

known, and the ﬁnancial reader can think of most of them by taking the

parallel of a portfolio strategy (see Section 1.4 and Chapter 2).

In general, the integral of a process H with respect to another one X

is well-deﬁned provided H is locally bounded and predictable and X is a

semimartingale with [ޅXt2 ] < ϱ for all t. The integral can then be thought

of as the limit of elementary sums

Ύ

t

0

Hs dXs “ סlim

nqϱ

Α Ht (Xt

i

iם1

Ϫ Xti )”

ti ʦ (n)

where for each n, (n) = (0 = t0 < t1 < иии < tpn = t) is a subdivision of

[0, t] whose mesh sup1 Յ i Յ pn (ti – tiϪ1 ) tends to 0 as n tends to ϱ. See [27],

[100], [103], or [104] for a rigorous deﬁnition.

1.3

9

Stochastic Calculus

Note an important property of the stochastic integral. Let X, Y be

semimartingales and H a predictable process such that Ύ Hs dYs is welldeﬁned; the following formula holds:

ͫ

Ύ

X,

.

Hs dYs

0

ͬ

t

ס

Ύ

t

Hs d[X, Y ]s

(1.7)

0

.

t

where Ύ0 Hs dYs denotes the process (Ύ0 Hs dYs , t Ն 0). The same formula

holds with [., .] replaced with ͗., .͘, provided the latter exists; this follows

from the linearity of the quadratic variation and the stochastic integral.

¯ Formula

Ito’s

We can now state the famous Ito’s

¯ formula. More details can be found in the

references mentioned previously. Let X = (X1 , . . . , Xn ) be a semimartingale

with values in ޒn and F be a function ޒn

ޒm of class C2 . Then F(X) is a

semimartingale, and

n

F(Xt ) סF(X0 ) םΑ

Ύ

t

iס1 0

n

ם

n

1

2 ΑΑ

iס1 jס1

Ύ

t

0

ѨF

(XsϪ )dXsi

Ѩxi

Ѩ 2F

(Xs )d[Xi , Xj ]cs

Ѩxi Ѩxj

Ά

n

םΑ F(Xs ) Ϫ F(XsϪ ) Ϫ Α

sՅt

iס1

(1.8)

ѨF

(XsϪ )⌬Xsi

Ѩxi

·

¯ formula is often written in the

where ⌬X is the jump process of X. Ito’s

differential form

n

dF(Xt ) ס

ѨF

Α Ѩxi (XtϪ )dXti

iס1

ם

1 n n Ѩ 2F

(Xt )d[Xi , Xj ]ct

Ѩx i Ѩx j

2 ΑΑ

iס1 jס1

(1.9)

םdZt

where

Ά

n

Zt סΑ F(Xs ) Ϫ F(XsϪ ) Ϫ Α

sՅt

iס1

ѨF

(XsϪ )⌬Xsi

Ѩ xi

·

derivatives

Theory and Applications

Marcus Overhaus

Andrew Ferraris

Thomas Knudsen

Ross Milward

Laurent Nguyen-Ngoc

Gero Schindlmayr

John Wiley & Sons, Inc.

Equity

derivatives

Founded in 1807, John Wiley & Sons is the oldest independent publishing

company in the United States. With ofﬁces in North America, Europe, Australia, and Asia, Wiley is globally committed to developing and marketing

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to e-commerce, risk management, ﬁnancial engineering, valuation, and

ﬁnancial instrument analysis, as well as much more. For a list of available

titles, please visit our web site at www.WileyFinance.com.

Equity

derivatives

Theory and Applications

Marcus Overhaus

Andrew Ferraris

Thomas Knudsen

Ross Milward

Laurent Nguyen-Ngoc

Gero Schindlmayr

John Wiley & Sons, Inc.

Copyright (c) 2002 by Marcus Overhaus. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada

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and speciﬁcally disclaim any implied warranties of merchantability or ﬁtness for a

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Wiley also publishes its books in a variety of electronic formats. Some content that

appears in print may not be available in electronic books.

Library of Congress Cataloging-in-Publication Data:

Overhaus, Marcus.

Equity derivatives: theory and applications / Marcus Overhaus.

p. cm.

Includes index.

ISBN 0-471-43646-1 (cloth : alk. paper)

1. Derivative securities. I. Title.

HG6024.A3 O94 2001

332.63’2-dc21

2001026547

Printed in the United States of America

10

9

8

7

6 5

4

3

2

1

about the authors

Marcus Overhaus is Managing Director and Global Head of Quantitative

Research at Deutsche Bank AG. He holds a Ph.D. in pure mathematics.

Andrew Ferraris is a Director in Global Quantitative Research at Deutsche

Bank AG. His work focuses on the software design of the model

library and its integration into client applications. He holds a D.Phil. in

experimental particle physics.

Thomas Knudsen is a Vice President in Global Quantitative Research at

Deutsche Bank AG. His work focuses on modeling volatility. He holds

a Ph.D. in pure mathematics.

Ross Milward is a Vice President in Global Quantitative Research at

Deutsche Bank AG. His work focuses on the architecture of analytics

services and web technologies. He holds a B.Sc. (Hons.) in computer

science.

Laurent Nguyen-Ngoc works in Global Quantitative Research at Deutsche

´

Bank AG. His work focuses on Levy

processes applied to volatility

modeling. He is completing a Ph.D. in probability theory.

Gero Schindlmayr is an Associate in Global Quantitative Research at

Deutsche Bank AG. His work focuses on ﬁnite difference techniques.

He holds a Ph.D. in pure mathematics.

v

preface

Equity derivatives and equity-linked structures—a story of success that still

continues. That is why, after publishing two books already, we decided

to publish a third book on this topic. We hope that the reader of this

book will participate and enjoy this very dynamic and proﬁtable business

and its associated complexity as much as we have done, still do, and will

continue to do.

Our approach is, as in our ﬁrst two books, to provide the reader with

a self-contained unit. Chapter 1 starts with a mathematical foundation for

all the remaining chapters. Chapter 2 is dedicated to pricing and hedging in

´

incomplete markets. In Chapter 3 we give a thorough introduction to Levy

processes and their application to ﬁnance, and we show how to push the

Heston stochastic volatility model toward a much more general framework:

the Heston Jump Diffusion model.

How to set up a general multifactor ﬁnite difference framework to

incorporate, for example, stochastic volatility, is presented in Chapter 4.

Chapter 5 gives a detailed review of current convertible bond models, and

expounds a detailed discussion of convertible bond asset swaps (CBAS) and

their advantages compared to convertible bonds.

Chapters 6, 7, and 8 deal with recent developments and new technologies in the delivery of pricing and hedging analytics over the Internet and

intranet. Beginning by outlining XML, the emerging standard for representing and transmitting data of all kinds, we then consider the technologies

available for distributed computing, focusing on SOAP and web services.

Finally, we illustrate the application of these technologies and of scripting

technologies to providing analytics to client applications, including web

browsers.

Chapter 9 describes a portfolio and hedging simulation engine and its

application to discrete hedging, to hedging in the Heston model, and to

CPPIs. We have tried to be as extensive as we could regarding the list of

references: Our only regret is that we are unlikely to have caught everything

that might have been useful to our readers.

We would like to offer our special thanks to Marc Yor for careful

reading of the manuscript and valuable comments.

The Authors

London, November 2001

vii

contents

CHAPTER 1

Mathematical Introduction

1.1

1.2

1.3

1.4

1.5

Probability Basis

1

Processes

2

Where in Time?

3

Martingales and Semimartingales

Markov Processes

7

Stochastic Calculus

8

Ito’s

9

¯ Formula

Girsanov’s Theorem

10

Financial Interpretations

11

Two Canonical Examples

11

1

4

CHAPTER 2

Incomplete Markets

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

15

Martingale Measures

15

Self-Financing Strategies, Completeness, and

No Arbitrage

17

Examples

21

Martingale Measures, Completeness, and

No Arbitrage

28

Completing the Market

30

Pricing in Incomplete Markets

37

Variance-Optimal Pricing and Hedging

43

Super Hedging and Quantile Hedging

46

CHAPTER 3

´ Processes

Financial Modeling with Levy

3.1

´ Processes

A Primer on Levy

52

First Properties

52

Measure Changes

58

Subordination

61

Levy

´ Processes with No Positive Jumps

51

66

ix

x

3.2

3.3

3.4

3.5

3.6

Contents

´ Processes

Modeling with Levy

68

Model Framework

69

The Choice of a Pricing Measure

69

European Options Pricing

70

Products and Models

72

Exotic Products

72

Some Particular Models

77

Model Calibration and Smile Replication

88

´ Processes

Numerical Methods for Levy

95

Fast Fourier Transform

95

Monte Carlo Simulation

95

Finite-Difference Methods

97

´

A Model Involving Levy Processes

98

CHAPTER 4

Finite-Difference Methods for Multifactor Models

4.1

4.2

4.3

4.4

4.5

4.6

Pricing Models and PDES

103

Multiasset Model

104

Stock-Spread Model

105

The Vasicek Model

106

The Heston Model

106

The Pricing PDE and Its Discretization

106

Explicit and Implicit Schemes

109

The ADI Scheme

110

Convergence and Performance

113

Dividend Treatment in Stochastic Volatility Models

Modeling Dividends

117

Stock Process with Dividends

117

Local Volatility Model with Dividends

122

Heston Model with Dividends

123

103

116

CHAPTER 5

Convertible Bonds and Asset Swaps

5.1

5.2

Convertible Bonds

125

Introduction

125

Deterministic Risk Premium in Convertible Bonds

Non–Black-Scholes Models for Convertible Bonds

Convertible Bond Asset Swaps

137

Introduction

137

Pricing and Analysis

140

125

127

132

xi

Contents

CHAPTER 6

Data Representation

6.1

6.2

6.3

6.4

XML

149

Tags and Elements

150

Attributes

151

Namespaces

151

Processing Instructions

152

Comments

152

Nesting

152

Parsing XML

153

Multiple Representation

154

XML Schema

154

XML Transformation

157

XML Document Transformation

158

Transformation into HTML

160

Representing Equity Derivative Market Data

147

162

CHAPTER 7

Application Connectivity

7.1

7.2

7.3

7.4

Components

166

Distributed Components

167

DCOM and CORBA

168

SOAP

168

SOAP Structure

171

SOAP Security

173

State and Scalability

175

Web Services

177

WSDL

177

UDDI

179

CHAPTER 8

Web-Based Quantitative Services

8.1

8.2

8.3

165

Web Pricing Servers

183

Thread Safety Issues in Web Servers

186

Model Integration into Risk Management and

Booking Systems

187

A Position Server

190

Web Applications and Dynamic Web Pages

191

Option Calculator Pages

193

Providing Pricing Applications to Clients

195

181

xii

Contents

CHAPTER 9

Portfolio and Hedging Simulation

9.1

9.2

9.3

9.4

9.5

9.6

Introduction

199

Algorithm and Software Design

199

Example: Discrete Hedging and Volatility

Misspeciﬁcation

201

Example: Hedging a Heston Market

205

Example: Constant Proportion Portfolio Insurance

Server Integration

209

199

206

REFERENCES

211

INDEX

219

Equity

derivatives

CHAPTER

1

Mathematical

Introduction

use of probability theory and stochastic calculus is now an established

T hestandard

in the ﬁeld of ﬁnancial derivatives. During the last 30 years, a

large amount of material has been published, in the form of books or papers,

on both the theory of stochastic processes and their applications to ﬁnance

problems. The goal of this chapter is to introduce notions on probability

theory and stochastic calculus that are used in the applications presented afterwards. The notations used here will remain identical throughout the book.

We hope that the reader who is not familiar with the theory of stochastic

processes will ﬁnd here an intuitive presentation, although rigorous enough

for our purposes, and a set of useful references about the underlying

mathematical theory. The reader acquainted with stochastic calculus will

ﬁnd here an introduction of objects and notations that are used constantly,

although maybe not very explicitly.

This chapter does not aim at giving a thorough treatment of the theory

of stochastic processes, nor does it give a detailed view of mathematical

ﬁnance theory in general. It recalls, rather, the main general facts that will

be used in the examples developed in the next chapters.

1.1

PROBABILITY BASIS

Financial models used for the evaluation of derivatives are mainly concerned

with the uncertainty of the future evolution of the stock prices. The theory

of probability and stochastic processes provides a framework with a form

of uncertainty, called randomness. A probability space ⍀ is assumed to be

given once and for all, interpreted as consisting of all the possible paths

of the prices of securities we are interested in. We will suppose that this

probability space is rich enough to carry all the random objects we wish

to construct and use. This assumption is not restrictive for our purposes,

because we could always enlarge the space ⍀ , for example, by considering

a product space. Note that ⍀ can be chosen to be a “canonical space,”

1

2

MATHEMATICAL INTRODUCTION

such as the space of continuous functions, or the space of cadlag (French

acronym for “continuous from the right, with left limits”) functions.

We endow the set ⍀ with a -ﬁeld Ᏺ which is also assumed to be ﬁxed

throughout this book, unless otherwise speciﬁed. Ᏺ represents all the events

that are or will eventually be observable.

Let ސbe a probability measure on the measurable space (⍀ , Ᏺ). The

(Lebesgue) integral with respect to ސof a random variable X (that is, a

measurable function from (⍀ , Ᏺ) to (ޒN , ᏮN ), where ᏮN is the Borel -ﬁeld

on ޒN ) is denoted by [ޅX] instead of Ύ⍀ X d ސand is called the expectation

of X. If we need to emphasize that the expectation operator ޅis relative to

ސ, we denote it by ސޅ. We assume that the reader is familiar with general

notions of probability theory such as independence, correlation, conditional

expectation, and so forth. For more details and references, we refer to [9],

[45], or [49].

The probability space (⍀ , Ᏺ, )ސis endowed with a ﬁltration (Ᏺt , t Ն 0),

that is, a family of sub- -ﬁelds of Ᏺ such that Ᏺs ʚ Ᏺt for all 0 Յ s Յ t.

The ﬁltration is said to be ސ-complete if for all t, all ސ-null sets belong to

every Ᏺt ; it is said to be right-continuous if for all t > 0,

Ᏺt ס

ʝᏲ

t⑀ם

⑀ Ͼ0

It will be implicit in the sequel that all the ﬁltrations we use have been

previously completed and made right-continuous (this is always possible).

The ﬁltration Ᏺt represents the “ﬂow of information” available; we will

often deal with the ﬁltration generated by some process (e.g., stock price

process), in which case Ᏺt represents past observations up to time t. For

detailed studies on ﬁltrations the reader can consult any book concerned

with stochastic calculus, such as [44], [63], and [103].

1.2

PROCESSES

We will be concerned with random quantities whose values depend on time.

Denote by ᐀ a subset of ;םޒ᐀ can be םޒitself, a bounded interval [0, T ], or

a discrete set ᐀ = {0, 1, . . .}. In general, given a measurable space (E, Ᏹ), a

process with values in E is an application X : ⍀ × ᐀

E that is measurable

with respect to the -ﬁelds Ᏹ and Ᏺ Ꮾ᐀ , where Ꮾ᐀ denotes the Borel

-ﬁeld on ᐀ .

In our applications we will need to consider only the case in which

E = ޒN and Ᏹ is the Borel -ﬁeld ᏮN . From now on, we make these

assumptions. A process will be denoted by X or (Xt , t ʦ ᐀ ); the (random)

value of the process at time t ʦ ᐀ will be denoted by Xt or X(t); we

may sometimes wish to emphasize the dependence on , in which case

1.2

3

Processes

we will use the notation Xt ( ) or X(t, ). The jump at time t of a process

X, is denoted by ⌬Xt and deﬁned by ⌬Xt = Xt – XtϪ , where XtϪ =

lim⑀Q0 XtϪ⑀ .

Where in Time?

Before we take on the study of processes themselves, we deﬁne a class of

random times that form a cornerstone in the theory of stochastic processes.

These are the times that are “suited” to the ﬁltration Ᏺt .

DEFINITION 1.1

A random time T , that is a random variable with values in םޒʜ {ϱ}, is

called an Ᏺt -stopping time if for all t ʦ ᐀

͕T Յ t͖ ʦ Ᏺt

This deﬁnition means that at each time t, based on the available information

Ᏺt , one is able to determine whether T is in the past or in the future.

Stopping times include constant times, as well as hitting times (i.e., random

times of the form = inf{t ʦ ᐀ : Xt ʦ B}, where B is a Borel set), among

others.

From a ﬁnancial point of view, the different quantities encountered are

constrained to depend only on the available information at the time they

are given a value. In mathematical words, we state the following:

DEFINITION 1.2

A process X is said to be adapted to the ﬁltration Ᏺt (or Ᏺt -adapted) if,

for all t ʦ ᐀ , Xt is Ᏺt -measurable.

A process used to model the price of an asset must be adapted to the ﬂow

of information available in the market. On the other hand, this information

consists mainly in the prices of different assets. Given a process X, we can

deﬁne a ﬁltration (ᏲtX ), where ᏲtX is the smallest sub- -ﬁeld of Ᏺ that makes

the variables (Xu , u Յ t) simultaneously measurable. The ﬁltration ᏲtX is

said to be generated by X, and X is clearly adapted to it. One also speaks of

X “in its own ﬁltration.”

Because we do not make the assumption that the processes we consider

have continuous paths, we need to introduce a ﬁne view of the “past.”

Continuous processes play a special role in this setting.

4

MATHEMATICAL INTRODUCTION

DEFINITION 1.3

1. The predictable -ﬁeld ᏼ is the -ﬁeld on ⍀ × ᐀ generated by

Ᏺt -adapted processes whose paths are continuous.

2. A process X is said to be predictable if it is measurable with respect

to ᏼ.

That is, ᏼ is the smallest -ﬁeld on ⍀ × ᐀ such that every process X, viewed

as a function of (, t), for which t ۋX(t) is continuous, is ᏼ-measurable.

It can be shown that ᏼ is also generated by random intervals (S, T ] where

S < T are stopping times.

A process that describes the number of shares in a trading strategy must

be predictable, because the investment decision is taken before the price has

a possible instantaneous shock.

In discrete time, the deﬁnition of a predictable process is much simpler,

since then a process (Xi , i ʦ )ގis predictable if for each n, Xn is ᏲnϪ1 measurable. However, we have the satisfactory property that if X is an

Ᏺt -adapted process, then the process of left limits (XtϪ , t Ն 0) is predictable.

For more details about predictable processes, see [27] or [63].

Let us also mention the optional -ﬁeld: It is the -ﬁeld ᏻ on ⍀ × ᐀

generated by Ᏺt -adapted processes with right-continuous paths. It will not

be, for our purposes, as crucial as the predictable -ﬁeld; see, however,

Chapter 2 for a situation where this is needed.

We end this discussion by introducing the notion of localization, which

is the key to establishing certain results in a general case.

DEFINITION 1.4

A localizing sequence (Tn ) is an increasing sequence of stopping times

ϱ as n

ϱ.

such that Tn

In this chapter, a property is said to hold locally if there exists a localizing

sequence such that the property holds on every interval [0, Tn ]. This notion

is important, because there are many interesting cases in which important

properties hold only locally (and not on a ﬁxed interval, [0, ϱ), for example).

Martingales and Semimartingales

Among the adapted processes deﬁned in the foregoing section, not all

are suitable for ﬁnancial modelling. The work of Harrison and Pliska

[60] shows that only a certain class of processes, called semimartingales

1.2

5

Processes

are good candidates. Indeed, the reader familiar with the theory of arbitrage

knows that the stock price process must be a local martingale under an

appropriate probability measure; Girsanov’s theorem then implies that

it must be a semimartingale under any (locally) equivalent probability

measure.

DEFINITION 1.5

A process X is called an Ᏺt -martingale if it is integrable (i.e., |[ޅXt| ] < ϱ

for all t), Ᏺt -adapted, and if it satisﬁes, for all 0 Յ s Յ t

[ޅXt ͉Ᏺs ] סXs

(1.1)

X is called a local martingale if there is a localizing sequence (Tn ) such

that for all n, (XtٙTn , t Ն 0) is a martingale. X is called a semimartingale

if it is Ᏺt -adapted and can be written

X t סX0 םM t םV t

(1.2)

where M is a local martingale, V has a.s. (almost surely) ﬁnite variation,

and M and V are null at time t = 0. If V can be chosen to be predictable,

X is called a special semimartingale and the decomposition with such V

is called the canonical decomposition.

If we need to emphasize the underlying probability measure ސ, we will say

that X is a ސ-(semi)martingale.

With a semimartingale X are associated two increasing processes,

called the quadratic variation and the conditional quadratic variation.

These processes are interesting because they allow us to compute the

decomposition of a semimartingale under a change of probability measure:

This is the famous Girsanov theorem (see Section 1.3). We give a brief

introduction to these processes here; for more details, see for example [27],

[44], [63], [100], [103], [104], [105].

We ﬁrst turn to the quadratic variation of semimartingale.

DEFINITION 1.6

Let X be a semimartingale such that [ޅXt2 ] < ϱ for all t. There exists an

increasing process, denoted by [X, X], and called the quadratic variation

of X, such that

[X, X]t סplim

Α (Xt Ϫ Xt

nqϱ t ʦ (n)

i

i

iϪ 1

)2

(1.3)

6

MATHEMATICAL INTRODUCTION

where for each n, (n) = (0 = t0 < t1 < иии < tpn = t) is a subdivision

of [0, t] whose mesh sup1 Յ i Յ pn (ti – tiϪ1 ) tends to 0 as n tends to ϱ.

The abbreviation “plim” stands for “limit in probability.” It can be shown

that the above deﬁnition is actually meaningful: The limit does not depend

on a particular sequence of subdivisions. Moreover, if X is a martingale,

the quadratic variation is a compensator of X2 ; that is, X2 – [X, X] is again

a martingale. More generally, given a process X, another process Y will be

called a compensator for X if X – Y is a local martingale. Because of the

properties of martingales, compensation is the key to many properties when

paths are not supposed to be continuous.

Given two semimartingales X and Y , we deﬁne the quadratic covariation

of X and Y by a polarization identity:

[X, Y ] ס

1

([X םY, X םY ] Ϫ [X, X] Ϫ [Y, Y ])

2

Let M be a martingale. It can be shown that there exist two uniquely

determined martingales Mc and Md such that: M = Mc + Md , Mc has continuous paths and Md is orthogonal to any continuous martingale; that is,

Md N is a martingale for any continuous martingale N. Mc is called the

martingale continuous part of M, while Md is called the purely discontinuous part. If X is a special semimartingale, with canonical decomposition

X = M + V , Xc denotes the martingale continuous part of M, that is

X c ϵ Mc .

Note that the jump at time t of the quadratic variation of a semimartingale X is simply given by ⌬[X, X]t = (⌬Xt )2 . We have the following

important property:

[Xc , Xc ]t [ סX, X]ct [ סX, X]t Ϫ Α ⌬[X, X]s

(1.4)

sՅt

where the last sum is actually meaningful (see [100]).

We now turn to the conditional quadratic variation.

DEFINITION 1.7

Let X be a semimartingale such that [ޅXt2 ] < ϱ for all t. If

plim

Αޅ

nqϱ t ʦ (n)

i

ͫ

(Xti Ϫ XtiϪ1 )2 ͉ᏲtiϪ1

ͬ

(1.5)

1.2

7

Processes

exists, where for each n, (n) = (0 = t0 < t1 < иии < tpn = t) is a subdivision of [0, t] whose mesh sup1 Յ i Յ pn ti – tiϪ1 tends to 0 as n tends to

ϱ, and the limit does not depend on a particular subdivision, this limit is

called the conditional quadratic variation of X and is denoted by ͗X, X͘t .

In that case, ͗X, X͘t is an increasing process.

In contrast to the quadratic variation, the limit in (1.5) may fail to exist for

some semimartingales X. However, it can be shown that the limit exists, and

that the process ͗X, X͘ is well-deﬁned, if X is a special semimartingale, in

´ process or a continuous semimartingale, for example.

particular for a Levy

Similar to the case of quadratic variation, the conditional quadratic

covariation is deﬁned as

͗X, Y ͘ ס

1

(͗X םY, X םY ͘ Ϫ ͗X, X͘ Ϫ ͗Y, Y ͘)

2

as soon as this expression makes sense.

It can also be proven that when it exists, the conditional quadratic

variation is the predictable compensator of the quadratic variation; that is,

͗X, X͘ is a predictable process and [X, X] – ͗X, X͘ is a martingale. It follows

that if X is a martingale, X2 – ͗X, X͘ is also a martingale, and the quadratic

variation is the predictable compensator of X2 . The (conditional) quadratic

variation has the following well-known properties, provided the quantities

considered exist:

Ⅲ The applications (X, Y ) [ ۋX, Y ] and (X, Y ) ͗ ۋX, Y ͘ are linear in X

and Y .

Ⅲ If X has ﬁnite variation, [X, Y ] = ͗X, Y ͘ = 0 for any semimartingale Y .

Moreover we have the following important identity (see [100]):

[Xc , Xc ] ͗ סXc , Xc ͘

so that if X has continuous paths, ͗X, X͘ is identical to [X, X]. The (conditional) quadratic variation will appear into the decomposition of F(X)

¯ formula, which lies at the heart of stochastic

for suitable F, given by Ito’s

calculus.

Markov Processes

We now introduce brieﬂy another class of processes that are memoryless at

stopping times.

8

MATHEMATICAL INTRODUCTION

DEFINITION 1.8

1. An Ᏺt -adapted process X is called a Markov process in the ﬁltration

(Ᏺt ) if for all t Ն 0, for every measurable and bounded functional F,

[ޅF(Xtםs , s Ն 0)͉Ᏺt ] [ޅ סF(Xtםs , s Ն 0)͉Xt ]

(1.6)

2. X is called a strong Markov process if (1.6) holds with t replaced by

any ﬁnite stopping time T .

In other words, for a Markov process, at each time t, the whole past is

summarized in the present value of the process Xt . For a strong Markov

process, this is true with a stopping time. In ﬁnancial words, an investment

decision is often made on the basis of the present state of the market, that

in some sense sums up its history.

A nice feature of Markov processes is the Feynman-Kac formula; this

formula links Markov processes to (integro-)partial differential equations

and makes available numerical techniques such as the ﬁnite difference

method explained in Chapter 4. We do not go further into Markov processes

and go on with stochastic calculus. Some relationships between Markov

processes and semimartingales are discussed in [28].

1.3

STOCHASTIC CALCULUS

With the processes deﬁned in the previous section (semimartingales), a

theory of (stochastic) integral calculus can be built and used to model

ﬁnancial time series. Accordingly, this section contains the two results of

¯ formula and the

probability theory that are most useful in ﬁnance: Ito’s

Girsanov theorem, both in a quite general form.

The construction and properties of the stochastic integral are well

known, and the ﬁnancial reader can think of most of them by taking the

parallel of a portfolio strategy (see Section 1.4 and Chapter 2).

In general, the integral of a process H with respect to another one X

is well-deﬁned provided H is locally bounded and predictable and X is a

semimartingale with [ޅXt2 ] < ϱ for all t. The integral can then be thought

of as the limit of elementary sums

Ύ

t

0

Hs dXs “ סlim

nqϱ

Α Ht (Xt

i

iם1

Ϫ Xti )”

ti ʦ (n)

where for each n, (n) = (0 = t0 < t1 < иии < tpn = t) is a subdivision of

[0, t] whose mesh sup1 Յ i Յ pn (ti – tiϪ1 ) tends to 0 as n tends to ϱ. See [27],

[100], [103], or [104] for a rigorous deﬁnition.

1.3

9

Stochastic Calculus

Note an important property of the stochastic integral. Let X, Y be

semimartingales and H a predictable process such that Ύ Hs dYs is welldeﬁned; the following formula holds:

ͫ

Ύ

X,

.

Hs dYs

0

ͬ

t

ס

Ύ

t

Hs d[X, Y ]s

(1.7)

0

.

t

where Ύ0 Hs dYs denotes the process (Ύ0 Hs dYs , t Ն 0). The same formula

holds with [., .] replaced with ͗., .͘, provided the latter exists; this follows

from the linearity of the quadratic variation and the stochastic integral.

¯ Formula

Ito’s

We can now state the famous Ito’s

¯ formula. More details can be found in the

references mentioned previously. Let X = (X1 , . . . , Xn ) be a semimartingale

with values in ޒn and F be a function ޒn

ޒm of class C2 . Then F(X) is a

semimartingale, and

n

F(Xt ) סF(X0 ) םΑ

Ύ

t

iס1 0

n

ם

n

1

2 ΑΑ

iס1 jס1

Ύ

t

0

ѨF

(XsϪ )dXsi

Ѩxi

Ѩ 2F

(Xs )d[Xi , Xj ]cs

Ѩxi Ѩxj

Ά

n

םΑ F(Xs ) Ϫ F(XsϪ ) Ϫ Α

sՅt

iס1

(1.8)

ѨF

(XsϪ )⌬Xsi

Ѩxi

·

¯ formula is often written in the

where ⌬X is the jump process of X. Ito’s

differential form

n

dF(Xt ) ס

ѨF

Α Ѩxi (XtϪ )dXti

iס1

ם

1 n n Ѩ 2F

(Xt )d[Xi , Xj ]ct

Ѩx i Ѩx j

2 ΑΑ

iס1 jס1

(1.9)

םdZt

where

Ά

n

Zt סΑ F(Xs ) Ϫ F(XsϪ ) Ϫ Α

sՅt

iס1

ѨF

(XsϪ )⌬Xsi

Ѩ xi

·

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