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Equity derivatives theory and applications

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

·


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