Lin X S -Introductory Stochastic Analysis For Finance And Insurance (2006)
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Lin X S -Introductory Stochastic Analysis For Finance And Insurance (2006)

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Introductory Stochastic 
Analysis for 
Finance and Insurance 
X. Sheldon Lin 
University of Toronto 
Department of Statistics 
Toronto, Ontario, Canada 
* 1949 
A JOHN WILEY & SONS, INC., PUBLICATION 
This Page Intentionally Left Blank
Introductory Stochastic 
Analysis for 
Finance and Insurance 
WILEY SERIES IN PROBABILITY AND STATISTICS 
Established by WALTERA. SHEWHART and SAMUEL S. WILKS 
Editors: David 1 Balding, Noel A . C. Cressie, Nicholas I. Fisher, 
Iain M. Johnstone, 1 B. Kadane, Geert Molenberghs, Louise M. Ryan, 
David W Scott, Adrian I? M. Smith, Jozef L. Teugels 
Editors Emeriti: Vic Barnett, 1 Stuart Hunter, David G. Kendall 
A complete list of the titles in this series appears at the end of this volume. 
Introductory Stochastic 
Analysis for 
Finance and Insurance 
X. Sheldon Lin 
University of Toronto 
Department of Statistics 
Toronto, Ontario, Canada 
* 1949 
A JOHN WILEY & SONS, INC., PUBLICATION 
Copyright 0 2006 by John Wiley & Sons, Inc. All rights reserved 
Published by John Wiley & Sons, Inc., Hoboken, New Jersey 
Published simultaneously in Canada. 
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form 
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Library of Congress Cataloging-in-Publication Data is available. 
ISBN-I 3 978-0-471 -71 642-6 
ISBN-I0 0-47 1-71 642-1 
Printed in the United States of America 
I 0 9 8 7 6 5 4 3 2 1 
To my family 
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CONTENTS 
List of Figures 
List of Tables 
Preface 
1 Introduction 
2 Overview of Probability Theory 
2.1 Probability Spaces and Information Structures 
2.2 Random Variables, Moments and Transforms 
2.3 Multivariate Distributions 
2.4 Conditional Probability and Conditional Distributions 
2.5 Conditional Expectation 
2.6 The Central Limit Theorem 
3 Discrete-Time Stochastic Processes 
3.1 Stochastic Processes and Information Structures 
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xiii 
xv 
1 
5 
6 
11 
20 
24 
34 
43 
45 
45 
vii 
Vii i CONTENTS 
3.2 Random Walks 
3.3 Discrete-Time Markov Chains 
3.4 Martingales and Change of Probability Measure 
3.5 Stopping Times 
3.6 Option Pricing with Binomial Models 
3.7 Binomial Interest Rate Models 
4 Continuous-Time Stochastic Processes 
4.1 General Description of Continuous-Time Stochastic Processes 
4.2 Brownian Motion 
4.3 The Reflection Principle and Barrier Hitting Probabilities 
4.4 The Poisson Process and Compound Poisson Process 
4.5 Martingales 
4.6 Stopping Times and the Optional Sampling Theorem 
5 Stochastic Calculus: Basic Topics 
5.1 Stochastic jIto) Integration 
5.2 Stochastic Differential Equations 
5.3 One-Dimensional Ito\u2019s Lemma 
5.4 Continuous-Time Interest Rate Models 
5.5 The Black-Scholes Model and Option Pricing Formula 
5.6 The Stochastic Version of Integration by Parts 
5.7 Exponential Martingales 
5.8 The Martingale Representation Theorem 
6 Stochastic Calculus: Advanced Topics 
6.1 The Feynman-Kac Formula 
6.2 The Black-Scholes Partial Differential Equation 
6.3 The Girsanov Theorem 
6.4 The Forward Risk Adjusted Measure and Bond Option Pricing 
6.5 Barrier Hitting Probabilities Revisited 
6.6 Two-Dimensional Stochastic Differential Equations 
7 Applications in insurance 
7.1 Deferred Variable Annuities and Equity-Indexed Annuities 
7.2 Guaranteed Annuity Options 
7.3 Universal Life 
47 
55 
60 
66 
72 
84 
97 
97 
98 
104 
112 
I I7 
122 
131 
131 
141 
! 44 
148 
155 
162 
165 
168 
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174 
175 
177 
181 
187 
191 
197 
198 
206 
210 
References 
Topic Index 
CONTENTS ix 
217 
221 
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LIST OF FIGURES 
2.1. 
2.2. 
2.3. 
3.1. 
3.2. 
3.3. 
3.4. 
3.5. 
3.6. 
3.7. 
3.8. 
3.9. 
The price of a stock over a two-day period. 
The probability tree of the stock price over a two-day period. 
The expectation tree of the stock price over a two-day period 
The tree of a standard random walk. 
The binomial model of the stock price. 
The binomial tree of the stock price. 
The returns of a stock and a bond. 
The payoff function of a call. 
The payoff function of a put. 
The payoff function of a strangle. 
1 1 
26 
39 
49 
53 
55 
73 
74 
74 
77 
Treasury yield curve, Treasury zero curve, and Treasury forward rate 
curve based on the quotes in Table 3.1. 
Constructing a short rate tree: step one. 
88 
92 
xi 
xii LIST OF FIGURES 
3.10. 
3.1 1 . 
4.1. 
4.2. 
4.3. 
4.4. 
4.5. 
4.6. 
4.7. 
4.8. 
4.9. 
Constructing a short rate tree: step two. 
The complete short rate tree. 
A sample path of standard Brownian motion ( p = 0 and 0 = 1). 
A sample path of Brownian motion with p = 1 and 0 = 1. 
A sample path of Brownian motion with p = - 1 and 
A sample path of Brownian motion with p = 0 and 
A sample path of Brownian motion with p = 0 and 0 = 0.5. 
A path of standard Brownian motion reflected after hitting. 
A path of standard Brownian motion reflected before hitting. 
A sample path of a compound Poisson process. 
A sample path of the shifted Poisson process { X T ( t ) } . 
= 1. 
= 2. 
93 
93 
103 
104 
105 
106 
107 
109 
110 
114 
116 
LIST OF TABLES 
3. I . 
3.2. The market term structure. 
5.1. 
A sample of quotes on U.S. Treasuries. 
The product rules in stochastic calculus. 
86 
91 
146 
xiii 
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PREFACE 
The aim of this book is to provide basic stochastic analysis techniques for 
mathematical finance. It is intended for those who want to study mathemati- 
cal finance but have a limited background in probability theory and stochastic 
analysis. The writing of the book started seven years ago after I taught mathe- 
matical finance to graduate students and practitioners for several years. Most 
of my students have a Master\u2019s degree in science or engineering but had only 
taken one or two entry level probability courses prior to my course. My 
initial approach for teaching such a course was to focus on financial mod- 
els and aspects of modelling techniques. I soon found that the lack