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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 or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 1 1 1 River Street, Hoboken, NJ 07030, (201) 748-601 1, fax (201) 748-6008, or online at http:Nwww.wiley.comlgo/permission. 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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 This Page Intentionally Left Blank 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 xi 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 ! 44 148 155 162 165 168 173 174 175 177 181 187 191 197 198 206 210 References Topic Index CONTENTS ix 217 221 This Page Intentionally Left Blank 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 This Page Intentionally Left Blank 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