Quantitative finance: its development, mathematical foundations, and current scope

399 Pages·2009·19 MB·English

by  T. Wake Epps

QUANTITATIVE FINANCE
QUANTITATIVE FINANCE
Its Development, Math em ati c al
Foundations, and Current Scope
T. W. Epps
University of Virginia
@
WILEY
A JOHN WILEY &SONS, INC., PUBLICATION
Copyright Q 2009 by John Wiley & Sons, Inc. All rights reserved.
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Library of Congress Cataloging-in-Publication Data:
Epps, T. W.
Quantitative finance : its development, mathematical foundations, and current scope / T.W. Epps.
p. crn.
Includes bibliographical references and index.
ISBN 978-0-470-43199-3 (cloth)
1. Finance-Mathematical models. 2. Investments-Mathematical models. I. Title.
HG106.E67 2009
332.0 1'51 95-dc22 2008041830
Printed in the United States of America
1 0 9 8 7 6 5 4 3 2 1
In loving memory of my mother and father
Jane Wakefield Epps, 1918-2008
Thomas L. Epps, 1920-1980
CONTENTS
Preface xv
Acronyms and Abbreviations xviii
PART I PERSPECTIVE AND PREPARATION
1 Introduction and Overview 3
1.1 An Elemental View of Assets and Markets 3
1.1.1 Assets as Bundles of Claims 4
1.1.2 Financial Markets as Transportation Agents 5
1.1.3 Why Is Transportation Desirable? 5
1.1.4 What Vehicles Are Available? 6
1.1.5 What Is There to Learn about Assets and Markets? 7
1.1.6 Why the Need for Quantitative Finance? 8
1.2 Where We Go from Here 8
2 Tools from Calculus and Analysis 11
2.1 Some Basics from Calculus 12
2.2 Elements of Measure Theory 15
vii
viii CONTENTS
2.2.1 Sets and Collections of Sets 15
2.2.2 Set Functions and Measures 16
2.3 Integration 18
2.3.1 Riemann-Stieltjes 19
2.3.2 LebesgueLebesgue-S tieltj es 20
2.3.3 Properties of the Integral 21
2.4 Changes of Measure 23
3 ProbabiI ity 25
3.1 Probability Spaces 25
3.2 Random Variables and Their Distributions 28
3.3 Independence of Random Variables 33
3.4 Expectation 34
3.4.1 Moments 36
3.4.2 Conditional Expectations and Moments 38
3.4.3 Generating Functions 40
3.5 Changes of Probability Measure 41
3.6 Convergence Concepts 42
3.7 Laws of Large Numbers and Central-Limit Theorems 45
3.8 Important Models for Distributions 46
3.8.1 Continuous Models 46
3.8.2 Discrete Models 51
PART II PORTFOLIOS AND PRICES
4 Interest and Bond Prices 55
4.1 Interest Rates and Compounding 55
4.2 Bond Prices, Yields, and Spot Rates 57
4.3 Forward Bond Prices and Rates 63
Exercises 66
Empirical Project 1 67
5 Models of Portfolio Choice 71
5.1 Models That Ignore Risk 72
5.2 Mean-Variance Portfolio Theory 75
5.2.1 Mean-Variance “Efficient” Portfolios 75
5.2.2 The Single-Index Model 79
Exercises 81
Empirical Project 2 82
CONTENTS ix
6 Prices in a Mean-Variance World 87
6.1 The Assumptions 87
6.2 The Derivation 88
6.3 Interpretation 91
6.4 Empirical Evidence 91
6.5 Some Reflections 94
Exercises 95
7 Rational Decisions under Risk 97
7.1 The Setting and the Axioms 98
7.2 The Expected-Utility (EU) Theorem 100
7.3 Applying EU Theory 103
7.3.1 Implementing EU Theory in Financial Modeling 104
7.3.2 Inferring Utilities and Beliefs 105
7.3.3 Qualitative Properties of Utility Functions 106
7.3.4 Measures of Risk Aversion 107
7.3.5 Examples of Utility Functions 108
7.3.6 Some Qualitative Implications of the EU Model 109
7.3.7 Stochastic Dominance 114
7.4 Is the Markowitz Investor Rational? 117
Exercises 121
Empirical Project 3 123
8 Observed Decisions under Risk 127
8.1 Evidence about Choices under Risk 128
8.1.1 Allais’ Paradox 128
8.1.2 Prospect Theory 129
8.1.3 Preference Reversals 131
8.1.4 Risk Aversion and Diminishing Marginal Utility 133
8.2 Toward “Behavioral” Finance 134
Exercises 136
9 Distributions of Returns 139
9.1 Some Background 140
9.2 The NormalLognormal Model 143
9.3 The Stable Model 147
9.4 Mixture Models 150
9.5 Comparison and Evaluation 152
Exercises 153
X CONTENTS
10 Dynamics of Prices and Returns 155
10.1 Evidence for First-Moment Independence 155
10.2 Random Walks and Martingales 160
10.3 Modeling Prices in Continuous Time 164
10.3.1 Poisson and Compound-Poisson Processes 165
10.3.2 Brownian Motions 167
10.3.3 Martingales in Continuous Time 171
Exercises 171
Empirical Project 4 173
11 Stochastic Calculus 177
11 .1 Stochastic Integrals 178
11 .1.1 It8 Integrals with Respect to a Brownian Motion (BM) 178
11 .1.2 From It8 Integrals to It6 Processes 180
11.1.3 Quadratic Variations of It8 Processes 182
11 .1.4 Integrals with Respect to It8 Processes 183
11.2 Stochastic Differentials 183
1 1.3 ItB’s Formula for Differentials 185
11 .3.1 Functions of a BM Alone 185
11.3.2 Functions of Time and a BM 186
11.3.3 Functions of Time and General It8 Processes 187
Exercises 189
12 Portfolio Decisions over Time 191
12.1 The Consumption-Investment Choice 192
12.2 Dynamic Portfolio Decisions 193
12.2.1 Optimizing via Dynamic Programming 194
12.2.2 A Formulation with Additively Separable Utility 198
Exercises 200
13 Optimal Growth 201
13.1 Optimal Growth in Discrete Time 203
13.2 Optimal Growth in Continuous Time 206
13.3 Some Qualifications 209
Exercises 21 1
Empirical Project 5 213
CONTENTS xi
14 Dynamic Models for Prices 21 7
14.1 Dynamic Optimization (Again) 218
14.2 Static Implications: The Capital Asset Pricing Model 219
14.3 Dynamic Implications: The Lucas Model 220
14.4 Assessment 223
14.4.1 The Puzzles 224
14.4.2 The Patches 225
14.4.3 Some Reflections 226
Exercises 227
15 Efficient Markets 229
15.1 Event Studies 230
15.1.1 Methods 23 1
15.1.2 A Sample Study 232
15.2 Dynamic Tests 234
15.2.1 Early History 234
15.2.2 Implications of the Dynamic Models 236
15.2.3 Excess Volatility 237
Exercises 24 1
PART 111 PARADIGMS FOR PRICING
16 Static Arbitrage Pricing 245
16.1 Pricing Paradigms: Optimization versus Arbitrage 246
16.2 The Arbitrage Pricing Theory (APT) 248
16.3 Arbitraging Bonds 252
16.4 Pricing a Simple Derivative Asset 254
Exercises 257
17 Dynamic Arbitrage Pricing 261
17.1 Dynamic Replication 262
17.2 Modeling Prices of the Assets 263
17.3 The Fundamental Partial Differential Equation (PDE) 264
17.3.1 The Feynman-Kac Solution to the PDE 266
17.3.2 Working out the Expectation 269
17.4 Allowing Dividends and Time-Varying Rates 27 1
Exercises 272
18 Properties of Option Prices 275
18.1 Bounds on Prices of European Options 275