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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 Quaatitative 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
2.2.1 Sets and Collections of Sets 15
2.2.2 Set Functions nnd Measures 16
2.3 Integration 18
2.3.1 Riemann—Stieltjes 19
2.3.2 Lebesgue/Lebesgue—Stiettjes 20
2.3.3 Properties of the tntegrat 21
2.4 Changes of Measure 23
3 ProbabIlity 25
3.1 Probability Spaces 25
3.2 Random Variables and Their Distribntions 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 tmportant 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 tnterest Rates and Componnding 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
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 Measnres 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 Allaie’ 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 Normal/Lngnormal Model 143
9.3 The Stable Model 147
9.4 Mixture Models 150
9.5 Comparison and Evaluation 152
Exercises 153
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 ItO Integrals with Respect to a Brownian Motion (BMj 178
11.1.2 From ItO Integrals to ItO Processes 180
11.1.3 Quadratic Variations of ItO Processes 182
11.1.4 Integrals with Respect to Ito Processes 183
11.2 Stochastic Differentials 183
11.3 ItO’s Formula for Differentials 185
11.3.1 FunctionsofaBMAlone 185
11.3.2 Functions of Time and a BM 186
11.3.3 Functions of Time snd General Ito 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 Optimicing via Dynamic Programming 194
12.2.2 A Formulation with Additively Separable Utility 198
Exercises 200
13 Optlmel Growth 201
13.1 Optimal Growth in Discrete Time 203
13.2 Optimal Growth in Continuous Time 206
13.3 Some Qualifications 209
Exercises 211
Empirical Project 5 213
14 Dynamic Models for Prices 217
14.1 Dynamic Optimization (Again) 218
14.2 Static Implications: The Capital Asset Pricing Model 219
14.3 Dynamic Implications: The Locas 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 Stadies 230
15.1.1 Methods 231
15,1.2 A Sample Study 232
t5.2 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 241
PART III 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 271
Exercises 272
18 Properties of Option Pricea 275
18.1 Bounds on Prices of Eucopean Options 275
18.2 Properties of Black-Scholes Prices 277
18.3 Delta Hedging 280
18.4 Does Black—Seholes Still Work? 282
18.5 American-Style Options 283
Exercises 285
Empirical Project 6 285
19 Martingale Pricing 289
19.1 Some Preparation 290
19.2 Fondameotat Theorem of Asset Pricing 291
19.3 Implications foe Pricing Derivatives 294
19.4 Applications 296
19.5 Martingale versus Equilibrium Pricing 298
19.6 Numeraires, Short Rates, and Equivalent Martingsle Measures 300
19.7 Replication and Uniqueness of the RMM 302
Exercises 304
20 Modeling Volatility 307
20.1 Models with Price-Dependent Volxtility 308
20.1.1 The Constant-Elasticity-of-Variance Model 308
20.1.2 The Hobson—Rogers Model 309
20.2 Autoregressive Condilional Heteroskednssieity Models 310
20.3 Stochastic Volatility 312
20.4 Is Replication Possible? 313
Exercises 314
21 Discontinuous Price Processes 317
21.1 Merton’s Jnmp—Diffosion Model 318
21.2 The Varianee—Gammn Model 322
21.3 Stock Prices as Branching Processes 324
21.4 Is Replication Possible? 326
Exercises 327
22 Options on Jump Processes 329
22.1 Options under Jump—Diffusions 330
22.2 A Primer on Characteristic Functions 336
22.3 Using Fourier Methods to Price Options 339
22.4 Applications to Jump Models 341
Exercises 344
23 Options on Stochastic Volatility Processes 347
23.1 Independent Price/Volatility Shocks 348
23.2 Dependent Price/Volatility Shocks 350
23.3 Stochastic Vnlatility with Jumps in Price 354
23.4 Further Advances 356
Exercises 357
Empirical Project 7 358
Solutions to Exercises 363
References 391
Index 397