Preface xvii
I THE SETTING: MARKETS, MODELS, INTEREST RATES,
UTILITY MAXIMIZATION, RISK 1
1 Financial Markets 3
1.1Bonds 3
1.1.1 Types of Bonds 5
1.1.2 Reasons for Trading Bonds 5
1.1.3 Risk of Trading Bonds 6
1.2Stocks 7
1.2.1 How Are Stocks Different from Bonds? 8
1.2.2 Going Long or Short 9
1.3 Derivatives 9
1.3.1 Futures and Forwards 10
1.3.2 Marking to Market 11
1.3.3 Reasons for Trading Futures 12
1.3.4Options 13
1.3.5 Calls and Puts 13
1.3.6 Option Prices 15
1.3.7 Reasons for Trading Options 16
1.3.8 Swaps 17
1.3.9 Mortgage-Backed Securities; Callable Bonds 19
1.4 Organization of Financial Markets 20
1.4.1Exchanges 20
1.4.2 Market Indexes 21
1.5 Margins 22
1.5.1 Trades That Involve Margin Requirements 23
1.6 Transaction Costs 24
Summary 25
Problems 26
Further Readings 29
2 Interest Rates 31
2.1 Computation of Interest Rates 31
2.1.1 Simple versus Compound Interest; Annualized Rates 32
2.1.2 Continuous Interest 34
viii Contents
2.2 Present Value 35
2.2.1 Present and Future Values of Cash Flows 36
2.2.2 Bond Yield 39
2.2.3 Price-Yield Curves 39
2.3 Term Structure of Interest Rates and Forward Rates 41
2.3.1 Yield Curve 41
2.3.2 Calculating Spot Rates; Rates Arbitrage 43
2.3.3 Forward Rates 45
2.3.4 Term-Structure Theories 47
Summary 48
Problems 49
Further Readings 51
3 Models of Securities Prices in Financial Markets 53
3.1 Single-Period Models 54
3.1.1 Asset Dynamics 54
3.1.2 Portfolio and Wealth Processes 55
3.1.3 Arrow-Debreu Securities 57
3.2 Multiperiod Models 58
3.2.1 General Model Specifications 58
3.2.2 Cox-Ross-Rubinstein Binomial Model 60
3.3 Continuous-Time Models 62
3.3.1 Simple Facts about the Merton-Black-Scholes Model 62
3.3.2 Brownian Motion Process 63
3.3.3 Diffusion Processes, Stochastic Integrals 66
3.3.4 Technical Properties of Stochastic Integrals∗ 67
3.3.5 Itˆo’s Rule 69
3.3.6 Merton-Black-Scholes Model 74
3.3.7 Wealth Process and Portfolio Process 78
3.4 Modeling Interest Rates 79
3.4.1 Discrete-Time Models 79
3.4.2 Continuous-Time Models 80
3.5 Nominal Rates and Real Rates 81
3.5.1 Discrete-Time Models 81
3.5.2 Continuous-Time Models 83
Contents ix
3.6 Arbitrage and Market Completeness 83
3.6.1 Notion of Arbitrage 84
3.6.2 Arbitrage in Discrete-Time Models 85
3.6.3 Arbitrage in Continuous-Time Models 86
3.6.4 Notion of Complete Markets 87
3.6.5 Complete Markets in Discrete-Time Models 88
3.6.6 Complete Markets in Continuous-Time Models∗ 92
3.7 Appendix 94
3.7.1 More Details for the Proof of Itˆo’s Rule 94
3.7.2 Multidimensional Itˆo’s Rule 97
Summary 97
Problems 98
Further Readings 101
4 Optimal Consumption/Portfolio Strategies 103
4.1 Preference Relations and Utility Functions 103
4.1.1 Consumption 104
4.1.2 Preferences 105
4.1.3 Concept of Utility Functions 107
4.1.4 Marginal Utility, Risk Aversion, and Certainty Equivalent 108
4.1.5 Utility Functions in Multiperiod Discrete-Time Models 112
4.1.6 Utility Functions in Continuous-Time Models 112
4.2 Discrete-Time Utility Maximization 113
4.2.1 Single Period 114
4.2.2 Multiperiod Utility Maximization: Dynamic Programming 116
4.2.3 Optimal Portfolios in the Merton-Black-Scholes Model 121
4.2.4 Utility from Consumption 122
4.3 Utility Maximization in Continuous Time 122
4.3.1 Hamilton-Jacobi-Bellman PDE 122
4.4 Duality/Martingale Approach to Utility Maximization 128
4.4.1 Martingale Approach in Single-Period Binomial Model 128
4.4.2 Martingale Approach in Multiperiod Binomial Model 130
4.4.3 Duality/Martingale Approach in Continuous Time∗ 133
4.5 Transaction Costs 138
4.6 Incomplete and Asymmetric Information 139
4.6.1 Single Period 139
x Contents
4.6.2 Incomplete Information in Continuous Time∗ 140
4.6.3 Power Utility and Normally Distributed Drift∗ 142
4.7 Appendix: Proof of Dynamic Programming Principle 145
Summary 146
Problems 147
Further Readings 150
5 Risk 153
5.1 Risk versus Return: Mean-Variance Analysis 153
5.1.1 Mean and Variance of a Portfolio 154
5.1.2 Mean-Variance Efficient Frontier 157
5.1.3 Computing the Optimal Mean-Variance Portfolio 160
5.1.4 Computing the Optimal Mutual Fund 163
5.1.5 Mean-Variance Optimization in Continuous Time∗ 164
5.2 VaR: Value at Risk 167
5.2.1 Definition of VaR 167
5.2.2 Computing VaR 168
5.2.3 VaR of a Portfolio of Assets 170
5.2.4 Alternatives to VaR 171
5.2.5 The Story of Long-Term Capital Management 171
Summary 172
Problems 172
Further Readings 175
II PRICING AND HEDGING OF DERIVATIVE SECURITIES 177
6 Arbitrage and Risk-Neutral Pricing 179
6.1 Arbitrage Relationships for Call and Put Options; Put-Call Parity 179
6.2 Arbitrage Pricing of Forwards and Futures 184
6.2.1 Forward Prices 184
6.2.2 Futures Prices 186
6.2.3 Futures on Commodities 187
6.3 Risk-Neutral Pricing 188
6.3.1 Martingale Measures; Cox-Ross-Rubinstein (CRR) Model 188
6.3.2 State Prices in Single-Period Models 192
6.3.3 No Arbitrage and Risk-Neutral Probabilities 193
Contents xi
6.3.4 Pricing by No Arbitrage 194
6.3.5 Pricing by Risk-Neutral Expected Values 196
6.3.6 Martingale Measure for the Merton-Black-Scholes Model 197
6.3.7 Computing Expectations by the Feynman-Kac PDE 201
6.3.8 Risk-Neutral Pricing in Continuous Time 202
6.3.9 Futures and Forwards Revisited∗ 203
6.4Appendix 206
6.4.1 No Arbitrage Implies Existence of a Risk-Neutral Probability∗ 206
6.4.2 Completeness and Unique EMM∗ 207
6.4.3 Another Proof of Theorem 6.4∗ 210
6.4.4 Proof of Bayes’ Rule∗∗ 211
Summary 211
Problems 213
Further Readings 215
7 Option Pricing 217
7.1 Option Pricing in the Binomial Model 217
7.1.1 Backward Induction and Expectation Formula 217
7.1.2 Black-Scholes Formula as a Limit of the Binomial
Model Formula 220
7.2 Option Pricing in the Merton-Black-Scholes Model 222
7.2.1 Black-Scholes Formula as Expected Value 222
7.2.2 Black-Scholes Equation 222
7.2.3 Black-Scholes Formula for the Call Option 225
7.2.4 Implied Volatility 227
7.3 Pricing American Options 228
7.3.1 Stopping Times and American Options 229
7.3.2 Binomial Trees and American Options 231
7.3.3 PDEs and American Options∗ 233
7.4 Options on Dividend-Paying Securities 235
7.4.1 Binomial Model 236
7.4.2 Merton-Black-Scholes Model 238
7.5 Other Types of Options 240
7.5.1 Currency Options 240
7.5.2 Futures Options 242
7.5.3 Exotic Options 243
xii Contents
7.6 Pricing in the Presence of Several Random Variables 247
7.6.1 Options on Two Risky Assets 248
7.6.2 Quantos 252
7.6.3 Stochastic Volatility with Complete Markets 255
7.6.4 Stochastic Volatility with Incomplete Markets; Market Price
of Risk∗ 256
7.6.5 Utility Pricing in Incomplete Markets∗ 257
7.7 Merton’s Jump-Diffusion Model∗ 260
7.8 Estimation of Variance and ARCH/GARCH Models 262
7.9 Appendix: Derivation of the Black-Scholes Formula 265
Summary 267
Problems 268
Further Readings 273
8 Fixed-Income Market Models and Derivatives 275
8.1 Discrete-Time Interest-Rate Modeling 275
8.1.1 Binomial Tree for the Interest Rate 276
8.1.2 Black-Derman-Toy Model 279
8.1.3 Ho-Lee Model 281
8.2 Interest-Rate Models in Continuous Time 286
8.2.1 One-Factor Short-Rate Models 287
8.2.2 Bond Pricing in Affine Models 289
8.2.3 HJM Forward-Rate Models 291
8.2.4 Change of Numeraire∗ 295
8.2.5 Option Pricing with Random Interest Rate∗ 296
8.2.6 BGM Market Model∗ 299
8.3 Swaps, Caps, and Floors 301
8.3.1 Interest-Rate Swaps and Swaptions 301
8.3.2 Caplets, Caps, and Floors 305
8.4 Credit/Default Risk 306
Summary 308
Problems 309
Further Readings 312
9 Hedging 313
9.1 Hedging with Futures 313
9.1.1 Perfect Hedge 313
Contents xiii
9.1.2 Cross-Hedging; Basis Risk 314
9.1.3 Rolling the Hedge Forward 316
9.1.4 Quantity Uncertainty 317
9.2 Portfolios of Options as Trading Strategies 317
9.2.1 Covered Calls and Protective Puts 318
9.2.2 Bull Spreads and Bear Spreads 318
9.2.3 Butterfly Spreads 319
9.2.4 Straddles and Strangles 321
9.3 Hedging Options Positions; Delta Hedging 322
9.3.1 Delta Hedging in Discrete-Time Models 323
9.3.2 Delta-Neutral Strategies 325
9.3.3 Deltas of Calls and Puts 327
9.3.4 Example: Hedging a Call Option 327
9.3.5 Other Greeks 330
9.3.6 Stochastic Volatility and Interest Rate 332
9.3.7 Formulas for Greeks 333
9.3.8 Portfolio Insurance 333
9.4 Perfect Hedging in a Multivariable Continuous-Time Model 334
9.5 Hedging in Incomplete Markets 335
Summary 336
Problems 337
Further Readings 340
10 Bond Hedging 341
10.1 Duration 341
10.1.1 Definition and Interpretation 341
10.1.2 Duration and Change in Yield 345
10.1.3 Duration of a Portfolio of Bonds 346
10.2Immunization 347
10.2.1 Matching Durations 347
10.2.2 Duration and Immunization in Continuous Time 350
10.3 Convexity 351
Summary 352
Problems 352
Further Readings 353
xiv Contents
11 Numerical Methods 355
11.1 Binomial Tree Methods 355
11.1.1 Computations in the Cox-Ross-Rubinstein Model 355
11.1.2 Computing Option Sensitivities 358
11.1.3 Extensions of the Tree Method 359
11.2 Monte Carlo Simulation 361
11.2.1 Monte Carlo Basics 362
11.2.2 Generating Random Numbers 363
11.2.3 Variance Reduction Techniques 364
11.2.4 Simulation in a Continuous-Time Multivariable Model 367
11.2.5 Computation of Hedging Portfolios by Finite Differences 370
11.2.6 Retrieval of Volatility Method for Hedging and
Utility Maximization∗ 371
11.3 Numerical Solutions of PDEs; Finite-Difference Methods 373
11.3.1 Implicit Finite-Difference Method 374
11.3.2 Explicit Finite-Difference Method 376
Summary 377
Problems 378
Further Readings 380
III EQUILIBRIUM MODELS 381
12 Equilibrium Fundamentals 383
12.1 Concept of Equilibrium 383
12.1.1 Definition and Single-Period Case 383
12.1.2 A Two-Period Example 387
12.1.3 Continuous-Time Equilibrium 389
12.2 Single-Agent and Multiagent Equilibrium 389
12.2.1 Representative Agent 389
12.2.2 Single-Period Aggregation 389
12.3 Pure Exchange Equilibrium 391
12.3.1 Basic Idea and Single-Period Case 392
12.3.2 Multiperiod Discrete-Time Model 394
12.3.3 Continuous-Time Pure Exchange Equilibrium 395
12.4 Existence of Equilibrium 398
12.4.1 Equilibrium Existence in Discrete Time 399
Contents xv
12.4.2 Equilibrium Existence in Continuous Time 400
12.4.3 Determining Market Parameters in Equilibrium 403
Summary 406
Problems 406
Further Readings 407
13 CAPM 409
13.1 Basic CAPM 409
13.1.1 CAPM Equilibrium Argument 409
13.1.2 Capital Market Line 411
13.1.3 CAPM formula 412
13.2 Economic Interpretations 413
13.2.1 Securities Market Line 413
13.2.2 Systematic and Nonsystematic Risk 414
13.2.3 Asset Pricing Implications: Performance Evaluation 416
13.2.4 Pricing Formulas 418
13.2.5 Empirical Tests 419
13.3 Alternative Derivation of the CAPM∗ 420
13.4 Continuous-Time, Intertemporal CAPM∗ 423
13.5 Consumption CAPM∗ 427
Summary 430
Problems 430
Further Readings 432
14 Multifactor Models 433
14.1 Discrete-Time Multifactor Models 433
14.2 Arbitrage Pricing Theory (APT) 436
14.3 Multifactor Models in Continuous Time∗ 438
14.3.1 Model Parameters and Variables 438
14.3.2 Value Function and Optimal Portfolio 439
14.3.3 Separation Theorem 441
14.3.4 Intertemporal Multifactor CAPM 442
Summary 445
Problems 445
Further Readings 445
xvi Contents
15 Other Pure Exchange Equilibria 447
15.1 Term-Structure Equilibria 447
15.1.1 Equilibrium Term Structure in Discrete Time 447
15.1.2 Equilibrium Term Structure in Continuous Time; CIR Model 449
15.2 Informational Equilibria 451
15.2.1 Discrete-Time Models with Incomplete Information 451
15.2.2 Continuous-Time Models with Incomplete Information 454
15.3 Equilibrium with Heterogeneous Agents 457
15.3.1 Discrete-Time Equilibrium with Heterogeneous Agents 458
15.3.2 Continuous-Time Equilibrium with Heterogeneous Agents 459
15.4 International Equilibrium; Equilibrium with Two Prices 461
15.4.1 Discrete-Time International Equilibrium 462
15.4.2 Continuous-Time International Equilibrium 463
Summary 466
Problems 466
Further Readings 467
16 Appendix: Probability Theory Essentials 469
16.1 Discrete Random Variables 469
16.1.1 Expectation and Variance 469
16.2 Continuous Random Variables 470
16.2.1 Expectation and Variance 470
16.3 Several Random Variables 471
16.3.1 Independence 471
16.3.2 Correlation and Covariance 472
16.4 Normal Random Variables 472
16.5 Properties of Conditional Expectations 474
16.6 Martingale Definition 476
16.7 Random Walk and Brownian Motion 476
References 479
Index 487

怎样可以有钱阿~~

想要下载这本书