Introduction to the Economics and Mathematics of Financial Markets
Jakˇsa Cvitani´c and Fernando Zapatero

2004

517 pages



因为字符超长， 目录后面的一些3级小目录被省去了


Contents
Preface
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


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


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
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


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

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

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

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

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


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


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

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


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

10 Bond Hedging 341
10.1 Duration 341
10.2Immunization 347

10.3 Convexity 351


11 Numerical Methods 355
11.1 Binomial Tree Methods 355
11.2 Monte Carlo Simulation 361
11.3 Numerical Solutions of PDEs; Finite-Difference Methods 373



III EQUILIBRIUM MODELS 381
12 Equilibrium Fundamentals 383
12.1 Concept of Equilibrium 383
12.2 Single-Agent and Multiagent Equilibrium 389
12.3 Pure Exchange Equilibrium 391
12.4 Existence of Equilibrium 398



12.4.2 Equilibrium Existence in Continuous Time 400
12.4.3 Determining Market Parameters in Equilibrium 403

13 CAPM 409
13.1 Basic CAPM 409

13.2 Economic Interpretations 413

13.3 Alternative Derivation of the CAPM∗ 420
13.4 Continuous-Time, Intertemporal CAPM∗ 423
13.5 Consumption CAPM∗ 427


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



15 Other Pure Exchange Equilibria 447
15.1 Term-Structure Equilibria 447
15.2 Informational Equilibria 451
15.3 Equilibrium with Heterogeneous Agents 457
15.4 International Equilibrium; Equilibrium with Two Prices 461


16 Appendix: Probability Theory Essentials 469
16.1 Discrete Random Variables 469

16.2 Continuous Random Variables 470
16.3 Several Random Variables 471
16.4 Normal Random Variables 472
16.5 Properties of Conditional Expectations 474
16.6 Martingale Definition 476