金融模型中的鞅方法 第2版 英文PDF电子书下载

PartⅠ Spot and Futures Markets 3

1 AnIntroduction to Financial Derivatives 3

1.1 Options 3

1.2 Futures Contracts and Options 5

1.3 Forward Contracts 6

1.4 Call and Put Spot Options 8

1.4.1 One-period Spot Market 9

1.4.2 Replicating Portfolios 10

1.4.3 Martingale Measure for a Spot Market 12

1.4.4 Absence of Arbitrage 13

1.4.5 Optimality of Replication 15

1.4.6 Change of a Numeraire 17

1.4.7 Put Option 18

1.5 Forward Contracts 19

1.5.1 Forward Price 20

1.6 Futures Call and Put Options 21

1.6.1 Futures Contracts and Futures Prices 21

1.6.2 One-period Futures Market 22

1.6.3 Martingale Measure for a Futures Market 23

1.6.4 Absence of Arbitrage 24

1.6.5 One-period Spot/Futures Market 26

1.7 Options of American Style 26

1.8 Universal No-arbitrage Inequalities 31

2 Discrete-time Security Markets 35

2.1 TheCox-Ross-Rubinstein Model 36

2.1.1 Binomial Lattice for the Stock Price 36

2.1.2 Recursive Pricing Procedure 38

2.1.3 CRR Option Pricing Formula 43

2.2 Martingale Properties of the CRR Model 46

2.2.1 Martingale Measures 47

2.2.2 Risk-neutral Valuation Formula 50

2.2.3 Change of a Numeraire 51

2.3 The Black-Scholes Option Pricing Formula 53

2.4 Valuation of American Options 58

2.4.1 American Call Options 58

2.4.2 American Put Options 60

2.4.3 American Claims 61

2.5 Options on a Dividend-paying Stock 63

2.6 Security Markets in Discrete Time 65

2.6.1 Finite Spot Markets 66

2.6.2 Self-financing Trading Strategies 66

2.6.3 Replication and Arbitrage Opportunities 68

2.6.4 Arbitrage Price 69

2.6.5 Risk-neutral Valuation Formula 70

2.6.6 Existence ofa Martingale Measure 73

2.6.7 Completeness of a Finite Market 75

2.6.8 Separating Hyperplane Theorem 77

2.6.9 Change of a Numeraire 78

2.6.10 Discrete-time Models with Infinite State Space 79

2.7 Finite Futures Markets 80

2.7.1 Self-financing Futures Strategies 81

2.7.2 Martingale Measures for a Futures Market 82

2.7.3 Risk-neutral Valuation Formula 84

2.7.4 Futures Prices Versus Forward Prices 85

2.8 American Contingent Claims 87

2.8.1 Optimal Stopping Problems 90

2.8.2 Valuation and Hedging of American Claims 97

2.8.3 American Call and Put 101

2.9 Game Contingent Claims 101

2.9.1 Dynkin Games 102

2.9.2 Valuation and Hedging ofGame Contingent Claims 108

3 Benchmark Models in Continuous Time 113

3.1 The Black-Scholes Model 114

3.1.1 Risk-free Bond 114

3.1.2 Stock Price 114

3.1.3 Self-financing Trading Strategies 118

3.1.4 Martingale Measure for the Black-Scholes Model 120

3.1.5 Black-Scholes Option Pricing Formula 125

3.1.6 Case of Time-dependent Coefficients 131

3.1.7 Merton's Model 132

3.1.8 Put-Call Parity for Spot Options 134

3.1.9 Black-Scholes PDE 134

3.1.10 A Riskless Portfolio Method 137

3.1.11 Black-Scholes Sensitivities 140

3.1.12 Market Imperfections 144

3.1.13 Numerical Methods 145

3.2 A Dividend-paying Stock 147

3.2.1 Case of a Constant Dividend Yield 148

3.2.2 Case of Known Dividends 151

3.3 Bachelier Model 154

3.3.1 Bachelier Option Pricing Formula 155

3.3.2 Bachelier's PDE 157

3.3.3 Bachelier Sensitivities 158

3.4 Black Model 159

3.4.1 Self-financing Futures Strategies 160

3.4.2 Martingale Measure for the Futures Market 160

3.4.3 Black's Futures Option Formula 161

3.4.4 Options on Forward Contracts 165

3.4.5 Forward and Futures Prices 167

3.5 Robustness of the Black-Scholes Approach 168

3.5.1 Uncertain Volatility 168

3.5.2 European Call and Put Options 169

3.5.3 Convex Path-independent European Claims 172

3.5.4 General Path-independent European Claims 177

4 Foreign Market Derivatives 181

4.1 Cross-currency Market Model 181

4.1.1 Domestic Martingale Measure 182

4.1.2 Foreign Martingale Measure 184

4.1.3 Foreign Stock Price Dynamics 185

4.2 Currency Forward Contracts and Options 186

4.2.1 Forward Exchange Rate 186

4.2.2 Currency Option Valuation Formula 187

4.3 Foreign Equity Forward Contracts 191

4.3.1 Forward Price of a Foreign Stock 191

4.3.2 Quanto Forward Contracts 192

4.4 Foreign Market Futures Contracts 194

4.5 Foreign Equity Options 197

4.5.1 Options Struck in a Foreign Currency 198

4.5.2 Options Struck in Domestic Currency 199

4.5.3 Quanto Options 200

4.5.4 Equity-linked Foreign Exchange Options 202

5 American Options 205

5.1 Valuation of American Claims 206

5.2 American Call and PutOptions 213

5.3 Early Exercise Representation of an American Put 216

5.4 Analytical Approach 219

5.5 Approximations ofthe American Put Price 222

5.6 Option on a Dividend-paying Stock 224

5.7 Game Contingent Claims 226

6 Exotic Options 229

6.1 Packages 230

6.2 Forward-start Options 231

6.3 Chooser Options 232

6.4 Compound Options 233

6.5 Digital Options 234

6.6 Barrier Options 235

6.7 Lookback Options 238

6.8 Asian Options 242

6.9 Basket Options 245

6.10 Quantile Options 249

6.11 Other Exotic Options 251

7 Volatility Risk 253

7.1 Implied Volatilities of Traded Options 254

7.1.1 Historical Volatility 255

7.1.2 Implied Volatility 255

7.1.3 Implied Volatility Versus Historical Volatility 256

7.1.4 Approximate Formulas 257

7.1.5 Implied Volatility Surface 259

7.1.6 Asymptotic Behavior of the Implied Volatility 261

7.1.7 Marked-to-Market Models 264

7.1.8 Vega Hedging 265

7.1.9 Correlated Brownian Motions 267

7.1.10 Forward-start Options 269

7.2 Extensions ofthe Black-Scholes Model 273

7.2.1 CEV Model 273

7.2.2 Shifted Lognormal Models 277

7.3 Local Volatility Models 278

7.3.1 Implied Risk-Neutral Probability Law 278

7.3.2 Local Volatility 281

7.3.3 Mixture Models 287

7.3.4 Advantages and Drawbacks of LV Models 290

7.4 Stochastic Volatility Models 291

7.4.1 PDE Approach 292

7.4.2 Examples of SV Models 293

7.4.3 Hull and White Model 294

7.4.4 Heston's Model 299

7.4.5 SABRModel 301

7.5 Dynamical Models of Volatility Surfaces 302

7.5.1 Dynamics ofthe Local Volatility Surface 303

7.5.2 Dynamics ofthe Implied Volatility Surface 303

7.6 Alternative Approaches 307

7.6.1 Modelling of Asset Returns 308

7.6.2 Modelling of Volatility and Realized Variance 313

8 Continuous-time Security Markets 315

8.1 Standard MarketModels 316

8.1.1 Standard Spot Market 316

8.1.2 Futures Market 325

8.1.3 Choice of a Numeraire 327

8.1.4 Existence of a Martingale Measure 330

8.1.5 Fundamental Theorem of Asset Pricing 332

8.2 Multidimensional Black-Scholes Model 333

8.2.1 Market Completeness 335

8.2.2 Variance-minimizing Hedging 337

8.2.3 Risk-minimizing Hedging 338

8.2.4 Market Imperfections 345

PartⅡ Fixed-income Markets 351

9 Interest Rates and Related Contracts 351

9.1 Zero-coupon Bonds 351

9.1.1 Term Structure of Interest Rates 352

9.1.2 Forward Interest Rates 353

9.1.3 Short-term Interest Rate 354

9.2 Coupon-bearing Bonds 354

9.2.1 Yield-to-Maturity 355

9.2.2 Market Conventions 357

9.3 Interest Rate Futures 358

9.3.1 Treasury Bond Futures 358

9.3.2 Bond Options 359

9.3.3 Treasury Bill Futures 360

9.3.4 Eurodollar Futures 362

9.4 Interest Rate Swaps 363

9.4.1 Forward Rate Agreements 364

9.5 Stochastic Models ofBond Prices 366

9.5.1 Arbitrage-free Family of Bond Prices 366

9.5.2 Expectations Hypotheses 367

9.5.3 Case of It？ Processes 368

9.5.4 Market Price for Interest Rate Risk 371

9.6 Forward Measure Approach 372

9.6.1 Forward Price 373

9.6.2 Forward Martingale Measure 375

9.6.3 Forward Processes 378

9.6.4 Choice of a Numeraire 379

10 Short-Term Rate Models 383

10.1 Single-factor Models 384

10.1.1 Time-homogeneous Models 384

10.1.2 Time-inhomogeneous Models 394

10.1.3 Model Choice 399

10.1.4 American Bond Options 401

10.1.5 Options on Coupon-beating Bonds 402

10.2 Multi-factor Models 402

10.2.1 State Variables 403

10.2.2 Affine Models 404

10.2.3 Yield Models 404

10.3 Extended CIR Model 406

10.3.1 Squared Bessel Process 407

10.3.2 Model Construction 407

10.3.3 Change ofa Probability Measure 408

10.3.4 Zero-coupon Bond 409

10.3.5 Case of Constant Coefficients 410

10.3.6 Case of Piecewise Constant Coefficients 411

10.3.7 Dynamics of Zero-coupon Bond 412

10.3.8 Transition Densities 414

10.3.9 Bond Option 415

11 Models of Instantaneous Forward Rates 417

11.1 Heath-Jarrow-Morton Methodology 418

11.1.1 Ho and Lee Model 419

11.1.2 Heath-Jarrow-Morton Model 419

11.1.3 Absence of Arbitrage 421

11.1.4 Short-term Interest Rate 427

11.2 Gaussian HJM Model 428

11.2.1 Markovian Case 430

11.3 European Spot Options 434

11.3.1 Bond Options 435

11.3.2 Stock Options 438

11.3.3 Option on a Coupon-bearing Bond 441

11.3.4 Pricing of General Contingent Claims 444

11.3.5 Replication of Options 446

11.4 Volatilities and Correlations 449

11.4.1 Volatilities 449

11.4.2 Correlations 451

11.5 Futures Price 452

11.5.1 Futures Options 453

11.6 PDE Approach to Interest Rate Derivatives 457

11.6.1 PDEs for Spot Derivatives 457

11.6.2 PDEs for Futures Derivatives 461

11.7 Recent Developments 465

12 Market LIBOR Models 469

12.1 Forward and Futures LIBORs 471

12.1.1 One-period Swap Settled in Arrears 471

12.1.2 One-period Swap Settled in Advance 473

12.1.3 Eurodollar Futures 474

12.1.4 LIBOR in the Gaussian HJM Model 475

12.2 Interest Rate Caps and Floors 477

12.3 Valuation in the Gaussian HJM Model 479

12.3.1 Plain-vanilla Caps and Floors 479

12.3.2 Exotic Caps 481

12.3.3 Captions 483

12.4 LIBOR Market Models 484

12.4.1 Black's Formula for Caps 484

12.4.2 Miltersen,Sandmann and Sondermann Approach 486

12.4.3 Brace,Gatarek and Musiela Approach 486

12.4.4 Musiela and Rutkowski Approach 489

12.4.5 SDEs for LIBORs under the Forward Measure 492

12.4.6 Jamshidian's Approach 495

12.4.7 Altemative Derivation of Jamshidian's SDE 498

12.5 Properties of the Lognormal LIBOR Model 500

12.5.1 Transition Density of the LIBOR 501

12.5.2 Transition Density of the Forward Bond Price 503

12.6 Valuation in the Lognormal LIBOR Model 506

12.6.1 Pricing of Caps and Floors 506

12.6.2 Hedging of Caps and Floors 508

12.6.3 Valuation of European Claims 510

12.6.4 Bond Options 513

12.7 Extensions of the LLM Model 515

13 Alternative Market Models 517

13.1 Swaps and Swaptions 518

13.1.1 Forward Swap Rates 518

13.1.2 Swaptions 522

13.1.3 Exotic Swap Derivatives 524

13.2 Valuarion in the Gaussian HJM Model 527

13.2.1 Swaptions 527

13.2.2 CMS Spread Options 527

13.2.3 Yield Curve Swaps 529

13.3 Co-terminal Forward Swap Rates 530

13.3.1 Jamshidian's Model 535

13.3.2 Valuation of Co-terminal Swaptions 538

13.3.3 Hedging of Swaptions 539

13.3.4 Bermudan Swaptions 540

13.4 Co.initial Forward Swap Rates 541

13.4.1 Valuation of Co-initial Swaptions 544

13.4.2 Valuation of Exotic Options 545

13.5 Co.sliding Forward Swap Rates 546

13.5.1 Modelling of Co-sliding Swap Rates 547

13.5.2 Valuation of Co-sliding Swaptions 551

13.6 Swap Rate Model Versus LIBOR Model 552

13.6.1 Swaptions in the LLM Model 553

13.6.2 Caplets in the Co-terminal Swap Market Model 557

13.7 Markov-functional Models 558

13.7.1 Terminal Swap Rate Model 559

13.7.2 Calibration of Markov-functional Models 562

13.8 Fiesaker and Hughston Approach 565

13.8.1 Rational Lognormal Model 568

13.8.2 Valuation of Caps and Swaptions 569

14 Cross-currency Derivatives 573

14.1 Arbitrage-free Cross-currency Markets 574

14.1.1 Forward Price of a Foreign Asset 576

14.1.2 Valuation of Foreign Contingent Claims 580

14.1.3 Cross-currency Rates 581

14.2 Gaussian Model 581

14.2.1 Currency Options 582

14.2.2 Foreign Equity Options 583

14.2.3 Cross-currency Swaps 588

14.2.4 Cross-currency Swaptions 599

14.2.5 Basket Caps 602

14.3 Model of Forward LIBOR Rates 603

14.3.1 Quamo Cap 604

14.3.2 Cross-currency Swap 606

14.4 Concluding Remarks 607

PartⅢ APPENDIX 611

A An Overview of It？ Stochastic Calculus 611

A.1 Conditional Expectation 611

A.2 Filtrations and Adapted Processes 615

A.3 Martingales 616

A.4 Standard Brownian Motion 617

A.5 Stopping Times and Martingales 621

A.6 It？ Stochastic Integral 622

A.7 Continuous Local Martingales 625

A.8 Continuous Semimartingales 628

A.9 It？'s Lemma 630

A.10 Lévy's Characterization Theorem 633

A.11 Martingale Representation Property 634

A.12 Stochastic Differential Equations 636

A.13 Stochastic Exponential 639

A.14 Radon-Nikodym Density 640

A.15 Girsanov's Theorem 641

A.16 Martingale Measures 645

A.17 Feynman-Kac Forrnula 646

A.18 First Passage Times 649

References 657

Index 707