外汇用的数学方法PDF电子书下载

Ⅰ Introduction 1

1 Foreign exchange markets 3

1.1 Introduction 3

1.2 Historical background 4

1.3 Forex as an asset class 7

1.4 Spot forex 8

1.5 Derivatives:forwards,futures,calls,puts,and all that 9

1.6 References and further reading 17

Ⅱ Mathematical preliminaries 19

2 Elements of probability theory 21

2.1 Introduction 21

2.2 Probability spaces 22

2.3 Random variables 26

2.4 Convergence of random variables and limit theorems 38

2.5 References and further reading 42

3 Discrete-time stochastic engines 43

3.1 Introduction 43

3.2 Time series 44

3.3 Binomial stochastic engines for single- and multi-period markets 46

3.4 Multinomial stochastic engines 55

3.5 References and further reading 57

4 Continuous-time stochastic engines 59

4.1 Introduction 59

4.2 Stochastic processes 61

4.3 Markov processes 63

4.4 Diffusions 65

4.5 Wiener processes 76

4.6 Poisson processes 81

4.7 SDE and Mappings 84

4.8 Linear SDEs 91

4.9 SDEs for jump-diffusions 96

4.10 Analytical solution of PDEs 98

4.10.1 Introduction 98

4.10.2 The reduction method 98

4.10.3 The Laplace transform method 103

4.10.4 The eigenfunction expansion method 104

4.11 Numerical solution of PDEs 106

4.11.1 Introduction 106

4.11.2 Explicit,implicit,and Crank-Nicolson schemes for solving one-dimensional problems 107

4.11.3 ADI scheme for solving two-dimensional problems 109

4.12 Numerical solution of SDEs 112

4.12.1 Introduction 112

4.12.2 Formulation of the problem 113

4.12.3 The Euler-Maruyama scheme 113

4.12.4 The Milstein scheme 114

4.13 References and further reading 116

Ⅲ Discrete-time models 119

5 Single-period markets 121

5.1 Introduction 121

5.2 Binomiai markets with nonrisky investments 123

5.3 Binomial markets without nonrisky investments 140

5.4 General single-period markets 145

5.5 Economic constraints 147

5.6 Pricing of contingent claims 154

5.7 Elementary portfolio theory 162

5.8 The optimal investment problem 166

5.9 Elements of equilibrium theory 168

5.10 References and further reading 169

6 Multi-period markets 171

6.1 Introduction 171

6.2 Stationary binomial markets 172

6.3 Non-stationary binomial markets 194

6.3.1 Introduction 194

6.3.2 The nonrecombining case 195

6.3.3 The recombining case 197

6.4 General multi-period markets 202

6.5 Contingent claims and their valuation and hedging 207

6.6 Portfolio theory 208

6.7 The optimal investment problem 210

6.8 References and further reading 212

Ⅳ Continuous-time models 213

7 Stochastic dynamics of forex 215

7.1 Introduction 215

7.2 Two-country markets with deterministic investments 216

7.3 Two-country markets without deterministic investments 223

7.4 Multi-country markets 227

7.5 The nonlinear diffusion model 230

7.6 The jump diffusion model 232

7.7 The stochastic volatility model 233

7.8 The general forex evolution model 236

7.9 References and further reading 237

8 European options:the group-theoretical approach 239

8.1 Introduction 239

8.2 The two-country homogeneous problem,Ⅰ 240

8.2.1 Formulation of the problem 240

8.2.2 Reductions of the pricing problem 245

8.2.3 Continuous hedging and the Greeks 248

8.3 Forwards,calls and puts 250

8.3.1 Definitions 250

8.3.2 Pricing via the Feynman-Kac formula 250

8.3.3 A naive pricing attempt 256

8.3.4 Pricing via the Fourier transform method 257

8.3.5 Pricing via the Laplace transform method 259

8.3.6 The limiting behavior of calls and puts 261

8.4 Contingent claims with arbitrary payoffs 263

8.4.1 Introduction 263

8.4.2 The decomposition formula 263

8.4.3 Call and put bets 264

8.4.4 Log contracts and modified log contracts 265

8.5 Dynamic asset allocation 266

8.6 The two-country homogeneous problem,Ⅱ 275

8.7 The multi-country homogeneous problem 277

8.7.1 Introduction 277

8.7.2 The homogeneous pricing problem 278

8.7.3 Reductions 279

8.7.4 Probabilistic pricing and hedging 280

8.8 Some representative multi-factor options 281

8.8.1 Introduction 281

8.8.2 Outperformance options 282

8.8.3 Options on the maximum or minimum of several FXRs 284

8.8.4 Basket options 286

8.8.5 Index options 290

8.8.6 The multi-factor decomposition formula 291

8.9 References and further reading 292

9 European options,the classical approach 293

9.1 Introduction 293

9.2 The classical two-country pricing problem,Ⅰ 294

9.2.1 The projection method 294

9.2.2 The classical method 296

9.2.3 The impact of the actual drift 297

9.3 Solution of the classical pricing problem 298

9.3.1 Nondimensionalization 298

9.3.2 Reductions 298

9.3.3 The pricing and hedging formulas for forwards,calls and puts 299

9.3.4 European options with exotic payoffs 306

9.4 The classical two-country pricing problem,Ⅱ 310

9.5 The multi-country classical pricing problem 315

9.5.1 Introduction 315

9.5.2 Derivation 315

9.5.3 Reductions 315

9.5.4 Pricing and hedging of multi-factor options 317

9.6 References and further reading 317

10 Deviations from the Black-Scholes paradigm Ⅰ:nonconstant volatility 319

10.1 Introduction 319

10.2 Volatility term structures and smiles 321

10.2.1 Introduction 321

10.2.2 The implied volatility 321

10.2.3 The local volatility 323

10.2.4 The inverse problem 325

10.2.5 How to deal with the smile 329

10.3 Pricing via implied t.p.d.f.'s 329

10.3.1 Implied t.p.d.f.'s and entropy maximization 329

10.3.2 Possible functional forms of t.p.d.f.'s 332

10.3.3 The chi-square pricing formula,Ⅰ 335

10.3.4 The Edgeworth-type pricing formulas 338

10.4 The sticky-strike and the sticky-delta models 341

10.5 The general local volatility model 344

10.5.1 Introduction 344

10.5.2 Possible functional forms of local volatility 345

10.5.3 The hyperbolic volatility model 348

10.5.4 The displaced diffusion model 350

10.6 Asymptotic treatment of the local volatility model 353

10.7 The CEV model 359

10.7.1 Introduction 359

10.7.2 Reductions of the pricing problem 360

10.7.3 Evaluation of the t.p.d.f 362

10.7.4 Derivative pricing 364

10.7.5 ATMF approximation 368

10.8 The jump diffusion model 371

10.8.1 Introduction 371

10.8.2 The pricing problem 371

10.8.3 Evaluation of the t.p.d.f 372

10.8.4 Risk-neutral pricing 373

10.9 The stochastic volatility model 375

10.9.1 Introduction 375

10.9.2 Basic equations 376

10.9.3 Evaluation of the t.p.d.f 379

10.9.4 The pricing formula 384

10.9.5 The case of zero correlation 386

10.10 Small volatility of volatility 387

10.10.1 Introduction 387

10.10.2 Basic equations 388

10.10.3 The martingale formulation 388

10.10.4 Perturbative expansion 389

10.10.5 Summary of ODEs 393

10.10.6 Solution of the leading order pricing problem 394

10.10.7 The square-root model 394

10.10.8 Computation of the implied volatility 397

10.11 Multi-factor problems 398

10.11.1 Introduction 398

10.11.2 The chi-square pricing formula,Ⅱ 399

10.12 References and further reading 404

11 American Options 405

11.1 Introduction 405

11.2 General considerations 407

11.2.1 The early exercise constraint 407

11.2.2 The early exercise premium 408

11.2.3 Some representative examples 410

11.2.4 Rational bounds 411

11.2.5 Parity and symmetry 414

11.3 The risk-neutral valuation 415

11.4 Alternative formulations of the valuation problem 416

11.4.1 Introduction 416

11.4.2 The inhomogeneous Black-Scholes problem formulation 416

11.4.3 The linear complementarity formulation 418

11.4.4 The linear program formulation 420

11.5 Duality between puts and calls 421

11.6 Application of Duhamel's principle 422

11.6.1 The value of the early exercise premium 422

11.6.2 The location of the early exercise boundary 423

11.7 Asymptotic analysis of the pricing problem 425

11.7.1 Short-dated options 425

11.7.2 Long-dated and perpetual options 430

11.8 Approximate solution of the valuation problem 434

11.8.1 Introduction 434

11.8.2 Bermudan approximation and extrapolation to the limit 434

11.8.3 Quadratic approximation 440

11.9 Numerical solution of the pricing problem 442

11.9.1 Bermudan approximation 442

11.9.2 Linear complementarity 443

11.9.3 Integral equation 444

11.9.4 Monte-Carlo valuation 444

11.10 American options in a non-Black-Scholes framework 445

11.11 Multi-factor American options 445

11.11.1 Formulation 445

11.11.2 Two representative examples 446

11.12 References and further reading 449

12 Path-dependent options Ⅰ:barrier options 451

12.1 Introduction 451

12.2 Single-factor,single-barrier options 452

12.2.1 Introduction 452

12.2.2 Pricing of single-barrier options via the method of images 453

12.2.3 Pricing of single-barrier options via the method of heat potentials 462

12.3 Static hedging 469

12.4 Single-factor,double-barrier options 472

12.4.1 Introduction 472

12.4.2 Formulation 473

12.4.3 The pricing problem without rebates 474

12.4.4 Pricing of no-rebate calls and puts and double-no-touch options 477

12.4.5 Pricing of calls and puts with rebate 482

12.5 Deviations from the Black-Scholes paradigm 484

12.5.1 Introduction 484

12.5.2 Barrier options in the presence of the term structure of volatility 484

12.5.3 Barrier options in the presence of constant elasticity of variance 486

12.5.4 Barrier options in the presence of stochastic volatility 492

12.6 Multi-factor barrier options 498

12.7 Options on one currency with barriers on the other currency 499

12.7.1 Introduction 499

12.7.2 Formulation 499

12.7.3 Solution via the Fourier method 501

12.7.4 Solution via the method of images 509

12.7.5 An alternative approach 513

12.8 Options with one barrier for each currency 514

12.8.1 General considerations 514

12.8.2 The Green's function 516

12.8.3 Two-factor,double-no-touch option 520

12.9 Four-barrier options 520

12.10 References and further reading 526

13 Path-dependent options Ⅱ: lookback,Asian and other options 527

13.1 Introduction 527

13.2 Path-dependent options and augmented SDEs 528

13.2.1 Description of path dependent options 528

13.2.2 The augmentation procedure 534

13.2.3 The pricing problem for augmented SDEs 537

13.3 Risk-neutral valuation of path-dependent options 538

13.4 Probabilistic pricing 539

13.5 Lookback calls and puts 542

13.5.1 Description 542

13.5.2 Pricing via the method of images 543

13.5.3 Similarity reductions 547

13.5.4 Pricing via the Laplace transform 548

13.5.5 Probabilistic pricing 550

13.5.6 Barriers 552

13.6 Asian options 553

13.6.1 Description 553

13.6.2 Geometric averaging 553

13.6.3 Arithmetic averaging 556

13.6.4 Exact solution via similarity reductions 558

13.6.5 Pricing via the Laplace transform 561

13.6.6 Approximate pricing of Asian calls revisited 562

13.6.7 Discretely sampled Asian options 565

13.7 Timer,fader and Parisian options 566

13.7.1 Introduction 566

13.7.2 Timer options 566

13.7.3 Fader options 573

13.7.4 Parisian options 573

13.8 Standard passport options 578

13.8.1 Description 578

13.8.2 Similarity reductions and splitting 579

13.8.3 Pricing via the Laplace transform 581

13.8.4 Explicit solution for zero foreign interest rate 582

13.9 More general passport options 586

13.9.1 General considerations 586

13.9.2 Explicit solution for zero foreign interest rate 587

13.10 Variance and volatility swaps 591

13.10.1 Introduction 591

13.10.2 Description of swaps 591

13.10.3 Pricing and hedging of swaps via convexity adjustments 593

13.10.4 Log contracts and robust pricing and hedging of variance swaps 599

13.11 The impact of stochastic volatility of path-dependent options 602

13.11.1 The general valuation formula 602

13.11.2 Evaluation of the t.p.d.f 603

13.11.3 A transformed valuation formula 608

13.12 Forward starting options(cliquets) 608

13.13 Options on volatility 611

13.13.1 The pricing problem 611

13.13.2 Pricing of variance swaps 613

13.13.3 Pricing of general swaps and swaptions 614

13.14 References and further reading 616

14 Deviations from the Black-Scholes paradigm Ⅱ:market frictions 617

14.1 Introduction 617

14.2 Imperfect hedging 618

14.2.1 P＆L distributions 618

14.2.2 Stop-loss start-gain hedging and local times 621

14.2.3 Parameter misspecification 622

14.3 The uncertain volatility model 627

14.4 Transaction costs 630

14.5 Liquidity risk 633

14.6 Default risk 635

14.6.1 Introduction 635

14.6.2 The pricing model 635

14.6.3 Pricing of defaultable European calls 637

14.6.4 Pricing of defaultable forward contracts 640

14.7 References and further reading 643

15 Future directions of research and conclusions 645

15.1 Introduction 645

15.2 Future directions 645

15.3 Conclusions 646

15.4 References and further reading 646

Bibliography 647

Index 669