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