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Ⅰ Introduction1

1 Foreign exchange markets3

1.1 Introduction3

1.2 Historical background4

1.3 Forex as an asset class7

1.4 Spot forex8

1.5 Derivatives:forwards,futures,calls,puts,and all that9

1.6 References and further reading17

Ⅱ Mathematical preliminaries19

2 Elements of probability theory21

2.1 Introduction21

2.2 Probability spaces22

2.3 Random variables26

2.4 Convergence of random variables and limit theorems38

2.5 References and further reading42

3 Discrete-time stochastic engines43

3.1 Introduction43

3.2 Time series44

3.3 Binomial stochastic engines for single- and multi-period markets46

3.4 Multinomial stochastic engines55

3.5 References and further reading57

4 Continuous-time stochastic engines59

4.1 Introduction59

4.2 Stochastic processes61

4.3 Markov processes63

4.4 Diffusions65

4.5 Wiener processes76

4.6 Poisson processes81

4.7 SDE and Mappings84

4.8 Linear SDEs91

4.9 SDEs for jump-diffusions96

4.10 Analytical solution of PDEs98

4.10.1 Introduction98

4.10.2 The reduction method98

4.10.3 The Laplace transform method103

4.10.4 The eigenfunction expansion method104

4.11 Numerical solution of PDEs106

4.11.1 Introduction106

4.11.2 Explicit,implicit,and Crank-Nicolson schemes for solving one-dimensional problems107

4.11.3 ADI scheme for solving two-dimensional problems109

4.12 Numerical solution of SDEs112

4.12.1 Introduction112

4.12.2 Formulation of the problem113

4.12.3 The Euler-Maruyama scheme113

4.12.4 The Milstein scheme114

4.13 References and further reading116

Ⅲ Discrete-time models119

5 Single-period markets121

5.1 Introduction121

5.2 Binomiai markets with nonrisky investments123

5.3 Binomial markets without nonrisky investments140

5.4 General single-period markets145

5.5 Economic constraints147

5.6 Pricing of contingent claims154

5.7 Elementary portfolio theory162

5.8 The optimal investment problem166

5.9 Elements of equilibrium theory168

5.10 References and further reading169

6 Multi-period markets171

6.1 Introduction171

6.2 Stationary binomial markets172

6.3 Non-stationary binomial markets194

6.3.1 Introduction194

6.3.2 The nonrecombining case195

6.3.3 The recombining case197

6.4 General multi-period markets202

6.5 Contingent claims and their valuation and hedging207

6.6 Portfolio theory208

6.7 The optimal investment problem210

6.8 References and further reading212

Ⅳ Continuous-time models213

7 Stochastic dynamics of forex215

7.1 Introduction215

7.2 Two-country markets with deterministic investments216

7.3 Two-country markets without deterministic investments223

7.4 Multi-country markets227

7.5 The nonlinear diffusion model230

7.6 The jump diffusion model232

7.7 The stochastic volatility model233

7.8 The general forex evolution model236

7.9 References and further reading237

8 European options:the group-theoretical approach239

8.1 Introduction239

8.2 The two-country homogeneous problem,Ⅰ240

8.2.1 Formulation of the problem240

8.2.2 Reductions of the pricing problem245

8.2.3 Continuous hedging and the Greeks248

8.3 Forwards,calls and puts250

8.3.1 Definitions250

8.3.2 Pricing via the Feynman-Kac formula250

8.3.3 A naive pricing attempt256

8.3.4 Pricing via the Fourier transform method257

8.3.5 Pricing via the Laplace transform method259

8.3.6 The limiting behavior of calls and puts261

8.4 Contingent claims with arbitrary payoffs263

8.4.1 Introduction263

8.4.2 The decomposition formula263

8.4.3 Call and put bets264

8.4.4 Log contracts and modified log contracts265

8.5 Dynamic asset allocation266

8.6 The two-country homogeneous problem,Ⅱ275

8.7 The multi-country homogeneous problem277

8.7.1 Introduction277

8.7.2 The homogeneous pricing problem278

8.7.3 Reductions279

8.7.4 Probabilistic pricing and hedging280

8.8 Some representative multi-factor options281

8.8.1 Introduction281

8.8.2 Outperformance options282

8.8.3 Options on the maximum or minimum of several FXRs284

8.8.4 Basket options286

8.8.5 Index options290

8.8.6 The multi-factor decomposition formula291

8.9 References and further reading292

9 European options,the classical approach293

9.1 Introduction293

9.2 The classical two-country pricing problem,Ⅰ294

9.2.1 The projection method294

9.2.2 The classical method296

9.2.3 The impact of the actual drift297

9.3 Solution of the classical pricing problem298

9.3.1 Nondimensionalization298

9.3.2 Reductions298

9.3.3 The pricing and hedging formulas for forwards,calls and puts299

9.3.4 European options with exotic payoffs306

9.4 The classical two-country pricing problem,Ⅱ310

9.5 The multi-country classical pricing problem315

9.5.1 Introduction315

9.5.2 Derivation315

9.5.3 Reductions315

9.5.4 Pricing and hedging of multi-factor options317

9.6 References and further reading317

10 Deviations from the Black-Scholes paradigm Ⅰ:nonconstant volatility319

10.1 Introduction319

10.2 Volatility term structures and smiles321

10.2.1 Introduction321

10.2.2 The implied volatility321

10.2.3 The local volatility323

10.2.4 The inverse problem325

10.2.5 How to deal with the smile329

10.3 Pricing via implied t.p.d.f.'s329

10.3.1 Implied t.p.d.f.'s and entropy maximization329

10.3.2 Possible functional forms of t.p.d.f.'s332

10.3.3 The chi-square pricing formula,Ⅰ335

10.3.4 The Edgeworth-type pricing formulas338

10.4 The sticky-strike and the sticky-delta models341

10.5 The general local volatility model344

10.5.1 Introduction344

10.5.2 Possible functional forms of local volatility345

10.5.3 The hyperbolic volatility model348

10.5.4 The displaced diffusion model350

10.6 Asymptotic treatment of the local volatility model353

10.7 The CEV model359

10.7.1 Introduction359

10.7.2 Reductions of the pricing problem360

10.7.3 Evaluation of the t.p.d.f362

10.7.4 Derivative pricing364

10.7.5 ATMF approximation368

10.8 The jump diffusion model371

10.8.1 Introduction371

10.8.2 The pricing problem371

10.8.3 Evaluation of the t.p.d.f372

10.8.4 Risk-neutral pricing373

10.9 The stochastic volatility model375

10.9.1 Introduction375

10.9.2 Basic equations376

10.9.3 Evaluation of the t.p.d.f379

10.9.4 The pricing formula384

10.9.5 The case of zero correlation386

10.10 Small volatility of volatility387

10.10.1 Introduction387

10.10.2 Basic equations388

10.10.3 The martingale formulation388

10.10.4 Perturbative expansion389

10.10.5 Summary of ODEs393

10.10.6 Solution of the leading order pricing problem394

10.10.7 The square-root model394

10.10.8 Computation of the implied volatility397

10.11 Multi-factor problems398

10.11.1 Introduction398

10.11.2 The chi-square pricing formula,Ⅱ399

10.12 References and further reading404

11 American Options405

11.1 Introduction405

11.2 General considerations407

11.2.1 The early exercise constraint407

11.2.2 The early exercise premium408

11.2.3 Some representative examples410

11.2.4 Rational bounds411

11.2.5 Parity and symmetry414

11.3 The risk-neutral valuation415

11.4 Alternative formulations of the valuation problem416

11.4.1 Introduction416

11.4.2 The inhomogeneous Black-Scholes problem formulation416

11.4.3 The linear complementarity formulation418

11.4.4 The linear program formulation420

11.5 Duality between puts and calls421

11.6 Application of Duhamel's principle422

11.6.1 The value of the early exercise premium422

11.6.2 The location of the early exercise boundary423

11.7 Asymptotic analysis of the pricing problem425

11.7.1 Short-dated options425

11.7.2 Long-dated and perpetual options430

11.8 Approximate solution of the valuation problem434

11.8.1 Introduction434

11.8.2 Bermudan approximation and extrapolation to the limit434

11.8.3 Quadratic approximation440

11.9 Numerical solution of the pricing problem442

11.9.1 Bermudan approximation442

11.9.2 Linear complementarity443

11.9.3 Integral equation444

11.9.4 Monte-Carlo valuation444

11.10 American options in a non-Black-Scholes framework445

11.11 Multi-factor American options445

11.11.1 Formulation445

11.11.2 Two representative examples446

11.12 References and further reading449

12 Path-dependent options Ⅰ:barrier options451

12.1 Introduction451

12.2 Single-factor,single-barrier options452

12.2.1 Introduction452

12.2.2 Pricing of single-barrier options via the method of images453

12.2.3 Pricing of single-barrier options via the method of heat potentials462

12.3 Static hedging469

12.4 Single-factor,double-barrier options472

12.4.1 Introduction472

12.4.2 Formulation473

12.4.3 The pricing problem without rebates474

12.4.4 Pricing of no-rebate calls and puts and double-no-touch options477

12.4.5 Pricing of calls and puts with rebate482

12.5 Deviations from the Black-Scholes paradigm484

12.5.1 Introduction484

12.5.2 Barrier options in the presence of the term structure of volatility484

12.5.3 Barrier options in the presence of constant elasticity of variance486

12.5.4 Barrier options in the presence of stochastic volatility492

12.6 Multi-factor barrier options498

12.7 Options on one currency with barriers on the other currency499

12.7.1 Introduction499

12.7.2 Formulation499

12.7.3 Solution via the Fourier method501

12.7.4 Solution via the method of images509

12.7.5 An alternative approach513

12.8 Options with one barrier for each currency514

12.8.1 General considerations514

12.8.2 The Green's function516

12.8.3 Two-factor,double-no-touch option520

12.9 Four-barrier options520

12.10 References and further reading526

13 Path-dependent options Ⅱ: lookback,Asian and other options527

13.1 Introduction527

13.2 Path-dependent options and augmented SDEs528

13.2.1 Description of path dependent options528

13.2.2 The augmentation procedure534

13.2.3 The pricing problem for augmented SDEs537

13.3 Risk-neutral valuation of path-dependent options538

13.4 Probabilistic pricing539

13.5 Lookback calls and puts542

13.5.1 Description542

13.5.2 Pricing via the method of images543

13.5.3 Similarity reductions547

13.5.4 Pricing via the Laplace transform548

13.5.5 Probabilistic pricing550

13.5.6 Barriers552

13.6 Asian options553

13.6.1 Description553

13.6.2 Geometric averaging553

13.6.3 Arithmetic averaging556

13.6.4 Exact solution via similarity reductions558

13.6.5 Pricing via the Laplace transform561

13.6.6 Approximate pricing of Asian calls revisited562

13.6.7 Discretely sampled Asian options565

13.7 Timer,fader and Parisian options566

13.7.1 Introduction566

13.7.2 Timer options566

13.7.3 Fader options573

13.7.4 Parisian options573

13.8 Standard passport options578

13.8.1 Description578

13.8.2 Similarity reductions and splitting579

13.8.3 Pricing via the Laplace transform581

13.8.4 Explicit solution for zero foreign interest rate582

13.9 More general passport options586

13.9.1 General considerations586

13.9.2 Explicit solution for zero foreign interest rate587

13.10 Variance and volatility swaps591

13.10.1 Introduction591

13.10.2 Description of swaps591

13.10.3 Pricing and hedging of swaps via convexity adjustments593

13.10.4 Log contracts and robust pricing and hedging of variance swaps599

13.11 The impact of stochastic volatility of path-dependent options602

13.11.1 The general valuation formula602

13.11.2 Evaluation of the t.p.d.f603

13.11.3 A transformed valuation formula608

13.12 Forward starting options(cliquets)608

13.13 Options on volatility611

13.13.1 The pricing problem611

13.13.2 Pricing of variance swaps613

13.13.3 Pricing of general swaps and swaptions614

13.14 References and further reading616

14 Deviations from the Black-Scholes paradigm Ⅱ:market frictions617

14.1 Introduction617

14.2 Imperfect hedging618

14.2.1 P＆L distributions618

14.2.2 Stop-loss start-gain hedging and local times621

14.2.3 Parameter misspecification622

14.3 The uncertain volatility model627

14.4 Transaction costs630

14.5 Liquidity risk633

14.6 Default risk635

14.6.1 Introduction635

14.6.2 The pricing model635

14.6.3 Pricing of defaultable European calls637

14.6.4 Pricing of defaultable forward contracts640

14.7 References and further reading643

15 Future directions of research and conclusions645

15.1 Introduction645

15.2 Future directions645

15.3 Conclusions646

15.4 References and further reading646

Bibliography647

Index669