Table of Contents

1 Products and Markets: Equities, Commodities, Exchange Rates, Forwards and Futures 1

1.1 Introduction 2

1.2 Equities 2

1.3 Commodities 9

1.4 Currencies 9

1.5 Indices 11

1.6 The time value of money 11

1.7 Fixed-income securities 17

1.8 Inflation-proof bonds 17

1.9 Forwards and futures 19

1.10 More about futures 22

1.11 Summary 24

2 Derivatives 27

2.1 Introduction 28

2.2 Options 28

2.3 Definition of common terms 33

2.4 Payoff diagrams 34

2.5 Writing options 39

2.6 Margin 39

2.7 Market conventions 39

2.8 The value of the option before expiry 40

2.9 Factors affecting derivative prices 41

2.10 Speculation and gearing 42

2.11 Early exercise 44

2.12 Put-call parity 44

2.13 Binaries or digitals 47

2.14 Bull and bear spreads 48

2.15 Straddles and strangles 50

2.16 Risk reversal 52

2.17 Butterflies and condors 53

2.18 Calendar spreads 53

2.19 LEAPS and FLEX 55

2.20 Warrants 55

2.21 Convertible bonds 55

2.22 Over the counter options 56

2.23 Summary 57

3 The Binomial Model 59

3.1 Introduction 60

3.2 Equities can go down as well as up 61

3.3 The option value 63

3.4 Which part of our ‘model’ didn’t we need? 65

3.5 Why should this ‘theoretical price’ be the ‘market price’? 65

3.6 How did I know to sell 1 2 of the stock for hedging? 66

3.7 How does this change if interest rates are non-zero? 67

3.8 Is the stock itself correctly priced? 68

3.9 Complete markets 69

3.10 The real and risk-neutral worlds 69

3.11 And now using symbols 73

3.12 An equation for the value of an option 75

3.13 Where did the probability p go? 77

3.14 Counter-intuitive? 77

3.15 The binomial tree 78

3.16 The asset price distribution 78

3.17 Valuing back down the tree 80

3.18 Programming the binomial method 85

3.19 The greeks 86

3.20 Early exercise 88

3.21 The continuous-time limit 90

3.22 Summary 90

4 The Random Behavior of Assets 95

4.1 Introduction 96

4.2 The popular forms of ‘analysis’ 96

4.3 Why we need a model for randomness: Jensen’s inequality 97

4.4 Similarities between equities, currencies, commodities and indices 99

4.5 Examining returns 100

4.6 Timescales 105

4.7 Estimating volatility 109

4.8 The random walk on a spreadsheet 109

4.9 The Wiener process 111

4.10 The widely accepted model for equities, currencies, commodities and indices 112

4.11 Summary 115

5 Elementary Stochastic Calculus 117

5.1 Introduction 118

5.2 A motivating example 118

5.3 The Markov property 120

5.4 The martingale property 120

5.5 Quadratic variation 120

5.6 Brownian motion 121

5.7 Stochastic integration 122

5.8 Stochastic differential equations 123

5.9 The mean square limit 124

5.10 Functions of stochastic variables and Itô’s lemma 124

5.11 Interpretation of Itô’s lemma 127

5.12 Itô and Taylor 127

5.13 Itô in higher dimensions 130

5.14 Some pertinent examples 130

5.15 Summary 136

6 The Black–Scholes Model 139

6.1 Introduction 140

6.2 A very special portfolio 140

6.3 Elimination of risk: delta hedging 142

6.4 No arbitrage 142

6.5 The Black–Scholes equation 143

6.6 The Black–Scholes assumptions 145

6.7 Final conditions 146

6.8 Options on dividend-paying equities 147

6.9 Currency options 147

6.10 Commodity options 148

6.11 Expectations and Black–Scholes 148

6.12 Some other ways of deriving the Black–Scholes equation 149

6.13 No arbitrage in the binomial, Black–Scholes and ‘other’ worlds 150

6.14 Forwards and futures 151

6.15 Futures contracts 152

6.16 Options on futures 153

6.17 Summary 153

7 Partial Differential Equations 157

7.1 Introduction 158

7.2 Putting the Black–Scholes equation into historical perspective 158

7.3 The meaning of the terms in the Black–Scholes equation 159

7.4 Boundary and initial/final conditions 159

7.5 Some solution methods 160

7.6 Similarity reductions 163

7.7 Other analytical techniques 163

7.8 Numerical solution 164

7.9 Summary 164

8 The Black–Scholes Formulæ and the ‘Greeks’ 169

8.1 Introduction 170

8.2 Derivation of the formulæ for calls, puts and simple digitals 170

8.3 Delta 182

8.4 Gamma 184

8.5 Theta 187

8.6 Speed 187

8.7 Vega 188

8.8 Rho 190

8.9 Implied volatility 191

8.10 A classification of hedging types 194

8.11 Summary 196

9 Overview of Volatility Modeling 203

9.1 Introduction 204

9.2 The different types of volatility 204

9.3 Volatility estimation by statistical means 205

9.4 Maximum likelihood estimation 211

9.5 Skews and smiles 215

9.6 Different approaches to modeling volatility 217

9.7 The choices of volatility models 221

9.8 Summary 221

10 How to Delta Hedge 225

10.1 Introduction 226

10.2 What if implied and actual volatilities are different? 227

10.3 Implied versus actual, delta hedging but using which volatility? 228

10.4 Case 1: Hedge with actual volatility, σ 228

10.5 Case 2: Hedge with implied volatility, ˜σ 231

10.6 Hedging with different volatilities 235

10.7 Pros and cons of hedging with each volatility 238

10.8 Portfolios when hedging with implied volatility 239

10.9 How does implied volatility behave? 241

10.10 Summary 245

11 An Introduction to Exotic and Path-dependent Options 247

11.1 Introduction 248

11.2 Option classification 248

11.3 Time dependence 249

11.4 Cashflows 250

11.5 Path dependence 252

11.6 Dimensionality 254

11.7 The order of an option 255

11.8 Embedded decisions 256

11.9 Classification tables 258

11.10 Examples of exotic options 258

11.11 Summary of math/coding consequences 266

11.12 Summary 267

12 Multi-asset Options 271

12.1 Introduction 272

12.2 Multidimensional lognormal random walks 272

12.3 Measuring correlations 274

12.4 Options on many underlyings 277

12.5 The pricing formula for European non-path-dependent options on dividend-paying assets 278

12.6 Exchanging one asset for another: a similarity solution 278

12.7 Two examples 280

12.8 Realities of pricing basket options 282

12.9 Realities of hedging basket options 283

12.10 Correlation versus cointegration 283

12.11 Summary 284

13 Barrier Options 287

13.1 Introduction 288

13.2 Different types of barrier options 288

13.3 Pricing methodologies 289

13.4 Pricing barriers in the partial differential equation framework 290

13.5 Examples 293

13.6 Other features in barrier-style options 300

13.7 Market practice: what volatility should I use? 302

13.9 Summary 307

14 Fixed-income Products and Analysis: Yield, Duration and Convexity 319

14.1 Introduction 320

14.2 Simple fixed-income contracts and features 320

14.3 International bond markets 324

14.4 Accrued interest 325

14.5 Day-count conventions 325

14.6 Continuously and discretely compounded interest 326

14.7 Measures of yield 327

14.8 The yield curve 329

14.9 Price/yield relationship 329

14.10 Duration 331

14.11 Convexity 333

14.12 An example 335

14.13 Hedging 335

14.14 Time-dependent interest rate 338

14.15 Discretely paid coupons 339

14.16 Forward rates and bootstrapping 339

14.17 Interpolation 344

14.18 Summary 346

15 Swaps 349

15.1 Introduction 350

15.2 The vanilla interest rate swap 350

15.3 Comparative advantage 351

15.4 The swap curve 353

15.5 Relationship between swaps and bonds 354

15.6 Bootstrapping 355

15.7 Other features of swaps contracts 356

15.8 Other types of swap 357

15.9 Summary 358

16 One-factor Interest Rate Modeling 359

16.1 Introduction 360

16.2 Stochastic interest rates 361

16.3 The bond pricing equation for the general model 362

16.4 What is the market price of risk? 365

16.5 Interpreting the market price of risk, and risk neutrality 366

16.6 Named models 366

16.7 Equity and FX forwards and futures when rates are stochastic 369

16.8 Futures contracts 370

16.9 Summary 372

17 Yield Curve Fitting 373

17.1 Introduction 374

17.2 Ho & Lee 374

17.3 The extended Vasicek model of Hull & White 375

17.4 Yield-curve fitting: For and against 376

17.5 Other models 380

17.6 Summary 380

18 Interest Rate Derivatives 383

18.1 Introduction 384

18.2 Callable bonds 384

18.3 Bond options 385

18.4 Caps and floors 389

18.5 Range notes 392

18.6 Swaptions, captions and floortions 392

18.7 Spread options 394

18.8 Index amortizing rate swaps 394

18.9 Contracts with embedded decisions 397

18.10 Some examples 398

18.11 More interest rate derivatives 400

18.12 Summary 401

19 The Heath, Jarrow & Morton and Brace, Gatarek & Musiela Models 403

19.1 Introduction 404

19.2 The forward rate equation 404

19.3 The spot rate process 404

19.4 The market price of risk 406

19.5 Real and risk neutral 407

19.6 Pricing derivatives 408

19.7 Simulations 408

19.8 Trees 410

19.9 The Musiela parameterization 411

19.10 Multi-factor HJM 411

19.11 Spreadsheet implementation 411

19.12 A simple one-factor example: Ho & Lee 412

19.13 Principal Component Analysis 413

19.14 Options on equities, etc. 416

19.15 Non-infinitesimal short rate 416

19.16 The Brace, Gatarek & Musiela model 417

19.17 Simulations 419

19.18 PVing the cashflows 419

19.19 Summary 420

20 Investment Lessons from Blackjack and Gambling 423

20.1 Introduction 424

20.2 The rules of blackjack 424

20.3 Beating the dealer 426

20.4 The distribution of profit in blackjack 428

20.5 The Kelly criterion 429

20.6 Can you win at roulette? 432

20.7 Horse race betting and no arbitrage 433

20.8 Arbitrage 434

20.9 How to bet 436

20.10 Summary 438

21 Portfolio Management 441

21.1 Introduction 442

21.2 Diversification 442

21.3 Modern portfolio theory 445

21.4 Where do I want to be on the efficient frontier? 447

21.5 Markowitz in practice 450

21.6 Capital Asset Pricing Model 451

21.7 The multi-index model 454

21.8 Cointegration 454

21.9 Performance measurement 455

21.10 Summary 456

22 Value at Risk 459

22.1 Introduction 460

22.2 Definition of Value at Risk 460

22.3 VaR for a single asset 461

22.4 VaR for a portfolio 463

22.5 VaR for derivatives 464

22.6 Simulations 466

22.7 Use of VaR as a performance measure 468

22.8 Introductory Extreme Value Theory 469

22.9 Coherence 470

22.10 Summary 470

23 Credit Risk 473

23.1 Introduction 474

23.2 The Merton model: equity as an option on a company’s assets 474

23.3 Risky bonds 475

23.4 Modeling the risk of default 476

23.5 The Poisson process and the instantaneous risk of default 477

23.6 Time-dependent intensity and the term structure of default 481

23.7 Stochastic risk of default 482

23.8 Positive recovery 484

23.9 Hedging the default 485

23.10 Credit rating 486

23.11 A model for change of credit rating 488

23.12 Copulas: pricing credit derivatives with many underlyings 488

23.13 Collateralized debt obligations 490

23.14 Summary 492

24 RiskMetrics and CreditMetrics 495

24.1 Introduction 496

24.2 The RiskMetrics datasets 496

24.3 Calculating the parameters the RiskMetrics way 496

24.4 The CreditMetrics dataset 498

24.5 The CreditMetrics methodology 501

24.6 A portfolio of risky bonds 501

24.7 CreditMetrics model outputs 502

24.8 Summary 502

25 CrashMetrics 505

25.1 Introduction 506

25.2 Why do banks go broke? 506

25.3 Market crashes 506

25.4 CrashMetrics 507

25.5 CrashMetrics for one stock 508

25.6 Portfolio optimization and the Platinum hedge 510

25.7 The multi-asset/single-index model 511

25.8 Portfolio optimization and the Platinum hedge in the multi-asset model 519

25.9 The multi-index model 520

25.10 Incorporating time value 521

25.11 Margin calls and margin hedging 522

25.12 Counterparty risk 524

25.13 Simple extensions to CrashMetrics 524

25.14 The CrashMetrics Index (CMI) 525

25.15 Summary 526

26 Derivatives **** Ups 527

26.1 Introduction 528

26.2 Orange County 528

26.3 Proctor and Gamble 529

26.4 Metallgesellschaft 532

26.5 Gibson Greetings 533

26.6 Barings 536

26.7 Long-Term Capital Management 537

26.8 Summary 540

27 Overview of Numerical Methods 541

27.1 Introduction 542

27.2 Finite-difference methods 542

27.3 Monte Carlo methods 544

27.4 Numerical integration 546

27.5 Summary 547

28 Finite-difference Methods for One-factor Models 549

28.1 Introduction 550

28.2 Grids 550

28.3 Differentiation using the grid 553

28.4 Approximating θ 553

28.5 Approximating 554

28.6 Approximating Ɣ 557

28.7 Example 557

28.8 Bilinear interpolation 558

28.9 Final conditions and payoffs 559

28.10 Boundary conditions 560

28.11 The explicit finite-difference method 562

28.12 The Code #1: European option 567

28.13 The Code #2: American exercise 571

28.14 The Code #3: 2-D output 573

28.15 Upwind differencing 575

28.16 Summary 578

29 Monte Carlo Simulation 581

29.1 Introduction 582

29.2 Relationship between derivative values and simulations: equities, indices, currencies, commodities 582

29.3 Generating paths 583

29.4 Lognormal underlying, no path dependency 584

29.5 Advantages of Monte Carlo simulation 585

29.6 Using random numbers 586

29.7 Generating Normal variables 587

29.8 Real versus risk neutral, speculation versus hedging 588

29.9 Interest rate products 590

29.10 Calculating the greeks 593

29.11 Higher dimensions: Cholesky factorization 594

29.12 Calculation time 596

29.13 Speeding up convergence 596

29.14 Pros and cons of Monte Carlo simulations 598

29.15 American options 598

29.16 Longstaff & Schwartz regression approach for American options 599

29.17 Basis functions 603

29.18 Summary 603

30 Numerical Integration 605

30.1 Introduction 606

30.2 Regular grid 606

30.3 Basic Monte Carlo integration 607

30.4 Low-discrepancy sequences 609

30.5 Advanced techniques 613

30.6 Summary 614

A All the Math You Need .and No More (An Executive Summary) 617

A. 1 Introduction 618

A. 2 e 618

A. 3 log 618

A. 4 Differentiation and Taylor series 620

A. 5 Differential equations 623

A. 6 Mean, standard deviation and distributions 623

A. 7 Summary 626

B Forecasting the Markets? A Small Digression 627

B. 1 Introduction 628

B. 2 Technical analysis 628

B. 3 Wave theory 637

B. 4 Other analytics 638

B. 5 Market microstructure modeling 640

B. 6 Crisis prediction 641

B. 7 Summary 641

C A Trading Game 643

C. 1 Introduction 643

C. 2 Aims 643

C. 3 Object of the game 643

C. 4 Rules of the game 643

C. 5 Notes 644

C. 6 How to fill in your trading sheet 645

D Contents of CD accompanying Paul Wilmott Introduces Quantitative Finance, second edition 649

E What you get if (when) you upgrade to PWOQF2 653

Bibliography 659

Index 683