Paul Wilmott Introduces Quantitative Finance doc

Paul Wilmott
Introduces
Quantitative Finance
Second Edition

Paul Wilmott
Introduces
Quantitative Finance
Second Edition
www.wilmott.com
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Library of Congress Cataloging-in-Publication Data
Wilmott, Paul.
Paul Wilmott introduces quantitative finance.—2nd ed.
p. cm.
ISBN 978-0-470-31958-1
1. Finance—Mathematical models. 2. Options (Finance)—Mathematical models. 3. Options (Finance)—Prices—
Mathematical models. I. Title. II Title: Quantitative finance.
HG173.W493 2007
332—dc22 2007015893
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN 978-0-470-31958-1 (PB)
Typeset in 10/12pt Helvetica by Laserwords Private Limited, Chennai, India
Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire
This book is printed on acid-free paper responsibly manufactured from sustainable forestry
in which at least two trees are planted for each one used for paper production.
To a rising Star

contents
Preface xxiii
1 Products and Markets: Equities, Commodities, Exchange Rates,
Forwards and Futures 1
1.1 Introduction 2
1.2 Equities 2
1.2.1 Dividends 7
1.2.2 Stock splits 8
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
1.9.1 A first example of no arbitrage 20
1.10 More about futures 22
1.10.1 Commodity futures 23
1.10.2 FX futures 23
1.10.3 Index futures 24
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
2.4.1 Other representations of value 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
viii contents
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
3.5.1 The role of expectations 66
3.6 How did I know to sell
1
2
of the stock for hedging? 66
3.6.1 The general formula for  67

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
3.10.1 Non-zero interest rates 72
3.11 And now using symbols 73
3.11.1 Average asset change 74
3.11.2 Standard deviation of asset price change 74
3.12 An equation for the value of an option 75
3.12.1 Hedging 75
3.12.2 No arbitrage 76
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
contents ix
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
4.6.1 The drift 107

4.6.2 The volatility 108
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 It
ˆ
o’s lemma 124
5.11 Interpretation of It
ˆ
o’s lemma 127
5.12 It
ˆ
o and Taylor 127
5.13 It
ˆ
o in higher dimensions 130
5.14 Some pertinent examples 130

5.14.1 Brownian motion with drift 131
5.14.2 The lognormal random walk 132
5.14.3 A mean-reverting random walk 134
5.14.4 And another mean-reverting random walk 135
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
x contents
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
6.12.1 The martingale approach 149
6.12.2 The binomial model 149
6.12.3 CAPM/utility 149
6.13 No arbitrage in the binomial, Black–Scholes and ‘other’ worlds 150
6.14 Forwards and futures 151
6.14.1 Forward contracts 151
6.15 Futures contracts 152
6.15.1 When interest rates are known, forward prices and futures
prices are the same 153
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
7.5.1 Transformation to constant coefficient diffusion equation 160
7.5.2 Green’s functions 161
7.5.3 Series solution 161
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
8.2.1 Formula for a call 175
8.2.2 Formula for a put 179
8.2.3 Formula for a binary call 181
8.2.4 Formula for a binary put 182
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
8.10.1 Why hedge? 194
contents xi

8.10.2 The two main classifications 194
8.10.3 Delta hedging 195
8.10.4 Gamma hedging 195
8.10.5 Vega hedging 195
8.10.6 Static hedging 195
8.10.7 Margin hedging 196
8.10.8 Crash (Platinum) hedging 196
8.11 Summary 196
9 Overview of Volatility Modeling 203
9.1 Introduction 204
9.2 The different types of volatility 204
9.2.1 Actual volatility 204
9.2.2 Historical or realized volatility 204
9.2.3 Implied volatility 205
9.2.4 Forward volatility 205
9.3 Volatility estimation by statistical means 205
9.3.1 The simplest volatility estimate: constant volatility/moving
window 205
9.3.2 Incorporating mean reversion 206
9.3.3 Exponentially weighted moving average 206
9.3.4 A simple GARCH model 207
9.3.5 Expected future volatility 207
9.3.6 Beyond close-close estimators: range-based estimation of
volatility 209
9.4 Maximum likelihood estimation 211
9.4.1 A simple motivating example: taxi numbers 211
9.4.2 Three hats 211
9.4.3 The math behind this: find the standard deviation 213
9.4.4 Quants’ salaries 214
9.5 Skews and smiles 215

9.5.1 Sensitivity of the straddle to skews and smiles 216
9.5.2 Sensitivity of the risk reversal to skews and smiles 216
9.6 Different approaches to modeling volatility 217
9.6.1 To calibrate or not? 217
9.6.2 Deterministic volatility surfaces 218
9.6.3 Stochastic volatility 219
9.6.4 Uncertain parameters 219
9.6.5 Static hedging 220
9.6.6 Stochastic volatility and mean-variance analysis 220
9.6.7 Asymptotic analysis of volatility 220
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
xii contents
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
10.5.1 The expected profit after hedging using implied volatility 232
10.5.2 The variance of profit after hedging using implied volatility 233
10.6 Hedging with different volatilities 235
10.6.1 Actual volatility = Implied volatility 236
10.6.2 Actual volatility > Implied volatility 236
10.6.3 Actual volatility < Implied volatility 238
10.7 Pros and cons of hedging with each volatility 238
10.7.1 Hedging with actual volatility 238
10.7.2 Hedging with implied volatility 239
10.7.3 Hedging with another volatility 239
10.8 Portfolios when hedging with implied volatility 239

10.8.1 Expectation 240
10.8.2 Variance 240
10.8.3 Portfolio optimization possibilities 241
10.9 How does implied volatility behave? 241
10.9.1 Sticky strike 242
10.9.2 Sticky delta 242
10.9.3 Time-periodic behavior 243
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
11.5.1 Strong path dependence 252
11.5.2 Weak path dependence 253
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
11.10.1 Compounds and choosers 258
11.10.2 Range notes 262
11.10.3 Barrier options 264
11.10.4 Asian options 265
11.10.5 Lookback options 265
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
contents xiii
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
12.8.1 Easy problems 283
12.8.2 Medium problems 283
12.8.3 Hard problems 283
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
13.3.1 Monte Carlo simulation 289
13.3.2 Partial differential equations 290
13.4 Pricing barriers in the partial differential equation framework 290
13.4.1 ‘Out’ barriers 291
13.4.2 ‘In’ barriers 292
13.5 Examples 293
13.5.1 Some more examples 294
13.6 Other features in barrier-style options 300
13.6.1 Early exercise 300
13.6.2 Repeated hitting of the barrier 301
13.6.3 Resetting of barrier 301

13.6.4 Outside barrier options 301
13.6.5 Soft barriers 301
13.6.6 Parisian options 301
13.7 Market practice: what volatility should I use? 302
13.8 Hedging barrier options 305
13.8.1 Slippage costs 306
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
14.2.1 The zero-coupon bond 320
14.2.2 The coupon-bearing bond 321
14.2.3 The money market account 321
14.2.4 Floating rate bonds 322
14.2.5 Forward rate agreements 323
14.2.6 Repos 323
14.2.7 STRIPS 324
xiv contents
14.2.8 Amortization 324
14.2.9 Call provision 324
14.3 International bond markets 324
14.3.1 United States of America 324
14.3.2 United Kingdom 325
14.3.3 Japan 325
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
14.7.1 Current yield 327
14.7.2 The yield to maturity (YTM) or internal rate of return (IRR) 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
14.16.1 Discrete data 340
14.16.2 On a spreadsheet 342
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
15.8.1 Basis rate swap 357
15.8.2 Equity swaps 357
15.8.3 Currency swaps 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
contents xv
16.6.1 Vasicek 366
16.6.2 Cox, Ingersoll & Ross 367
16.6.3 Ho & Lee 368
16.6.4 Hull & White 369
16.7 Equity and FX forwards and futures when rates are stochastic 369
16.7.1 Forward contracts 369
16.8 Futures contracts 370
16.8.1 The convexity adjustment 371
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
17.4.1 For 376
17.4.2 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
18.3.1 Market practice 387
18.4 Caps and floors 389
18.4.1 Cap/floor parity 389
18.4.2 The relationship between a caplet and a bond option 391

18.4.3 Market practice 391
18.4.4 Collars 391
18.4.5 Step-up swaps, caps and floors 392
18.5 Range notes 392
18.6 Swaptions, captions and floortions 392
18.6.1 Market practice 392
18.7 Spread options 394
18.8 Index amortizing rate swaps 394
18.8.1 Other features in the index amortizing rate swap 396
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
19.3.1 The non-Markov nature of HJM 406
xvi contents
19.4 The market price of risk 406
19.5 Real and risk neutral 407
19.5.1 The relationship between the risk-neutral forward rate drift
and volatility 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
19.13.1 The power method 415
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
20.3.1 Summary of winning at blackjack 427
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
20.7.1 Setting the odds in a sporting game 433
20.7.2 The mathematics 434
20.8 Arbitrage 434
20.8.1 How best to profit from the opportunity? 436
20.9 How to bet 436
20.10 Summary 438
21 Portfolio Management 441
21.1 Introduction 442
21.2 Diversification 442
21.2.1 Uncorrelated assets 443
21.3 Modern portfolio theory 445
21.3.1 Including a risk-free investment 447
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
21.6.1 The single-index model 451
21.6.2 Choosing the optimal portfolio 453
contents xvii
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
22.5.1 The delta approximation 464
22.5.2 The delta/gamma approximation 464
22.5.3 Use of valuation models 466
22.5.4 Fixed-income portfolios 466
22.6 Simulations 466
22.6.1 Monte Carlo 467
22.6.2 Bootstrapping 467
22.7 Use of VaR as a performance measure 468
22.8 Introductory Extreme Value Theory 469
22.8.1 Some EVT results 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
23.5.1 A note on hedging 481
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
23.12.1 The copula function 489
23.12.2 The mathematical definition 489
23.12.3 Examples of copulas 490
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
xviii contents
24.3 Calculating the parameters the RiskMetrics way 496
24.3.1 Estimating volatility 497
24.3.2 Correlation 498
24.4 The CreditMetrics dataset 498
24.4.1 Yield curves 499
24.4.2 Spreads 500
24.4.3 Transition matrices 500
24.4.4 Correlations 500
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
25.6.1 Other ‘cost’ functions 511
25.7 The multi-asset/single-index model 511
25.7.1 Assuming Taylor series for the moment 517
25.8 Portfolio optimization and the Platinum hedge in the multi-asset
model 519
25.8.1 The marginal effect of an asset 520
25.9 The multi-index model 520
25.10 Incorporating time value 521
25.11 Margin calls and margin hedging 522
25.11.1 What is margin? 522
25.11.2 Modeling margin 522
25.11.3 The single-index model 524
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

26.4.1 Basis risk 533
26.5 Gibson Greetings 533
26.6 Barings 536
26.7 Long-Term Capital Management 537
26.8 Summary 540
contents xix
27 Overview of Numerical Methods 541
27.1 Introduction 542
27.2 Finite-difference methods 542
27.2.1 Efficiency 543
27.2.2 Program of study 543
27.3 Monte Carlo methods 544
27.3.1 Efficiency 545
27.3.2 Program of study 545
27.4 Numerical integration 546
27.4.1 Efficiency 546
27.4.2 Program of study 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  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

28.11.1 The Black–Scholes equation 565
28.11.2 Convergence of the explicit method 565
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
29.7.1 Box–Muller 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
xx contents
29.12 Calculation time 596
29.13 Speeding up convergence 596
29.13.1 Antithetic variables 597
29.13.2 Control variate technique 597
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.2.1 Plotting 628
B.2.2 Support and resistance 628
B.2.3 Trendlines 629
B.2.4 Moving averages 629
B.2.5 Relative strength 632
B.2.6 Oscillators 632
B.2.7 Bollinger bands 632
B.2.8 Miscellaneous patterns 633
B.2.9 Japanese candlesticks 635
B.2.10 Point and figure charts 635
B.3 Wave theory 637

B.3.1 Elliott waves and Fibonacci numbers 638
B.3.2 Gann charts 638
B.4 Other analytics 638
B.5 Market microstructure modeling 640
B.5.1 Effect of demand on price 640
contents xxi
B.5.2 Combining market microstructure and option theory 641
B.5.3 Imitation 641
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
C.6.1 During a trading round 645
C.6.2 At the end of the game 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