Paul Wilmott on Quantitative Finance

Paul Wilmott On Quantitative Finance......Page 4
contents of volume one......Page 10
contents of volume two......Page 12
contents of volume three......Page 14
Visual Basic Code......Page 16
Prolog to the Second Edition......Page 18
PART ONE MATHEMATICAL AND FINANCIAL FOUNDATIONS; BASIC THEORY OF DERIVATIVES; RISK AND RETURN......Page 22
1.2 The time value of money......Page 26
1.3 Equities......Page 28
1.3.1 Dividends......Page 33
1.3.2 Stock splits......Page 34
1.4 Commodities......Page 35
1.5 Currencies......Page 36
1.7 Fixed-income securities......Page 38
1.8 Inflation-proof bonds......Page 40
1.9 Forwards and futures......Page 42
1.9.1 A first example of no arbitrage......Page 43
1.10 Summary......Page 45
2.2 Options......Page 46
2.3 Definition of common terms......Page 52
2.4 Payoff diagrams......Page 54
2.4.1 Other representations of value......Page 57
2.7 Market conventions......Page 58
2.9 Factors affecting derivatives prices......Page 59
2.10 Speculation and gearing......Page 61
2.12 Put-call parity......Page 62
2.13 Binaries or digitals......Page 64
2.14 Bull and bear spreads......Page 65
2.15 Straddles and strangles......Page 67
2.16 Risk reversal......Page 68
2.18 Calendar spreads......Page 70
2.22 Over the counter options......Page 72
2.23 Summary......Page 74
3.2 The popular forms of ‘analysis’......Page 76
3.3 Why we need a model for randomness: Jensen’s Inequality......Page 77
3.5 Examining returns......Page 79
3.6 Timescales......Page 83
3.7 Estimating volatility......Page 86
3.9 The Wiener process......Page 88
3.11 Summary......Page 90
4.2 A motivating example......Page 92
4.5 Quadratic variation......Page 94
4.6 Brownian motion......Page 95
4.7 Stochastic integration......Page 96
4.8 Stochastic differential equations......Page 97
4.9 The mean square limit......Page 98
4.10 Functions of stochastic variables and Ito’s lemma......Page 99
4.11 Interpretation of Ito’s lemma......Page 101
4.12 Ito and Taylor......Page 102
4.12.1 The intuition......Page 103
4.13 Ito in higher dimensions......Page 104
4.14.1 Brownian motion with drift......Page 105
4.14.2 The lognormal random walk......Page 106
4.14.3 A mean-reverting random walk......Page 107
4.14.4 And another mean-reverting random walk......Page 109
4.15 Summary......Page 111
5.2 A very special portfolio......Page 112
5.4 No arbitrage......Page 114
5.5 The Black–Scholes equation......Page 115
5.6 The Black–Scholes assumptions......Page 116
5.7 Final conditions......Page 117
5.9 Currency options......Page 118
5.12.2 The binomial model......Page 119
5.13 Summary......Page 120
6.2 Putting the Black–Scholes equation into historical perspective......Page 122
6.3 The meaning of the terms in the Black–Scholes equation......Page 123
6.5.1 Transformation to constant coefficient diffusion equation......Page 125
6.5.3 Series solution......Page 126
6.6 Similarity reductions......Page 127
6.7 Other analytical techniques......Page 128
6.9 Summary......Page 129
7.1 Introduction......Page 130
7.2 Derivation of the formulae for calls, puts and simple digitals......Page 131
7.2.1 Formula for a call......Page 135
7.2.2 Formula for a put......Page 139
7.2.3 Formula for a binary call......Page 140
7.2.4 Formula for a binary put......Page 141
7.3 Delta......Page 142
7.4 Gamma......Page 144
7.5 Theta......Page 145
7.6 Speed......Page 147
7.7 Vega......Page 148
7.8 Rho......Page 150
7.9 Implied volatility......Page 151
7.10.4 Gamma hedging......Page 155
7.10.7 Margin hedging......Page 156
7.11 Summary......Page 157
8.2 Dividends, foreign interest and cost of carry......Page 160
8.4 Dividend payments and no arbitrage......Page 161
8.5 The behavior of an option value across a dividend date......Page 162
8.6 Commodities......Page 164
8.6.3 Convenience yield......Page 165
8.7 Stock borrowing and repo......Page 166
8.8 Time-dependent parameters......Page 168
8.10 The log contract......Page 170
8.11 Summary......Page 171
9.2 The perpetual American put......Page 172
9.3 Perpetual American call with dividends......Page 176
9.4 Mathematical formulation for general payoff......Page 177
9.5 Local solution for call with constant dividend yield......Page 180
9.6 Other dividend structures......Page 181
9.7 One-touch options......Page 182
9.8 Other features in American-style contracts......Page 183
9.8.2 Make your mind up......Page 184
9.9.2 Free-boundary problems......Page 186
9.9.3 Numerical solution......Page 188
9.10 Summary......Page 189
10.2 The transition probability density function......Page 190
10.3 A trinomial model for the random walk......Page 191
10.4 The forward equation......Page 192
10.5 The steady-state distribution......Page 194
10.6 The backward equation......Page 195
10.7 First-exit times......Page 196
10.8 Cumulative distribution functions for first-exit times......Page 197
10.9 Expected first-exit times......Page 198
10.10 Another example of optimal stopping......Page 199
10.11 Expectations and Black–Scholes......Page 200
10.13 Summary......Page 202
11.2 Multi-dimensional lognormal random walks......Page 204
11.3 Measuring correlations......Page 205
11.4 Options on many underlyings......Page 207
11.5 The pricing formula for European non-path-dependent options on dividend-paying assets......Page 208
11.6 Exchanging one asset for another: A similarity solution......Page 209
11.7 Quantos......Page 210
11.9 Other features......Page 212
11.11 Realities of hedging basket options......Page 215
11.13 Summary......Page 216
12.1 Introduction......Page 218
12.2 What if implied and actual volatilities are different?......Page 219
12.3 Implied versus actual; delta hedging but using which volatility?......Page 220
12.4 Case 1: Hedge with actual volatility, σ......Page 221
12.5 Case 2: Hedge with implied volatility,  σ......Page 223
12.5.1 The expected profit after hedging using implied volatility......Page 224
12.5.2 The variance of profit after hedging using implied volatility......Page 227
12.5.3 Hedging with different volatilities......Page 231
12.6.1 Expectation......Page 233
12.6.3 Portfolio optimization possibilities......Page 234
12.7.2 Case 2: Hedge with implied volatility,  σ?......Page 236
12.8.2 Sticky delta......Page 237
12.8.3 Time-periodic behavior......Page 238
12.9 Summary......Page 240
13.2.1 The zero-coupon bond......Page 246
13.2.2 The coupon-bearing bond......Page 247
13.2.5 Forward rate agreements......Page 249
13.3.1 United States of America......Page 250
13.5 Day-count conventions......Page 251
13.7.1 Current yield......Page 252
13.7.2 The yield to maturity (YTM) or internal rate of return (IRR)......Page 253
13.10 Duration......Page 254
13.11 Convexity......Page 258
13.13 Hedging......Page 260
13.14 Time-dependent interest rate......Page 262
13.16 Forward rates and bootstrapping......Page 263
13.16.1 Discrete data......Page 264
13.16.2 On a spreadsheet......Page 267
13.18 Summary......Page 270
14.2 The vanilla interest rate swap......Page 272
14.3 Comparative advantage......Page 274
14.5 Relationship between swaps and bonds......Page 275
14.7 Other features of swaps contracts......Page 278
14.8.1 Basis rate swap......Page 279
14.9 Summary......Page 280
15.1 Introduction......Page 282
15.2 Equities can go down as well as up......Page 283
15.3 The option value......Page 285
15.4 Which part of our ‘model’ didn’t we need?......Page 287
15.6 How did I know to sell 12 of the stock for hedging?......Page 288
15.6.1 The general formula for......Page 289
15.8 Is the stock itself correctly priced?......Page 290
15.9 Complete markets......Page 291
15.10 The real and risk-neutral worlds......Page 293
15.11 And now using symbols......Page 295
15.11.1 Average asset change......Page 296
15.12.1 Hedging......Page 297
15.12.2 No arbitrage......Page 298
15.13 Where did the probability p go?......Page 299
15.15 The binomial tree......Page 300
15.17 Valuing back down the tree......Page 301
15.18 Programming the binomial method......Page 307
15.19 The greeks......Page 308
15.20 Early exercise......Page 310
15.22 No arbitrage in the binomial, Black–Scholes and ‘other’ worlds......Page 312
15.23 Summary......Page 313
16.2 Why we like the Normal distribution: the Central Limit Theorem......Page 316
16.3 Normal versus lognormal......Page 317
16.4 Does my tail look fat in this?......Page 318
16.5 Use a different distribution, perhaps......Page 319
16.7 Summary......Page 320
17.1 Introduction......Page 322
17.2 The rules of blackjack......Page 323
17.3 Beating the dealer......Page 324
17.3.1 Summary of winning at blackjack......Page 325
17.4 The distribution of profit in blackjack......Page 326
17.5 The Kelly criterion......Page 327
17.6 Can you win at roulette?......Page 330
17.7.2 The mathematics......Page 331
17.8 Arbitrage......Page 332
17.8.1 How best to profit from the opportunity?......Page 333
17.9 How to bet......Page 334
17.10 Summary......Page 336
18.1 Introduction......Page 338
18.2 Diversification......Page 339
18.3 Modern Portfolio Theory......Page 340
18.3.1 Including a risk-free investment......Page 343
18.4 Where do I want to be on the efficient frontier?......Page 344
18.6.1 The single-index model......Page 346
18.7 The multi-index model......Page 348
18.8 Cointegration......Page 349
18.9 Performance measurement......Page 350
18.10 Summary......Page 351
19.2 Definition of value at risk......Page 352
19.3 VaR for a single asset......Page 353
19.4 VaR for a portfolio......Page 355
19.5.2 Which volatility do I use?......Page 356
19.5.3 The delta-gamma approximation......Page 357
19.5.5 Fixed-income portfolios......Page 358
19.6.2 Bootstrapping......Page 359
19.8 Introductory extreme value theory......Page 360
19.9 Coherence......Page 362
19.10 Summary......Page 363
20.2 Technical analysis......Page 364
20.2.1 Plotting......Page 365
20.2.6 Oscillators......Page 366
20.2.8 Miscellaneous patterns......Page 369
20.2.9 Japanese candlesticks......Page 370
20.3 Wave theory......Page 374
20.4 Other analytics......Page 376
20.5.1 Effect of demand on price......Page 377
20.7 Summary......Page 378
21.4 Rules of the game......Page 380
21.6.1 During a trading round......Page 382
21.6.2 At the end of the game......Page 383
PART TWO EXOTIC CONTRACTS AND PATH DEPENDENCY......Page 386
22.1 Introduction......Page 388
22.3 Time dependence......Page 389
22.4 Cashflows......Page 390
22.5.2 Weak path dependence......Page 392
22.6 Dimensionality......Page 393
22.7 The order of an option......Page 394
22.8 Embedded decisions......Page 395
22.10.1 Compounds and choosers......Page 396
22.10.2 Range notes......Page 400
22.10.4 Asian options......Page 402
22.11 Summary of math/coding consequences......Page 404
22.12 Summary......Page 405
23.1 Introduction......Page 406
23.2 Different types of barrier options......Page 407
23.3.1 Monte Carlo simulation......Page 408
23.4.1 ‘Out’ barriers......Page 409
23.4.3 Some formulae when volatility is constant......Page 410
23.4.4 Some more examples......Page 414
23.5.1 Early exercise......Page 418
23.5.2 The intermittent barrier......Page 419
23.5.4 Resetting of barrier......Page 420
23.6 First-exit time......Page 421
23.7 Market practice: What volatility should I use?......Page 422
23.8 Hedging barrier options......Page 426
23.9 Slippage costs......Page 427
23.10 Summary......Page 428
24.1 Introduction......Page 438
24.2.1 Examples......Page 439
24.3.1 Example......Page 441
24.4.1 Examples......Page 442
24.5 Discrete sampling: The pricing equation......Page 443
24.5.1 Examples......Page 444
24.7 Pricing via expectations......Page 446
24.9 Summary......Page 447
25.1 Introduction......Page 448
25.3 Types of averaging......Page 449
25.4.1 Monte Carlo simulation......Page 450
25.5.1 Continuously sampled averages......Page 451
25.5.2 Discretely sampled averages......Page 453
25.5.3 Exponentially weighted and other averages......Page 456
25.8 Similarity reductions......Page 458
25.8.1 Put-call parity for the European average strike......Page 460
25.10 Term-structure effects......Page 461
25.11 Some formulae......Page 462
25.12 Summary......Page 463
26.3 Continuous measurement of the maximum......Page 466
26.4 Discrete measurement of the maximum......Page 469
26.5 Similarity reduction......Page 470
26.6 Some formulae......Page 471
26.7 Summary......Page 473
27.2 Perfect trader and passport options......Page 474
27.2.1 Similarity solution......Page 476
27.3 Limiting the number of trades......Page 477
27.4 Limiting the time between trades......Page 478
27.6 Summary......Page 480
28.2 Forward-start options......Page 482
28.3 Shout options......Page 484
28.4 Capped lookbacks and Asians......Page 485
28.5.1 The maximum of the asset and the average of the asset......Page 486
28.5.2 The average of the asset and the maximum of the average......Page 487
28.6 The volatility option......Page 488
28.6.1 The continuous-time limit......Page 491
28.6.2 Hedging variance swaps with vanilla options......Page 492
28.8 Ladders......Page 493
28.9 Parisian options......Page 495
28.9.1 Examples......Page 497
28.11 Summary......Page 500
29.2 Contingent premium put......Page 502
29.3.1 Simple basket option......Page 504
29.3.2 Basket option with averaging over time......Page 506
29.4.1 Double knockout......Page 508
29.4.2 Instalment knockout......Page 511
29.5 Range notes......Page 515
29.5.1 A really simple range note......Page 516
29.6 Lookbacks......Page 519
29.7 Cliquet option......Page 522
29.7.1 Path dependency, constant volatility......Page 523
29.9 Decomposition of exotics into vanillas......Page 527
PART THREE FIXED-INCOME MODELING AND DERIVATIVES......Page 528
30.2 Stochastic interest rates......Page 530
30.3 The bond pricing equation for the general model......Page 531
30.4 What is the market price of risk?......Page 533
30.6 Tractable models and solutions of the bond pricing equation......Page 534
30.7 Solution for constant parameters......Page 537
30.8 Named models......Page 538
30.8.1 Vasicek......Page 539
30.8.2 Cox, Ingersoll & Ross......Page 540
30.9 Equity and FX forwards and futures when rates are stochastic......Page 542
30.9.1 Forward contracts......Page 543
30.9.3 The convexity adjustment......Page 544
30.10 Summary......Page 545
31.2 Ho & Lee......Page 546
31.3 The extended Vasicek model of Hull & White......Page 547
31.4.1 For......Page 548
31.4.2 Against......Page 549
31.5 Other models......Page 551
31.6 Summary......Page 552
32.2 Callable bonds......Page 554
32.3.1 Market practice......Page 555
32.4 Caps and floors......Page 557
32.4.3 Market practice......Page 560
32.5 Range notes......Page 561
32.6.1 Market practice......Page 562
32.8 Index amortizing rate swaps......Page 563
32.8.1 Other features in the index amortizing rate swap......Page 566
32.9 Contracts with embedded decisions......Page 567
32.10 When the interest rate is not the spot rate......Page 568
32.11 Some examples......Page 569
32.12 More interest rate derivatives......Page 571
32.13 Summary......Page 572
33.2 Convertible bond basics......Page 574
33.2.3 Why issue a convertible?......Page 575
33.2.5 Some statistics......Page 576
33.4 Converts as options......Page 577
33.5 Pricing CBs with known interest rate......Page 580
33.5.1 Call and put features......Page 582
33.6 Two-factor modeling: Convertible bonds with stochastic interest rate......Page 585
33.7 A special model......Page 588
33.9 Dilution......Page 589
33.10 Credit risk issues......Page 590
33.11 Summary......Page 591
34.2 Individual mortgages......Page 592
34.2.2 Prepayment......Page 593
34.3.1 The issuers......Page 594
34.4.1 The statistics of repayment......Page 595
34.4.2 The PSA model......Page 596
34.4.3 More realistic models......Page 598
34.5 Valuing MBSs......Page 599
34.6 Summary......Page 600
35.2 Theoretical framework for two factors......Page 602
35.2.1 Special case: Modeling a long-term rate......Page 604
35.3 Popular models......Page 605
35.5 The phase plane in the absence of randomness......Page 608
35.6 The yield curve swap......Page 611
35.7.1 Tractable affine models......Page 612
35.8 Summary......Page 614
36.1 Introduction......Page 616
36.2 Popular one-factor spot-rate models......Page 617
36.3 Implied modeling: One factor......Page 618
36.4 The volatility structure......Page 619
36.5 The drift structure......Page 620
36.6 The slope of the yield curve and the market price of risk......Page 622
36.7 What the slope of the yield curve tells us......Page 625
36.8 Properties of the forward rate curve ‘on average’......Page 626
36.9 Implied modeling: Two factor......Page 627
36.10 Summary......Page 629
37.2 The forward rate equation......Page 630
37.3.1 The non-Markov nature of HJM......Page 632
37.5 Real and risk neutral......Page 633
37.7 Simulations......Page 634
37.9 The Musiela parameterization......Page 635
37.11 Spreadsheet implementation......Page 636
37.12 A simple one-factor example: Ho & Lee......Page 637
37.13 Principal Component Analysis......Page 638
37.13.1 The power method......Page 640
37.15 Non-in.nitesimal short rate......Page 641
37.16 The Brace, Gatarek and Musiela model......Page 642
37.17 Simulations......Page 644
37.19 Summary......Page 645
38.2.1 Optimal choice of ranges?......Page 648
38.2.2 Pricing......Page 649
38.2.3 Differences between optimal for the writer and the buyer......Page 650
38.3 Index amortizing rate swap......Page 652
38.3.2 The code......Page 654
PART FOUR CREDIT RISK......Page 658
39.1 Introduction......Page 660
39.2 The Merton model: Equity as an option on a company’s assets......Page 661
39.2.3 Stochastic interest rates......Page 662
39.3 Modeling with measurable parameters and variables......Page 663
39.4 Calculating the value of the firm......Page 666
39.5 Summary......Page 667
40.2 Risky bonds......Page 670
40.3 Modeling the risk of default......Page 671
40.4 The Poisson process and the instantaneous risk of default......Page 672
40.5 Time-dependent intensity and the term structure of default......Page 675
40.6 Stochastic risk of default......Page 676
40.7 Positive recovery......Page 678
40.9 A case study: The Argentine Par bond......Page 679
40.10 Hedging the default......Page 682
40.11.1 Implied hazard rate and duration......Page 683
40.12 Credit rating......Page 684
40.13 A model for change of credit rating......Page 686
40.13.1 The forward equation......Page 687
40.14 The pricing equation......Page 688
40.15 Credit risk in CBs......Page 689
40.16 Modeling liquidity risk......Page 690
40.17 Summary......Page 693
41.2 What are Credit Derivatives?......Page 696
41.3 Popular credit derivatives......Page 698
41.3.1 Asset swap......Page 699
41.3.2 Total return swaps or total rate of return swaps......Page 700
41.4.1 Basic definitions......Page 701
41.4.3 Credit default swap......Page 702
41.4.5 First to default......Page 703
41.5.3 Credit spread options......Page 704
41.6 Payment on change of rating......Page 705
41.7.1 Exploiting your credit risk view......Page 707
41.8.2 Binary payoff bond......Page 708
41.8.4 Basket credit-linked note......Page 709
41.9 Pricing credit derivatives......Page 710
41.10 An exchange option......Page 711
41.11.2 Hedging......Page 713
41.12 Payoff on change of rating......Page 714
41.13 Multi-factor derivatives......Page 715
41.14.2 The mathematical definition......Page 716
41.14.3 Examples of copulas......Page 717
41.15 Collateralized Debt Obligations......Page 718
41.16 Summary......Page 720
42.1 Introduction......Page 722
42.3.1 Estimating volatility......Page 723
42.3.2 Correlation......Page 724
42.4.2 Spreads......Page 726
42.5 The CreditMetrics methodology......Page 727
42.6 A portfolio of risky bonds......Page 728
42.8 Summary......Page 729
43.2 Why do banks go broke?......Page 730
43.3 Market crashes......Page 731
43.5 CrashMetrics for one stock......Page 732
43.6 Portfolio optimization and the Platinum Hedge......Page 734
43.7 The multi-asset/single-index model......Page 736
43.7.1 Assuming Taylor series for the moment......Page 741
43.8 Portfolio optimization and the Platinum Hedge in the multi-asset model......Page 743
43.9 The multi-index model......Page 744
43.10 Incorporating time value......Page 745
43.11.2 Modeling margin......Page 746
43.13 Simple extensions to CrashMetrics......Page 748
43.14 The CrashMetrics Index (CMI)......Page 749
43.15 Summary......Page 750
44.2 Orange County......Page 752
44.3 Proctor and Gamble......Page 754
44.4 Metallgesellschaft......Page 756
44.4.1 Basis risk......Page 757
44.5 Gibson Greetings......Page 758
44.6 Barings......Page 760
44.7 Long-Term Capital Management......Page 761
44.8 Summary......Page 765
PART FIVE ADVANCED TOPICS......Page 766
45.1 Introduction......Page 770
45.2 Warning: Modeling as it is currently practiced......Page 771
45.2.2 The find-and-replace school of mathematical modeling......Page 772
45.3 Summary......Page 774
46.1 Introduction......Page 776
46.5 Deterministic volatility surfaces......Page 777
46.7 Uncertain parameters......Page 778
46.11 Jump diffusion......Page 779
46.14 Optimal static hedging......Page 780
46.15 The feedback effect of hedging in illiquid markets......Page 781
46.19 Serial autocorrelation in returns......Page 782
46.20 Summary......Page 783
47.2 Motivating example: The trinomial model......Page 784
47.3 A model for a discretely hedged position......Page 785
47.4 A higher-order analysis......Page 788
47.4.1 Choosing the best......Page 789
47.4.2 The hedging error......Page 790
47.4.4 Pricing the option......Page 793
47.4.5 The adjusted   and option value......Page 794
47.5 The real distribution of returns and the hedging error......Page 795
47.6 Total hedging error for the real distribution of returns......Page 797
47.8 Summary......Page 798
48.2 The effect of costs......Page 804
48.3 The model of Leland (1985)......Page 805
48.4 The model of Hoggard, Whalley & Wilmott (1992)......Page 806
48.5 Non-single-signed gamma......Page 811
48.6 The marginal effect of transaction costs......Page 813
48.8 Hedging to a bandwidth: The model of Whalley & Wilmott (1993) and Henrotte (1993)......Page 814
48.9.1 The model of Hodges & Neuberger (1989)......Page 815
48.9.3 The asymptotic analysis of Whalley & Wilmott (1993)......Page 816
48.9.4 Arbitrary cost structure......Page 817
48.10.1 Nonlinearity......Page 818
48.10.3 Existence of solutions......Page 819
48.12 Empirical testing......Page 821
48.12.1 Black–Scholes and Leland hedging......Page 822
48.12.2 Market movement or delta-tolerance strategy......Page 823
48.12.3 The utility strategy......Page 825
48.12.5 And the winner is. . .......Page 827
48.13 Transaction costs and discrete hedging put together......Page 828
48.14 Summary......Page 829
49.2 The different types of volatility......Page 834
49.2.4 Forward volatility......Page 835
49.3.2 Incorporating mean reversion......Page 836
49.3.5 Expected future volatility......Page 837
49.3.6 Beyond close-close estimators: Range-based estimation of volatility......Page 839
49.4.1 A simple motivating example: Taxi numbers......Page 841
49.4.2 Three hats......Page 842
49.4.3 The maths behind this: Find the standard deviation......Page 843
49.4.4 Quants’ salaries......Page 844
49.5.2 Sensitivity of the risk reversal to skews and smiles......Page 845
49.6.2 Deterministic volatility surfaces......Page 847
49.6.3 Stochastic volatility......Page 848
49.6.6 Static hedging......Page 849
49.6.9 Volatility case study: The cliquet option......Page 850
49.8 Summary......Page 851
50.2 Implied volatility......Page 854
50.3 Time-dependent volatility......Page 856
50.4 Volatility smiles and skews......Page 860
50.5 Volatility surfaces......Page 861
50.6 Backing out the local volatility surface from European call option prices......Page 862
50.7 A simple volatility surface parameterization......Page 866
50.9 Volatility information contained in an at-the-money straddle......Page 867
50.10 Volatility information contained in a risk-reversal......Page 868
50.12 A market convention......Page 869
50.14 Summary......Page 870
51.2 Random volatility......Page 874
51.3 A stochastic differential equation for volatility......Page 875
51.4 The pricing equation......Page 876
51.5 The market price of volatility risk......Page 877
51.5.1 Aside: The market price of risk for traded assets......Page 878
51.7 An example......Page 879
51.9 Named/popular models......Page 881
51.9.2 The REGARCH model and its diffusion limit......Page 883
51.10 A note on biases......Page 885
51.11 Stochastic implied volatility: The model of Sch¨onbucher......Page 886
51.12 Summary......Page 887
52.1 Introduction......Page 890
52.2 Best and worst cases......Page 892
52.2.1 Uncertain volatility: The model of Avellaneda, Levy & Par´as (1995) and Lyons (1995)......Page 893
52.2.2 Example: An up-and-out call......Page 894
52.2.4 Uncertain dividends......Page 898
52.4 Nonlinearity......Page 900
52.5 Summary......Page 901
53.2 Stochastic volatility and uncertain parameters revisited......Page 902
53.3 Deriving an empirical stochastic volatility model......Page 903
53.4 Estimating the volatility of volatility......Page 904
53.5 Estimating the drift of volatility......Page 905
53.7.1 Option pricing with stochastic volatility......Page 907
53.7.3 Stochastic volatility, certainty bands and confidence limits......Page 908
53.8 Summary......Page 909
54.2 The model for the asset and its volatility......Page 910
54.4 Analysis of the variance......Page 911
54.6 The mean and variance equations......Page 912
54.7 How to interpret and use the mean and variance......Page 913
54.9 Example: Valuing and hedging an up-and-out call......Page 915
54.10 Static hedging......Page 917
54.12 Summary......Page 919
55.2 Fast mean reversion and high volatility of volatility......Page 922
55.3 Conditions on the models......Page 924
55.4.2 The Heston/Ball–Roma model......Page 925
55.5 Notation......Page 926
55.6 Asymptotic analysis......Page 927
55.7 Vanilla options: Asymptotics for values......Page 929
55.8 Vanilla options: Implied volatilities......Page 931
55.9 Summary......Page 934
56.2 The subtle nature of the cliquet option......Page 936
56.3 Path dependency, constant volatility......Page 938
56.4 Results......Page 939
56.4.2 Uncertain volatility......Page 940
56.5 Code: Cliquet with uncertain volatility, in similarity variables......Page 944
56.6 Summary......Page 947
57.2 Evidence for jumps......Page 948
57.3 Poisson processes......Page 952
57.4 Hedging when there are jumps......Page 953
57.5 Hedging the diffusion......Page 954
57.6 Hedging the jumps......Page 955
57.8 The downside of jump-diffusion models......Page 956
57.9 Jump volatility......Page 957
57.11 Summary......Page 958
58.2 Value at risk......Page 960
58.3 A simple example: The hedged call......Page 961
58.4 A mathematical model for a crash......Page 962
58.4.1 Case I: Black–Scholes Hedging......Page 964
58.4.2 Case II: Crash Hedging......Page 965
58.5 An example......Page 966
58.6 Optimal static hedging: VaR reduction......Page 967
58.7 Continuous-time limit......Page 968
58.9.1 Limiting the total number of crashes......Page 969
58.10 Crashes in a multi-asset world......Page 970
58.11 Fixed and floating exchange rates......Page 971
58.12 Summary......Page 972
59.1 Introduction......Page 974
59.2.1 The present value of expected payoff......Page 975
59.2.2 Standard deviation......Page 976
59.3 More sophisticated models for the return on an asset......Page 978
59.3.2 Jump drift......Page 980
59.4 Early closing......Page 983
59.5 To hedge or not to hedge?......Page 986
59.7 Summary......Page 988
60.2 Static replicating portfolio......Page 990
60.3 Matching a ‘target’ contract......Page 991
60.4 Vega matching......Page 992
60.5 Static hedging: Non-linear governing equation......Page 993
60.7.1 Example: Non-linear model, unhedged......Page 994
60.7.2 Static hedging: A first attempt......Page 995
60.8 Optimal static hedging: The theory......Page 997
60.10 Hedging path-dependent options with vanilla options, non-linear models......Page 999
60.10.2 Barrier options......Page 1000
60.10.3 Pricing and optimally hedging a portfolio of barrier options......Page 1001
60.11 The mathematics of optimization......Page 1002
60.11.1 Downhill simplex method......Page 1003
60.12 Summary......Page 1007
61.1 Introduction......Page 1010
61.2 The trading strategy for option replication......Page 1011
61.4 Incorporating the trading strategy......Page 1012
61.5 The influence of replication......Page 1014
61.6 The forward equation......Page 1017
61.6.1 The boundaries......Page 1018
61.7 Numerical results......Page 1019
61.7.1 Time-independent trading strategy......Page 1020
61.7.2 Put replication trading......Page 1022
61.9 Summary......Page 1023
62.2 Ranking events......Page 1026
62.5 Special utility functions......Page 1028
62.6 Certainty equivalent wealth......Page 1029
62.7 Maximization of expected utility......Page 1031
62.8 Summary......Page 1032
63.1 Introduction......Page 1034
63.2 What Derivatives Week published......Page 1035
63.5 And finally, the paper . . .......Page 1036
63.6 Introduction......Page 1037
63.7 Preliminary: Pricing and Hedging......Page 1039
63.8 Utility-Maximizing Exercise Time......Page 1040
63.8.2 Hyperbolic absolute risk aversion......Page 1043
63.9 Profit from Selling American Options......Page 1044
63.10 Concluding Remarks......Page 1047
63.11 Who wins and who loses?......Page 1048
63.13 Another situation where the same idea applies: Passport options......Page 1050
63.13.2 Utility maximization in the passport option......Page 1051
63.14 Summary......Page 1054
64.2 Why do we need dividend models?......Page 1056
64.3 Effects of dividends on asset prices......Page 1058
64.3.2 Term structure of dividends......Page 1059
64.6 Uncertainty in dividend amount and timing......Page 1061
64.7 Summary......Page 1064
65.2 Evidence......Page 1066
65.3 The Telegraph equation......Page 1068
65.4 Pricing and hedging derivatives......Page 1070
65.5 Summary......Page 1071
66.2 One risk-free and one risky asset......Page 1072
66.2.2 Maximizing expected utility......Page 1073
66.2.3 Stochastic control and the Bellman equation......Page 1074
66.2.4 Constant relative risk aversion......Page 1075
66.2.5 Constant absolute risk aversion......Page 1076
66.3 Many assets......Page 1077
66.4 Maximizing long-term growth......Page 1078
66.5 A brief look at transaction costs......Page 1079
66.6 Summary......Page 1081
67.1 Introduction......Page 1082
67.2 Optimal Portfolios under the Threat of a Crash: The single stock case......Page 1083
67.3 Maximizing Growth Rate under the Threat of a Crash: n stocks......Page 1091
67.4.1 Arbitrary upper bound for the number of crashes......Page 1094
67.5 Summary......Page 1096
68.1 Introduction......Page 1098
68.3 A non-probabilistic model for the behavior of the short-term interest rate......Page 1099
68.4 Worst-case scenarios and a non-linear equation......Page 1100
68.4.1 Let’s see that again in slow motion......Page 1101
68.5 Examples of hedging: Spreads for prices......Page 1103
68.5.1 Hedging with one instrument......Page 1104
68.5.2 Hedging with multiple instruments......Page 1106
68.6 Generating the ‘Yield Envelope’......Page 1108
68.7 Swaps......Page 1112
68.8 Caps and floors......Page 1115
68.9.1 Identifying arbitrage opportunities......Page 1116
68.9.5 A remark on the validity of the model......Page 1117
68.10 Summary......Page 1118
69.2 A real portfolio......Page 1120
69.3.1 Pricing the European option on a zero-coupon bond......Page 1124
69.3.2 Pricing and hedging American options......Page 1126
69.4 Contracts with embedded decisions......Page 1129
69.5 The index amortizing rate swap......Page 1131
69.6 Convertible bonds......Page 1135
69.7 Summary......Page 1137
70.2 Fitting forward rates......Page 1138
70.3 Economic cycles......Page 1139
70.4 Interest rate bands......Page 1140
70.4.1 Estimating   from past data......Page 1141
70.5.1 A maximum number of crashes......Page 1143
70.5.2 A maximum frequency of crashes......Page 1145
70.5.3 Estimating   from past data......Page 1146
70.6 Liquidity......Page 1147
70.7 Summary......Page 1149
71.2.2 Inflation caps and floors......Page 1150
71.3 Pricing, first thoughts......Page 1151
71.5 Pricing, second thoughts......Page 1152
71.6 Analyzing the data......Page 1154
71.7 Can we model inflation independently of interest rates?......Page 1156
71.9 Non-linear pricing methods......Page 1159
71.10 Summary......Page 1160
72.2 The energy market......Page 1162
72.3 What’s so special about the energy markets?......Page 1163
72.4 Why can’t we apply Black–Scholes theory to energy derivatives?......Page 1166
72.6 The Pilopovi´c two-factor model......Page 1167
72.7.2 Asian options......Page 1169
72.7.5 Basis spreads......Page 1170
72.8 Summary......Page 1171
73.2 Financial options and Real options......Page 1172
73.3 An introductory example: Abandonment of a machine......Page 1173
73.4 Optimal investment: Simple example #2......Page 1174
73.5 Temporary suspension of the project, costless......Page 1175
73.7 Sequential and incremental investment......Page 1176
73.8 Ashanti: Gold mine case study......Page 1178
73.9 Summary......Page 1180
74.2 Life expectancy......Page 1182
74.4 Death seen as default......Page 1184
74.5 Pricing a single policy......Page 1187
74.5.1 Internal rate of return......Page 1189
74.6 Pricing portfolios......Page 1191
74.6.1 Extension risk......Page 1193
74.7 Summary......Page 1194
75.2.1 Bonus depending on the Sharpe ratio......Page 1196
75.2.2 Numerical results......Page 1198
75.3 The skill factor......Page 1201
75.4 Putting skill into the equation......Page 1205
75.4.1 Example......Page 1206
75.5 Summary......Page 1207
PART SIX NUMERICAL METHODS AND PROGRAMS......Page 1210
76.2 Finite-difference methods......Page 1212
76.2.2 Program of study......Page 1214
76.3 Monte Carlo methods......Page 1215
76.4 Numerical integration......Page 1216
76.5 Summary......Page 1217
77.1 Introduction......Page 1220
77.3 Grids......Page 1221
77.5 Approximating θ......Page 1223
77.6 Approximating......Page 1224
77.6.1 One-sided differences......Page 1226
77.8 Example......Page 1227
77.10 Boundary conditions......Page 1228
77.10.1 Other boundary conditions......Page 1230
77.11 The explicit finite-difference method......Page 1231
77.12 Convergence of the explicit method......Page 1234
77.13 The Code # 1: European option......Page 1236
77.14 The Code # 2: American exercise......Page 1240
77.15 The Code # 3: 2-D output......Page 1242
77.16 Bilinear interpolation......Page 1243
77.17 Upwind differencing......Page 1245
77.18 Summary......Page 1247
78.2 Implicit finite-difference methods......Page 1248
78.3 The Crank–Nicolson method......Page 1250
78.3.1 Boundary condition type I: V k+1 0 given......Page 1252
78.3.2 Boundary condition type II: Relationship between V k+1 0 and V k+ 1......Page 1253
78.3.4 The matrix equation......Page 1254
78.3.5 LU decomposition......Page 1255
78.3.6 Successive over-relaxation, SOR......Page 1257
78.4 Comparison of finite-difference methods......Page 1260
78.6 Douglas schemes......Page 1262
78.7 Three time-level methods......Page 1263
78.8 Richardson extrapolation......Page 1264
78.9 Free boundary problems and American options......Page 1265
78.9.2 Early exercise and Crank–Nicolson......Page 1266
78.10.1 A discrete cashflow......Page 1267
78.10.2 Discretely paid dividend......Page 1268
78.11.1 Discretely sampled quantities......Page 1270
78.12 Summary......Page 1271
79.2 Two-factor models......Page 1274
79.3 The explicit method......Page 1276
79.4 Calculation time......Page 1279
79.5 Alternating Direction Implicit......Page 1280
79.6 The Hopscotch method......Page 1281
79.7 Summary......Page 1282
80.2 Relationship between derivative values and simulations: Equities, indices, currencies, commodities......Page 1284
80.3 Generating paths......Page 1285
80.4 Lognormal underlying, no path dependency......Page 1287
80.6 Using random numbers......Page 1288
80.7 Generating normal variables......Page 1289
80.8 Real versus risk neutral, speculation versus hedging......Page 1290
80.9 Interest rate products......Page 1291
80.10 Calculating the greeks......Page 1295
80.11 Higher dimensions: Cholesky factorization......Page 1296
80.12 Calculation time......Page 1297
80.13.2 Control variate technique......Page 1298
80.15 American options......Page 1299
80.16 Longstaff & Schwartz regression approach for American options......Page 1300
80.18 Summary......Page 1304
81.2 Regular grid......Page 1306
81.3 Basic Monte Carlo integration......Page 1307
81.4 Low-discrepancy sequences......Page 1309
81.5 Advanced techniques......Page 1313
81.6 Summary......Page 1314
82.2 Kolmogorov equation......Page 1316
82.3 Explicit one-factor model for a convertible bond......Page 1317
82.4 American call, implicit......Page 1318
82.5 Explicit Parisian option......Page 1320
82.6 Passport options......Page 1321
82.7 Chooser passport option......Page 1322
82.8 Explicit stochastic volatility......Page 1324
82.10 Crash modeling......Page 1325
82.11 Explicit Epstein–Wilmott solution......Page 1326
82.12 Risky-bond calculator......Page 1328
83.2 Monte Carlo pricing of a basket......Page 1332
83.3 Quasi Monte Carlo pricing of a basket......Page 1334
83.4 Monte Carlo for American options......Page 1335
A.2 The different types of mathematics found in finance......Page 1338
A.2.1 Modeling approaches......Page 1339
A.2.2 The tools......Page 1340
A.3 e......Page 1341
A.5 Differentiation and Taylor series......Page 1342
A.6 Expectation and variance......Page 1345
A.7 Another look at Black–Scholes......Page 1346
A.8 Summary......Page 1348
Bibliography......Page 1350
Index......Page 1372