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COMPUTATIONAL FINANCE
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COMPUTATIONAL FINANCE
Numerical Methods for Pricing Financial Instruments
George Levy
AMSTERDAM BOSTON HEIDELBERG LONDON NEWYORK OXFORD
PARIS SANDIEGO SANFRANCISCO SINGAPORE SYDNEY TOKYO
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Butterworth-Heinemann
Elsevier
LinacreHouse,JordanHill,OxfordOX28DP
200WheelerRoad,Burlington,MA01803
Firstpublished2004
Copyright#2004,GeorgeLevy.Allrightsreserved
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To Kathryn
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Contents
Preface xi
Part I Using NumericalSoftware Components within Microsoft Windows 1
1 Introduction 3
2 DynamicLink Libraries (DLLs) 6
2.1 Visual Basicand ExcelVBA 6
2.2 VB.NET 16
2.3 C# 21
3 ActiveX andCOM 28
3.1 Introduction 28
3.2 The COM interface IDispatch 30
3.3 Type libraries 31
3.4 Using IDispatch 31
3.5 ActiveX controls and the Internet 33
3.6 Using ActiveX components on a Web page 34
4 Afinancial derivative pricing example 38
4.1 Interactive user-interface 38
4.2 Languageuser-interface 38
4.3 Use within Delphi 41
5 ActiveX components and numerical optimization 44
5.1 Ray tracing example 44
5.2 Portfolio allocation example 49
5.3 Numerical optimization within Microsoft Excel 51
6 XML and transformation using XSL 54
6.1 Introduction 54
6.2 XML 55
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viii Contents
6.3 XML schema 57
6.4 XSL 59
6.5 Stock market data example 60
7 Epilogue 64
7.1 Wrapping Cwith Cþþ forOO numerics in.NET 64
7.2 Final remarks 73
PartII Pricing Assets 75
8 Introduction 77
8.1 Anintroduction to options andderivatives 77
8.2 Brownian motion 78
8.3 ABrownian model ofasset price movements 81
8.4 Ito’s lemma in one dimension 83
8.5 Ito’s lemma in manydimensions 84
9 Analytic methods and single asset Europeanoptions 87
9.1 Introduction 87
9.2 Put–callparity 88
9.3 Vanillaoptions and the Black–Scholes model 90
9.4 Barrieroptions 110
10 Numeric methodsand singleasset American options 116
10.1 Introduction 116
10.2 Perpetual options 116
10.3 Approximations forvanillaAmerican options 121
10.4 Lattice methods forvanillaoptions 137
10.5 Implied lattice methods 159
10.6 Grid methods forvanillaoptions 177
10.7 PricingAmerican optionsusing a stochastic lattice 212
11 Monte Carlosimulation 221
11.1 Introduction 221
11.2 Pseudorandom and quasirandom sequences 222
11.3 Generation of multivariate distributions: independentvariates 229
11.4 Generation of multivariate distributions: correlated variates 234
12 MultiassetEuropean and American options 247
12.1 Introduction 247
12.2 The multiasset Black–Scholes equation 247
12.3 Multidimensional Monte Carlo methods 248
12.4 Multidimensional lattice methods 253
12.5 Two asset options 257
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Contents ix
12.6 Three asset options 267
12.7 Four asset options 272
13 Dealingwith missingdata 274
13.1 Introduction 274
13.2 Iterative multiple linear regression, MREG 275
13.3 The EM algorithm 278
Part III Financial Econometrics 285
14 Introduction 287
14.1 Assetreturns 289
14.2 Nonsynchronoustrading 291
14.3 Bid-ask spread 293
14.4 Models of volatility 294
14.5 Stochastic autoregressive volatility, ARV 296
14.6 Generalized hyperbolic Levy motion 297
15 GARCHmodels 301
15.1 Box Jenkins models 301
15.2 Gaussian Linear GARCH 303
15.3 The IGARCHmodel 309
15.4 The GARCH-M model 309
15.5 Regression-GARCH and AR-GARCH 310
16 Nonlinear GARCH 311
16.1 AGARCH-I 313
16.2 AGARCH-II 316
16.3 GJR–GARCH 317
17 GARCHconditional probabilitydistributions 319
17.1 Gaussian distribution 319
17.2 Student’s tdistribution 321
17.3 General error distribution 323
18 Maximum likelihoodparameter estimation 327
18.1 The conditionallog likelihood 327
18.2 The covariance matrix ofthe parameter estimates 328
18.3 Numerical optimization 332
18.4 Scaling the data 334
19 Analytic derivativesof thelog likelihood 336
19.1 The first derivatives 336
19.2 The second derivatives 339
