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Computational_Finance_Numerical_Methods.pdf

Contents
Preface xi
Part I Using Numerical Software Components within Microsoft Windows 1
1 Introduction 3
2 Dynamic Link Libraries(DLLs) 6
2.1 Visual Basic and Excel VBA 6
2.2 VB.NET 16
2.3 C# 21
3 ActiveX and COM 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 A financial derivative pricing example 38
4.1 Interactive user-interface 38
4.2 Language user-interface 38
4.3 Use within Delphi 41
5 ActiveX componentsand 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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6.3 XML schema 57
6.4 XSL 59
6.5 Stock market data example 60
7 Epilogue 64
7.1 Wrapping C with Cþþ for OO numerics in .NET 64
7.2 Final remarks 73
Part II Pricing Assets 75
8 Introduction 77
8.1 An introduction to options and derivatives 77
8.2 Brownian motion 78
8.3 A Brownian model of asset price movements 81
8.4 Ito’s lemma in one dimension 83
8.5 Ito’s lemma in many dimensions 84
9 Analytic methods and single asset European options 87
9.1 Introduction 87
9.2 Put–call parity 88
9.3 Vanilla options and the Black–Scholes model 90
9.4 Barrier options 110
10 Numeric methods and single asset American options 116
10.1 Introduction 116
10.2 Perpetual options 116
10.3 Approximations for vanilla American options 121
10.4 Lattice methods for vanilla options 137
10.5 Implied lattice methods 159
10.6 Grid methods for vanilla options 177
10.7 Pricing American options using a stochastic lattice 212
11 Monte Carlo simulation 221
11.1 Introduction 221
11.2 Pseudorandomand quasirandomsequences 222
11.3 Generation of multivariate distributions: independent variates 229
11.4 Generation of multivariate distributions: correlated variates 234
12 Multiasset European 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
viii Contents
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12.6 Three asset options 267
12.7 Four asset options 272
13 Dealing with missing data 274
13.1 Introduction 274
13.2 Iterative multiple linear regression, MREG 275
13.3 The EM algorithm278
Part III Financial Econometrics285
14 Introduction 287
14.1 Asset returns 289
14.2 Nonsynchronous trading 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 GARCH models301
15.1 Box Jenkins models 301
15.2 Gaussian Linear GARCH 303
15.3 The IGARCH model 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 GARCH conditional probability distributions 319
17.1 Gaussian distribution 319
17.2 Student’s t distribution 321
17.3 General error distribution 323
18 Maximum likelihood parameter estimation 327
18.1 The conditional log likelihood 327
18.2 The covariance matrix of the parameter estimates 328
18.3 Numerical optimization 332
18.4 Scaling the data 334
19 Analytic derivativesof the log likelihood 336
19.1 The first derivatives 336
19.2 The second derivatives 339
Contents ix
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20 GJR–GARCH algorithms344
20.1 Initial estimates and pre-observed values 344
20.2 Gaussian distribution 346
20.3 Student’s t distribution 350
21 GARCH software 353
21.1 Expected sofware capabilities 353
21.2 Testing GARCH software 354
22 GARCH process identification 360
22.1 Likelihood ratio test 360
22.2 Significance of the estimated parameters 360
22.3 The independence of the standardized residuals 360
22.4 The distribution of the standardized residuals 361
22.5 Modelling the S&P 500 index 362
22.6 Excel demonstration 364
22.7 Internet Explorer demonstration 368
23 Multivariate time series 371
23.1 Principal component GARCH 371
Appendices375
A Computer code for Part I 377
A.1 The ODL file for the derivative pricing control 377
B Some more option pricing formulae 379
B.1 Binary options 379
B.2 Option to exchange one asset for another 379
B.3 Lookback options 380
C Derivation of the Greeksfor vanilla European options 381
C.1 Introduction 381
C.2 Gamma 382
C.3 Delta 383
C.4 Theta 383
C.5 Rho 384
C.6 Vega 385
D Multiasset binomial lattices 386
D.1 Truncated two asset binomial lattice 386
D.2 Recursive two asset binomial lattice 388
D.3 Four asset jump probabilities 391
x Contents
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E Derivation of the conditional mean and covariance for a
multivariate normal distribution 393
F Standard statistical results 395
F.1 The law of large numbers 395
F.2 The central limit theorem 395
F.3 The mean and variance of linear functions of random variables 396
F.4 Standard algorithms for the mean and variance 397
F.5 The Hanson and West algorithmfor the mean and variance 399
F.6 Jensen’s inequality 401
G Derivation of barrier option integrals403
G.1 The down and out call 403
G.2 The up and out call 406
H Algorithmsfor an AGARCH-I process 410
H.1 Gaussian distribution 410
H.2 Student’s t distribution 413
I The general error distribution 417
I.1 Value of 
 for variance hi 417
I.2 The kurtosis 417
I.3 The distribution when the shape parameter, a is very large 418
J The Student’s t distribution 420
J.1 The kurtosis 420
K Mathematical reference 423
K.1 Standard integrals 423
K.2 Gamma function 423
K.3 The cumulative normal distribution function 424
K.4 Arithmetic and geometric progressions 425
L The stability of the Black–Scholes finite-difference schemes 426
L.1 The general case 426
L.2 The log transformation and a uniform grid 426
Glossary of terms 429
Computing reading list 430
Mathematics and finance references 432
Index 439