Levy, George.
Computational Finance Using C and C# / George Levy.
p. cm. – (Quantitative finance)
Includes bibliographical references and index.
ISBN-13: 978-0-7506-6919-1 (alk. paper) 1. Finance-Mathematical models. I. Title.
HG106.L484 2008
332.0285’5133-dc22
2008000470


content
Preface xi
1 Overview of financial derivatives 1
2 Introduction to stochastic processes 5
2.1 Brownian motion 5
2.2 A Brownian model of asset price movements 9
2.3 Ito’s formula (or lemma) 10
2.4 Girsanov’s theorem 12
2.5 Ito’s lemma for multiasset geometric Brownian motion 13
2.6 Ito product and quotient rules in two dimensions 15
2.7 Ito product in n dimensions 18
2.8 The Brownian bridge 19
2.9 Time-transformed Brownian motion 21
2.10 Ornstein–Uhlenbeck process 24
2.11 The Ornstein–Uhlenbeck bridge 27
2.12 Other useful results 31
2.13 Selected problems 33
3 Generation of random variates 37
3.1 Introduction 37
3.2 Pseudo-random and quasi-random sequences 38
3.3 Generation of multivariate distributions: independent variates 41
3.4 Generation of multivariate distributions: correlated variates 47
4 European options 59
4.1 Introduction 59
4.2 Pricing derivatives using a martingale measure 59
4.3 Put call parity 60
4.4 Vanilla options and the Black–Scholes model 62
4.5 Barrier options 85
5 Single asset American options 97
5.1 Introduction 97
5.2 Approximations for vanilla American options 97
5.3 Lattice methods for vanilla options 114
viii Computational Finance Using C and C#
5.4 Grid methods for vanilla options 135
5.5 Pricing American options using a stochastic lattice 172
6 Multiasset options 181
6.1 Introduction 181
6.2 The multiasset Black–Scholes equation 181
6.3 Multidimensional Monte Carlo methods 183
6.4 Introduction to multidimensional lattice methods 185
6.5 Two asset options 190
6.6 Three asset options 201
6.7 Four asset options 205
7 Other financial derivatives 209
7.1 Introduction 209
7.2 Interest rate derivatives 209
7.3 Foreign exchange derivatives 228
7.4 Credit derivatives 232
7.5 Equity derivatives 237
8 C# portfolio pricing application 245
8.1 Introduction 245
8.2 Storing and retrieving the market data 254
8.3 The PricingUtils class and the Analytics_MathLib 262
8.4 Equity deal classes 267
8.5 FX deal classes 280
Appendix A: The Greeks for vanilla European options 289
A.1 Introduction 289
A.2 Gamma 290
A.3 Delta 291
A.4 Theta 292
A.5 Rho 293
A.6 Vega 294
Appendix B: Barrier option integrals 295
B.1 The down and out call 295
B.2 The up and out call 298
Appendix C: Standard statistical results 303
C.1 The law of large numbers 303
C.2 The central limit theorem 303
C.3 The variance and covariance of random variables 305
C.4 Conditional mean and covariance of normal distributions 310
C.5 Moment generating functions 311