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Monte Carlo Methods in Finance

Monte Carlo Methods in Finance

Wiley Finance Series

Aus der Reihe

Fr. 168.00

inkl. gesetzl. MwSt.
Versandkostenfrei

Beschreibung

Details

Einband

Gebundene Ausgabe

Erscheinungsdatum

03.04.2002

Verlag

John Wiley & Sons Inc

Seitenzahl

222

Maße (L/B/H)

25.6/17.4/2 cm

Beschreibung

Details

Einband

Gebundene Ausgabe

Erscheinungsdatum

03.04.2002

Verlag

John Wiley & Sons Inc

Seitenzahl

222

Maße (L/B/H)

25.6/17.4/2 cm

Gewicht

590 g

Auflage

1. Auflage

Sprache

Englisch

ISBN

978-0-471-49741-7

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  • Monte Carlo Methods in Finance

  • Preface xi

    Acknowledgements xiii

    Mathematical Notation xv

    1 Introduction 1

    2 The Mathematics Behind Monte Carlo Methods 5

    2.1 A Few Basic Terms in Probability and Statistics 5

    2.2 Monte Carlo Simulations 7

    2.2.1 Monte Carlo Supremacy 8

    2.2.2 Multi-dimensional Integration 8

    2.3 Some Common Distributions 9

    2.4 Kolmogorov's Strong Law 18

    2.5 The Central Limit Theorem 18

    2.6 The Continuous Mapping Theorem 19

    2.7 Error Estimation for Monte Carlo Methods 20

    2.8 The Feynman-Kac Theorem 21

    2.9 The Moore-Penrose Pseudo-inverse 21

    3 Stochastic Dynamics 23

    3.1 Brownian Motion 23

    3.2 Itô's Lemma 24

    3.3 Normal Processes 25

    3.4 Lognormal Processes 26

    3.5 The Markovian Wiener Process Embedding Dimension 26

    3.6 Bessel Processes 27

    3.7 Constant Elasticity Of Variance Processes 28

    3.8 Displaced Diffusion 29

    4 Process-driven Sampling 31

    4.1 Strong versus Weak Convergence 31

    4.2 Numerical Solutions 32

    4.2.1 The Euler Scheme 32

    4.2.2 The Milstein Scheme 33

    4.2.3 Transformations 33

    4.2.4 Predictor-Corrector 35

    4.3 Spurious Paths 36

    4.4 Strong Convergence for Euler and Milstein 37

    5 Correlation and Co-movement 41

    5.1 Measures for Co-dependence 42

    5.2 Copulæ 45

    5.2.1 The Gaussian Copula 46

    5.2.2 The t-Copula 49

    5.2.3 Archimedean Copulae 51

    6 Salvaging a Linear Correlation Matrix 59

    6.1 Hypersphere Decomposition 60

    6.2 Spectral Decomposition 61

    6.3 Angular Decomposition of Lower Triangular Form 62

    6.4 Examples 63

    6.5 Angular Coordinates on a Hypersphere of Unit Radius 65

    7 Pseudo-random Numbers 67

    7.1 Chaos 68

    7.2 The Mid-square Method 72

    7.3 Congruential Generation 72

    7.4 Ran0 To Ran3 74

    7.5 The Mersenne Twister 74

    7.6 Which One to Use? 75

    8 Low-discrepancy Numbers 77

    8.1 Discrepancy 78

    8.2 Halton Numbers 79

    8.3 Sobol' Numbers 80

    8.3.1 Primitive Polynomials Modulo Two 81

    8.3.2 The Construction of Sobol' Numbers 82

    8.3.3 The Gray Code 83

    8.3.4 The Initialisation of Sobol' Numbers 85

    8.4 Niederreiter (1988) Numbers 88

    8.5 Pairwise Projections 88

    8.6 Empirical Discrepancies 91

    8.7 The Number of Iterations 96

    8.8 Appendix 96

    8.8.1 Explicit Formula for the L2-norm Discrepancy on the Unit Hypercube 96

    8.8.2 Expected L2-norm Discrepancy of Truly Random Numbers 97

    9 Non-uniform Variates 99

    9.1 Inversion of the Cumulative Probability Function 99

    9.2 Using a Sampler Density 101

    9.2.1 Importance Sampling 103

    9.2.2 Rejection Sampling 104

    9.3 Normal Variates 105

    9.3.1 The Box-Muller Method 105

    9.3.2 The Neave Effect 106

    9.4 Simulating Multivariate Copula Draws 109

    10 Variance Reduction Techniques 111

    10.1 Antithetic Sampling 111

    10.2 Variate Recycling 112

    10.3 Control Variates 113

    10.4 Stratified Sampling 114

    10.5 Importance Sampling 115

    10.6 Moment Matching 116

    10.7 Latin Hypercube Sampling 119

    10.8 Path Construction 120

    10.8.1 Incremental 120

    10.8.2 Spectral 122

    10.8.3 The Brownian Bridge 124

    10.8.4 A Comparison of Path Construction Methods 128

    10.8.5 Multivariate Path Construction 131

    10.9 Appendix 134

    10.9.1 Eigenvalues and Eigenvectors of a Discrete-time Covariance Matrix 134

    10.9.2 The Conditional Distribution of the Brownian Bridge 137

    11 Greeks 139

    11.1 Importance Of Greeks 139

    11.2 An Up-Out-Call Option 139

    11.3 Finite Differencing with Path Recycling 140

    11.4 Finite Differencing with Importance Sampling 143

    11.5 Pathwise Differentiation 144

    11.6 The Likelihood Ratio Method 145

    11.7 Comparative Figures 147

    11.8 Summary 153

    11.9 Appendix 153

    11.9.1 The Likelihood Ratio Formula for Vega 153

    11.9.2 The Likelihood Ratio Formula for Rho 156

    12 Monte Carlo in the BGM/J Framework 159

    12.1 The Brace-Gatarek-Musiela/Jamshidian Market Model 159

    12.2 Factorisation 161

    12.3 Bermudan Swaptions 163

    12.4 Calibration to European Swaptions 163

    12.5 The Predictor-Corrector Scheme 169

    12.6 Heuristics of the Exercise Boundary 171

    12.7 Exercise Boundary Parametrisation 174

    12.8 The Algorithm 176

    12.9 Numerical Results 177

    12.10 Summary 182

    13 Non-recombining Trees 183

    13.1 Introduction 183

    13.2 Evolving the Forward Rates 184

    13.3 Optimal Simplex Alignment 187

    13.4 Implementation 190

    13.5 Convergence Performance 191

    13.6 Variance Matching 192

    13.7 Exact Martingale Conditioning 195

    13.8 Clustering 196

    13.9 A Simple Example 199

    13.10 Summary 200

    14 Miscellanea 201

    14.1 Interpolation of the Term Structure of Implied Volatility 201

    14.2 Watch Your CPU Usage 202

    14.3 Numerical Overflow and Underflow 205

    14.4 A Single Number or a Convergence Diagram? 205

    14.5 Embedded Path Creation 206

    14.6 How Slow is Exp()? 207

    14.7 Parallel Computing And Multi-threading 209

    Bibliography 213

    Index 219