Monte Carlo Methods in Finance
Description
The book is packed with numerous examples using real world data and is supplied with a CD to aid in the use of the examples.Dieses Buch ist ein handlicher und praktischer Leitfaden zur Monte Carlo Simulation (MCS). Er gibt eine Einführung in Standardmethoden und fortgeschrittene Verfahren, um die zunehmende Komplexität derivativer Portfolios besser zu erfassen. Das hier behandelte Spektrum von MCS-Anwendungen reicht von der Preisbestimmung komplexerer Derivate, z.B. von amerikanischen und asiatischen Optionen, bis hin zur Messung des Value at Risk und zur Modellierung komplexer Marktdynamik. Anhand einer Vielzahl praktischer Beispiele wird erläutert, wie man Monte Carlo Methoden einsetzt. Dabei gehen die Autoren zunächst auf die Grundlagen und danach auf fortgeschrittene Techniken ein. Darüber hinaus geben sie nützliche Tipps und Hinweise für das Entwickeln und Arbeiten mit MCS-Methoden. Die Autoren sind Experten auf dem Gebiet der Monte Carlo Simulation und verfügen über langjährige Erfahrung im Umgang mit MCS-Methoden. Die Begleit-CD enthält Excel Muster Spreadsheets sowie VBA und C++ Code Snippets, die der Leser installieren und so mit den im Buch beschriebenen Beispiele frei experimentieren kann. "Monte Carlo Methods in Finance" - ein unverzichtbares Nachschlagewerk für quantitative Analysten, die bei der Bewertung von Optionspreisen und Riskmanagement auf Modelle zurückgreifen müssen.
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
Peter Jackel currently works at Commerzbank Securities in London as a quant in the front office product development and derivatives modelling group. Prior to that he worked within the NatWest Group/Royal Bank of Scotland Quantitative Research Centre. He started his career in finance with his employment at Nikko Securities' London operation. Monte Carlo Methods in Finance is an important reference for those working in investment banks, insurance and strategic management consultancy. Of particular importance are the many known variance reduction methods, and they are duly covered, not only in their own right, but also with respect to their potential combinations, and in the direct context of realistic applications. Most notably, the issue of the reliability of low-discrepancy numbers in high dimensions is discussed in detail. The book also contains an introduction to the theory of copule as an extension to the modelling of correlation of financial securities. An entire chapter is dedicated to the evaluation of interest rate derivatives in the Brace-Gatarek-Musiela/Jamshidian framework by the aid of fast-convergence Monte Carlo simulations. What's more, for the first time, this book also gives a description of the construction of non-recombining trees. Whilst non-recombining trees are usually not viable in a production environment, they often are the very tool of last resort when Monte Carlo approximations to problems such as Bermudan swaptions are to be tested, and the tricks for the construction of non-recombining trees presented in this book are invaluable for that purpose. "There is no book on the market to compare with Dr Jackel's. All the techniques, the tricks, the pitfalls of this important methodology are covered in detail and with great insight. This is no book on abstract theory, Dr Jackel is a practitioner who has implemented every single one of these ideas. He has done all the hard work, so you don't have to." -Paul Wilmott"Few expert practitioners also have the academic expertise to match Peter Jackel's in this area, let alone take the trouble to write a most accessible, comprehensive and yet self contained text. This book is a delight to read and contains a wealth of information that is essential for anyone involved with implementing Monte Carlo methods in finance." -Professor Carol Alexander, ISMA Centre, University of Reading, UK
" This book is a very welcome addition to the growing literature on applied quantitative methods in finance. Dr Jackel has done the field a service in combining both a thorough review of 'standard' material with techniques that were learned on the job as a quant at top financial institutions." -Michael Curran, Quantin' Leap
Based on the author's own experience, Monte Carlo Methods in Finance adopts a practical flavour throughout, the emphasis being on financial modelling and derivatives pricing. Numerous real world examples help the reader foster an intuitive grasp of the mathematical and numerical techniques needed to solve particular financial problems. At the same time, the book tries to give a detailed explanation of the theoretical foundations of the various methods and algorithms presented.
Monte Carlo methods have been used in the financial community for many years for addressing complex financial calculations. Recent advances by both practitioners and academic researchers in the area of fast convergence methods, together with the improvements achieved by the manufacturers of computer hardware, make Monte Carlo simulations more and more frequently the method of choice. In this long needed book on modern Monte Carlo methods in finance, Peter Jackel provides an introduction to many of the leading edge techniques available.
PUBLISHER:
Wiley
ISBN-13:
9780471497417
BINDING:
Hardback
BISAC:
BUSINESS & ECONOMICS
BOOK DIMENSIONS:
Dimensions: 165.10(W) x Dimensions: 246.40(H) x Dimensions: 22.90(D)
AUDIENCE TYPE:
General/Adult
LANGUAGE:
English