Simulation and Monte Carlo With applications in finance and MCMC pdf

Simulation and Monte Carlo With applications in finance and MCMC J. S. Dagpunar School of Mathematics University of Edinburgh, UK b 21 2006 il /S i ffi Simulation and Monte Carlo b 21 2006 il /S ii ffi b 21 2006 il /S iii ffi Simulation and Monte Carlo With applications in finance and MCMC J. S. Dagpunar School of Mathematics University of Edinburgh, UK b 21 2006 il /S i ffi Copyright © 2007 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone +44 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wiley.com All Rights Reserved. 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Library of Congress Cataloging in Publication Data British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13: 978-0-470-85494-5 (HB) 978-0-470-85495-2 (PB) ISBN-10: 0-470-85494-4 (HB) 0-470-85495-2 (PB) Typeset in 10/12pt Times by Integra Software Services Pvt. Ltd, Pondicherry, India Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production. b 21 2006 il /S ffi To the memory of Jim Turner, Veterinary surgeon, 1916–2006 b 21 2006 il /S i ffi Contents Preface xi Glossary xiii 1 Introduction to simulation and Monte Carlo 1 1.1 Evaluating a definite integral 2 1.2 Monte Carlo is integral estimation 4 1.3 An example 5 1.4 A simulation using Maple 7 1.5 Problems 13 2 Uniform random numbers 17 2.1 Linear congruential generators 18 2.1.1 Mixed linear congruential generators 18 2.1.2 Multiplicative linear congruential generators 22 2.2 Theoretical tests for random numbers 25 2.2.1 Problems of increasing dimension 26 2.3 Shuffled generator 28 2.4 Empirical tests 29 2.4.1 Frequency test 29 2.4.2 Serial test 30 2.4.3 Other empirical tests 30 2.5 Combinations of generators 31 2.6 The seed(s) in a random number generator 32 2.7 Problems 32 3 General methods for generating random variates 37 3.1 Inversion of the cumulative distribution function 37 3.2 Envelope rejection 40 3.3 Ratio of uniforms method 44 3.4 Adaptive rejection sampling 48 3.5 Problems 52 4 Generation of variates from standard distributions 59 4.1 Standard normal distribution 59 4.1.1 Box–Müller method 59 4.1.2 An improved envelope rejection method 61 4.2 Lognormal distribution 62 viii Contents 4.3 Bivariate normal density 63 4.4 Gamma distribution 64 4.4.1 Cheng’s log-logistic method 65 4.5 Beta distribution 67 4.5.1 Beta log-logistic method 67 4.6 Chi-squared distribution 69 4.7 Student’s t distribution 69 4.8 Generalized inverse Gaussian distribution 71 4.9 Poisson distribution 73 4.10 Binomial distribution 74 4.11 Negative binomial distribution 74 4.12 Problems 75 5 Variance reduction 79 5.1 Antithetic variates 79 5.2 Importance sampling 82 5.2.1 Exceedance probabilities for sums of i.i.d. random variables 86 5.3 Stratified sampling 89 5.3.1 A stratification example 92 5.3.2 Post stratification 96 5.4 Control variates 98 5.5 Conditional Monte Carlo 101 5.6 Problems 103 6 Simulation and finance 107 6.1 Brownian motion 108 6.2 Asset price movements 109 6.3 Pricing simple derivatives and options 111 6.3.1 European call 113 6.3.2 European put 114 6.3.3 Continuous income 115 6.3.4 Delta hedging 115 6.3.5 Discrete hedging 116 6.4 Asian options 118 6.4.1 Naive simulation 118 6.4.2 Importance and stratified version 119 6.5 Basket options 123 6.6 Stochastic volatility 126 6.7 Problems 130 7 Discrete event simulation 135 7.1 Poisson process 136 7.2 Time-dependent Poisson process 140 7.3 Poisson processes in the plane 141 7.4 Markov chains 142 7.4.1 Discrete-time Markov chains 142 7.4.2 Continuous-time Markov chains 143 [...]... population Simulation and Monte Carlo: With applications in finance and MCMC © 2007 John Wiley & Sons, Ltd J S Dagpunar 2 Introduction to simulation and Monte Carlo Since simulations provide an estimate of a parameter of interest, there is always some error, and so a quantification of the precision is essential, and forms an important part of the design and analysis of the experiment 1.1 Evaluating a definite... grounding in the principles of simulation Chapter 8 deals with the other burgeoning area of simulation, namely Markov chain Monte Carlo and its use in Bayesian statistics Here, I have been influenced by the works of Robert and Casella (2004) and Gilks et al (1996) I have also included several examples from the reliability area since the repair and maintenance of systems is another area that interests... approach is to generate pseudo-random numbers at run-time, using a specified deterministic recurrence equation on integers This allows fast generation, eliminates the storage problem, and gives a reproducible sequence However, great care is needed in selecting an appropriate recurrence, to make the sequence appear random Simulation and Monte Carlo: With applications in finance and MCMC © 2007 John Wiley &... exotic derivatives and in Bayesian estimation In a stroke this has caused a renaissance in simulation In Chapter 6, I have been influenced by the work of Glasserman (2004), particularly his work combining importance and stratified sampling I hope in Sections 6.4.2 and 6.5 that I have provided a more direct and accessible way of deriving and applying such variance reduction methods to Asian and basket options... Regenerative analysis Simulating a G/G/1 queueing system using the three-phase method Simulating a hospital ward Problems 144 146 149 151 8 Markov chain Monte Carlo 8.1 Bayesian statistics 8.2 Markov chains and the Metropolis–Hastings (MH) algorithm 8.3 Reliability inference using an independence sampler 8.4 Single component Metropolis–Hastings and Gibbs sampling 8.4.1 Estimating multiple failure rates... acceptance probabilities 229 Appendix 4: Random variate generators (standard distributions) 233 Appendix 5: Variance reduction 239 Appendix 6: Simulation and finance 249 Appendix 7: Discrete event simulation 283 Appendix 8: Markov chain Monte Carlo 299 References 325 Index 329 Preface This book provides an introduction to the theory and practice of Monte Carlo and Simulation methods It arises from a 20... very small 14 Introduction to simulation and Monte Carlo 5 An intoxicated beetle moves over a cardboard unit circle x2 + y2 < 1 The x y plane is horizontal and the cardboard is suspended above a wide open jar of treacle In the √ √ time interval t t + t it moves by amounts x = Z1 1 t and y = Z2 2 t along the x and y axes where Z1 and Z2 are independent standard normal random variables and 1 and 2 are specified... particularly in finance and statistical inference In general, simulation may be appropriate when there is a problem that is too difficult to solve analytically In a simulation a controlled sampling experiment is conducted on a computer using random numbers Statistics arising from the sampling experiments (examples are sample mean, sample proportion) are used to estimate some parameters of interest in the original... uses U 0 1 random numbers that are statistically independent A Monte Carlo method is a method of estimating the value of an integral (or a sum) using the realized values from a simulation It exploits the connection between an integral (or a sum) and the expectation of a function of a(some) random variable(s) 1.3 An example Let us now examine how a Monte Carlo approach can be used in the following problem... starts with the Maple prompt ‘>’ The procedure is written within an execution group Each line of code is terminated by a semicolon However, anything appearing after the ‘#’ symbol is not executed This allows programmer comments to be added Use the ‘shift’ and ‘return’ keys to obtain a fresh line within the procedure The procedure terminates with a semicolon and successful entry of the procedure results in . some parameters of interest in the original problem, system, or population. Simulation and Monte Carlo: With applications in finance and MCMC J. S. Dagpunar ©. /S i ffi Simulation and Monte Carlo b 21 2006 il /S ii ffi b 21 2006 il /S iii ffi Simulation and Monte Carlo With applications in finance and MCMC J. S.