Simulation and Monte Carlo With applications in finance and MCMC pdf

Introduction to simulation and Monte Carlo A simulation is an experiment, usually conducted on a computer, involving the use of random numbers.. A random number stream is a sequence of s

Simulation and Monte Carlo

With applications in finance and MCMC

J S Dagpunar

School of Mathematics University of Edinburgh, UK

Simulation and Monte Carlo

Simulation and Monte Carlo

With applications in finance and MCMC

J S Dagpunar

School of Mathematics University of Edinburgh, UK

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ISBN-13: 978-0-470-85494-5 (HB) 978-0-470-85495-2 (PB)

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Contents

4.3 Bivariate normal density 63

Contents ix

7.6 Simulating a G/G/1 queueing system using the three-phase method 146

8.2 Markov chains and the Metropolis–Hastings (MH) algorithm 159

8.4 Single component Metropolis–Hastings and Gibbs sampling 165

The intention is that this be a practical book that encourages readers to write andexperiment with actual simulation models The choice of programming environment,Maple, may seem strange, perhaps even perverse It arises from the fact that at Edinburghall mathematics students are conversant with it from year 1 I believe this is true of manyother mathematics departments The disadvantage of slow numerical processing in Maple

is neutralized by the wide range of probabilistic, statistical, plotting, and list processingfunctions available A large number of specially written Maple procedures are available

on the website accompanying this book (www.wiley.com/go/dagpunar_simulation) Theyare also listed in the Appendices.1

The content of the book falls broadly into two halves, with Chapters 1 to 5 mostlycovering the theory and probabilistic aspects, while Chapters 6 to 8 cover three applicationareas Chapter 1 gives a brief overview of the breadth of simulation All problems at theend of this chapter involve the writing of Maple procedures, and full solutions are given

in Appendix 1 Chapter 2 concerns the generation and assessment of pseudo-randomnumbers Chapter 3 discusses three main approaches to the sampling (generation) ofrandom variates from distributions These are: inversion of the distribution function, theenvelope rejection method, and the ratio of uniforms method It is recognized that manyother methods are available, but these three seem to be the most frequently used, andthey have the advantage of leading to easily programmed algorithms Readers interested

in the many other methods are directed to the excellent book by Devroye (1986) or anearlier book of mine (Dagpunar, 1988a) Two short Maple procedures in Appendix 3allow readers to quickly ascertain the efficiency of rejection type algorithms Chapter 4deals with the generation of variates from standard distributions The emphasis is onshort, easily implemented algorithms Where such an algorithm appears to be fasterthan the corresponding one in the Maple statistics package, I have given a listing inAppendix 4 Taken together, I hope that Chapters 3 and 4 enable readers to understandhow the generators available in various packages work and how to write algorithms fordistributions that either do not appear in such packages or appear to be slow in execution.Chapter 5 introduces variance reduction methods Without these, many simulations areincapable of giving precise estimates within a reasonable amount of processing time.Again, the emphasis is on an empirical approach and readers can use the procedures in

1 The programs are provided for information only and may not be suitable for all purposes Neither the author nor the publisher is liable, to the fullest extent permitted by law, for any failure of the programs to meet the user’s requirements or for any inaccuracies or defects in the programs.

Appendix 5 to illustrate the efficacy of the various designs, including importance andstratified sampling.

Chapters 6 and 8, on financial mathematics and Markov chain Monte Carlo methodsrespectively, would not have been written 10 years ago Their inclusion is a result ofthe high-dimensional integrations to be found in the pricing of exotic derivatives and inBayesian estimation In a stroke this has caused a renaissance in simulation In Chapter 6, Ihave been influenced by the work of Glasserman (2004), particularly his work combiningimportance and stratified sampling I hope in Sections 6.4.2 and 6.5 that I have provided amore direct and accessible way of deriving and applying such variance reduction methods

to Asian and basket options Another example of high-dimensional integrations arises

in stochastic volatility and Section 6.6 exposes the tip of this iceberg Serious financialengineers would not use Maple for simulations Nevertheless, even with Maple, it isapparent from the numerical examples in Chapter 6 that accurate results can be obtained

in a reasonable amount of time when effective variance reduction designs are employed

I also hope that Maple can be seen as an effective way of experimenting with variousmodels, prior to the final construction of an efficient program in C++ or Java, say TheMaple facility to generate code in, say, C++ or Fortran is useful in this respect.Chapter 7 introduces discrete event simulation, which is perhaps best known tooperational researchers It starts with methods of simulating various Markov processes,both in discrete and continuous time It includes a discussion of the regenerative method

of analysing autocorrelated simulation output The simulation needs of the operationalresearcher, the financial engineer, and the Bayesian statistician overlap to a certain extent,but it is probably true to say that no single computing environment is ideal for allapplication fields An operational researcher might progress from Chapter 7 to make use

of one of the powerful purpose-built discrete event simulation languages such as SimscriptII.5 or Witness If so, I hope that the book provides a good grounding in the principles

of simulation

Chapter 8 deals with the other burgeoning area of simulation, namely Markov chainMonte 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 anotherarea that interests me Maple has been quite adequate for the examples discussed in thischapter For larger hierarchical systems a purpose-built package such as BUGS is theanswer

There are problems at the end of each chapter and solutions are given to selectedones A few harder problems have been designated accordingly In the text and problems,numerical answers are frequently given to more significant figures than the data wouldwarrant This is done so that independent calculations may be compared with the onesappearing here

I am indebted to Professor Alastair Gillespie, head of the School of Mathematics,Edinburgh University, for granting me sabbatical leave for the first semester of the2005–2006 session I should also like to acknowledge the several cohorts of simulationstudents that provided an incentive to write this book Finally, my thanks to Angie forher encouragement and support, and for her forbearance when I was not there

beta  1 A random variable that is beta distributed with p.d.f f x=

 +  x−11− x −1/     1≥ x ≥ 0, where  > 0

 > 0

binomial n p A binomially distributed random variable

≥ 0 Standard Brownian motion

c.d.f Cumulative distribution function

CovfX Y  The covariance of X and Y where f x y is the joint p.d.f./p.m.f

of X and Y (the subscript is often dropped)

Exp   A r.v that has the p.d.f f x= e− x x≥ 0, where > 0

EfX The expectation of a random variable X that has the p.d.f or p.m.f

f (the subscript is often dropped)

fXx The p.d.f or p.m.f of a random variable X (the subscript is often

dropped)

FXx The c.d.f of a random variable X

FXx Complementary cumulative distribution function = 1 − FXx gamma   A gamma distributed r.v with the p.d.f f x= x−1e− x/

i.i.d Identically and independently distributed

negbinom k p A negative binomial r.v with the p.d.f f x∝ px1− pk x=

p.d.f Probability density function

Poisson  A r.v distributed as a Poisson with the p.m.f f x= xe−x/x! x =

0 1    , where  > 0

P X < x Probability that the random variable X is less than x

P X= x Probability that a (discrete) random variable equals x

1 This can also refer to the distribution itself This applies to all corresponding random variable names in this list.

support f  ∈  f x = 0

U 0 1 A continuous r.v that is uniformly distributed in the interval (0, 1).VarfX The variance of a random variable X that has the p.d.f or p.m.f f

(the subscript is often dropped)

v.r.r Variance reduction ratio

Weibull   A Weibull distributed random variable with the p.d.f f x=

 x−1e− ... some parameters of

interest in the original problem, system, or population

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