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Simulation and Monte Carlo: With Applications in Finance and MCMC

J. S. Dagpunar
John Wiley
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Ita Cirovic Donev
, on

Simulation and Monte Carlo methods have long roots in finance. Today, with the introduction of more complex financial instruments and contracts, the need for more precise estimates is even greater. There is quite a publishing stream of books that deal with computational techniques in finance. This book is an addition to this list. It is an excellent addition. It seems that quite a number of such books go into publishing for other purposes than to educate the readers and actually allow them to make use of these techniques in the field. Dagnopar's book is an exception. It is not an advanced book, which is why I think it will be a great contribution. There are some excellent books on this topic but they are all somewhat advanced, which leaves the novice audience out.

This book is excellent for students and practitioners who don't have previous experience with simulation methods. Since it is an applied book, it is natural that it is divided into two parts, one dealing with the theory while the other presents some examples from the chosen field of application, in this case finance.

Part I of the book includes chapters 1 to 5. Chapter 1 gives brief introduction to the subject. It is a bit sparse, but a rough general understanding of the topic should be obtained, especially if the reader tries out the actual computational examples. Next the role of uniform random variables is explained, along with the methods for generating random variates. These sections are quite detailed, with Maple code to illustrate some concepts. If the reader has an appropriate background in statistics, then things should fall in place with one reading. Readers who don't feel comfortable with basic statistical distributions should first review that material in some other statistics book. The author presents examples throughout the text with very detailed solutions. The proofs of the theorems are also quite detailed and easy to follow. Part I concludes with methods for variance reduction.

Part II presents some examples in finance and the concept of Markov Chain Monte Carlo (MCMC). Option theory is one area that uses quite a lot of simulation; it is the most natural choice for the presentation of applications. It would be advantageous for the reader to have some prior knowledge of the basics of option theory (at least what options are and the mathematical definitions of options) and the connection between stochastic processes and option theory, as the author doesn't spend much time explaining these bits of theory. He goes on directly, after a short introduction, to problems in options pricing. Overall, the applications, including MCMC, are explained with enough detail to obtain good understanding of the concepts. With the completion of Maple examples within the text things should be much clearer.

Exercises are provided at the end of each chapter. Mostly they are computational (based on Maple ). These exercises should be of great help to students, especially as the solutions are provided at the end of the book.

One of the drawbacks of the book is the choice of the computing environment, which is Maple, which is not really a popular computational environment within institutions in finance, especially for this type of computations. However, if the reader is so inclined, he/she can rewrite the code in some other more appropriate language, such as C++, Java, or other. It would be good practice.

The book is suitable for a wide audience. Students (undergraduate and first year graduate) should benefit a lot from this book, especially if they do the computational part. For novice practitioners in finance this book can be a little gem. Once readers finish this book I believe they would be well prepared to embark on some more advanced book such as Monte Carlo Methods in Financial Engineering by P. Glasserman or Monte Carlo Statistical Methods by C.P. Robert and G. Casella. I think this book could also serve as a main textbook in an introductory course on simulation and Monte Carlo methods.

Ita Cirovic Donev is a PhD candidate at the University of Zagreb. She hold a Masters degree in statistics from Rice University. Her main research areas are in mathematical finance; more precisely, statistical mehods of credit and market risk. Apart from the academic work she does consulting work for financial institutions.


1 Introduction to simulation and Monte Carlo.

1.1 Introduction.

1.2 Evaluating a de.nite integral.

1.3 Monte Carlo is integral estimation.

1.4 An example.

1.5 A simulation using Maple.

1.6 Problems.

2 Uniform random numbers.

2.1 Linear congruential generators.

2.1.1 Mixed linear congruential generators.

2.1.2 Multiplicative linear congruential generators.

2.2 Theoretical tests for random numbers.

2.2.1 Problems of increasing dimension.

2.3 Shu­ ed generator.


2.4 Empirical tests.

2.4.1 Frequency test.

2.4.2 Serial test.

2.4.3 Other empirical tests.

2.5 Combinations of generators.

2.6 The seed(s) in a random number generator.

2.7 Problems.

3 General methods for generating random variates.

3.1 Inversion of the cumulative distribution function.

3.2 Envelope rejection.

3.3 Ratio of uniforms method.

3.4 Adaptive rejection sampling.

3.5 Problems.

4 Generation of variates from standard distributions.

4.1 Standard normal distribution.

4.1.1 Box-Müller method.

4.1.2 An improved envelope rejection method.

4.2 Lognormal distribution.

4.3 Bivariate normal density.

4.4 Gamma distribution.

4.4.1 Cheng.s log-logistic method.

4.5 Beta distribution.

4.5.1 Beta log-logistic method.

4.6 Chi-squared distribution.

4.7 Student.s t-distribution.

4.8 Generalized inverse Gaussian distribution.

4.9 Poisson distribution.

4.10 Binomial distribution.

4.11 Negative binomial distribution.

4.12 Problems.

5 Variance reduction.

5.1 Antithetic variates.

5.2 Importance sampling.

5.2.1 Exceedance probabilities for sums of i.i.d. randomvari-ables.

5.3 Strati.ed sampling.

5.3.1 A Strati.cation example.

5.3.2 Post strati.cation.

5.4 Control variates.

5.5 Conditional Monte Carlo.

5.6 Problems.

6 Simulation and.nance.

6.1 Brownian motion.

6.2 Asset price movements.

6.3 Pricing simple derivatives and options.

6.3.1 European call.

6.3.2 European put.

6.3.3 Continuous income.

6.3.4 Delta hedging.

6.3.5 Discrete hedging.

6.4 Asian options.

6.4.1 Naive simulation.

6.4.2 Importance and strati.ed version.

6.5 Basket options.

6.6 Stochastic volatility.

6.7 Problems.

7 Discrete event simulation.

7.1 Poisson process.

7.2 Time dependent Poisson process.

7.3 Poisson processes in the plane.

7.4 Markov chains.

7.4.1 Discrete time Markov chains.

7.4.2 Continuous time Markov chains.

7.5 Regenerative analysis.

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

7.7 Simulating a hospital ward.

7.8 Problems.

8 Markov chain Monte Carlo.

8.1 Bayesian statistics.

8.2 Markov chains and the Metropolis-Hastings 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.

8.4.2 Capture-recapture.

8.4.3 Minimal repair.

8.5 Other aspects of Gibbs sampling.

8.5.1 Slice sampling.

8.5.2 Completions.

8.6 Problems.

9 Solutions.

9.1 Solutions 1.

9.2 Solutions 2.

9.3 Solutions 3.

9.4 Solutions 4.

9.5 Solutions 5.

9.6 Solutions 6.

9.7 Solutions 7.

9.8 Solutions 8.

10 References.