085 Explorations in Monte Carlo Methods.pdf

Undergraduate Texts in Mathematics

Editorial Board

S. Axler

K.A. Ribet

For other titles published in this series, go to

http://www.springer.com/series/666

Ronald W. Shonkwiler

Franklin Mendivil

Explorations in

Monte Carlo Methods

Ronald W. Shonkwiler

Franklin Mendivil

School of Mathematics

Department of Mathematics and Statistics

Acadia University

Georgia Institute of Technology

Atlanta, GA 30332-0160

Wolfville, NS B4P 2R6

USA

Canada

shonkwiler@math.gatech.edu

franklin.mendivil@acadiau.ca

Editorial Board

S. Axler

K.A. Ribet

Mathematics Department

San Francisco State University

University of California at Berkeley

San Francisco, CA 94132

Berkeley, CA 94720-3840

USA

axler@sfsu.edu

ribet@math.berkeley.edu

ISSN 0172-6056

ISBN 978-0-387-87836-2

e-ISBN 978-0-387-87837-9

DOI 10.1007/978-0-387-87837-9

Springer Dordrecht Heidelberg London New York

Library of Congress Control Number: 2009932312

Mathematics Subject Classification (2000): 65C05 (primary), 68T05, 60J20 (secondary), 60G40 (tertiary)

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Preface

The Monte Carlo method is a technique for analyzing phenomena by

means of computer algorithms that employ, in an essential way, the

generation of random numbers.

A solution by Monte Carlo methods was one of the very rst uses

made of the newly invented digital computer. In a very real sense the

method was born with the computer. From the start, Monte Carlo

methods have been used to solve di cult problems for which no other

solution method was available at the time. This is still the case. In some

cases better methods arose and displaced Monte Carlo; as it should be.

Yet, in many applications Monte Carlo is unsurpassed. It still enjoys

almost exclusive dominion over its original application, simulating com-

plex interactions in any area where quantitative models are possible.

In the meantime Monte Carlo has moved ahead as well, nding new

areas of application and enjoying new resurgence in former areas as a

result of increased computer power. Today Monte Carlo methods are

more widespread than ever.

Monte Carlo methods originated in their modern form with, and

were named by, Stanislaw Ulam and John von Neumann. Later Ulam

went on to expand on the method and champion its use. He inspired

many subsequent adherents who themselves developed and extended

the method. To Stan, as he was known to his friends, Monte Carlo

was just one technique for performing mathematical experiments on

the computer, an idea in which he fervently believed.

In this book we show how Monte Carlo can be used to solve prob-

lems in science, engineering, business, and industry. But most of all,

we intend to have fun. We will use the computer to explore the hidden

and sometimes surprising nature of quantitative systems. From throw-

Preface

ing needles on cracks to playing blackjack, from following sea birds

searching for food to using computer creatures to solve optimization

problems, we will explore what Monte Carlo can do.

Along the way we will learn and improve skills in probability, statis-

tics, programming, mathematics, and even Monte Carlo methods. For

example, the central limit theorem plays a central role in many parts

of the book. And we will learn that computer random numbers are not

random at all; nevertheless, a source of good pseudorandom numbers

is essential to the method.

The initial mathematical skills the student should have are calculus

through integration and matrix arithmetic. A familiarity with program-

ming is helpful, but the programming in this text starts at an elemen-

tary level and proceeds gradually in scope. A computer inclined mind

is probably more important. Our demonstration programs are amply

commented.

Programming is at the heart of Monte Carlo methods and will be the

primary activity of our work in this text. All that we learn and discover

emerges in the display of the results of our programs. Toward that end

we provide a large number of problems for computer solution at the end

of each chapter. The statement of each problem is prexed by a number

in parentheses indicating the relative di culty or time requirement for

that problem. These are only estimates, however; your perception may

be diﬀerent, but hopefully not by too much. Keep in mind that creating

and debugging code always takes more time than expected. Hofstadters

Law states: it always takes longer than you expect, even when you take

Hofstadters Law into account.

With regard to programming, quick, short, and simple solutions is

a major feature of Monte Carlo methods. For example the program on

page 4 for simulating the Buﬀon needle problem mentioned above is

just seven lines. Generally, programs in the earlier chapters tend to be

small and simple. Later on, they become a little more elaborate. Thus

the program for pricing options via simulation on page 178 is a bit

longer at 24 lines.

Besides the numerical computations, the results have to be graph-

ically displayed. Most convenient is a software package that can do

both. We have chosen to use Matlab for illustration purposes through-

out. Rather than using pseudocode for this purpose, Matlab is itself

easy to understand, and the code presented can be directly run within

Matlab. The included code produced many of the gures in the text.

However, while Matlab is good as a mathematics and graphics pack-

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