Elementary Calculus of Financial Mathematics (Monographs on Mathematical Modeling & Computation) (Monographs on Mathematical Modeling and Computation)

140 Pages·2008·1.72 MB·English

by  A. J. Roberts

Elementary
Calculus
of Financial
Mathematics
Mathematical Modeling
Editor-in-Chief
and Computation
Richard Haberman
Southern Methodist
About the Series University
The SIAM series on Mathematical Modeling and Computation draws attention
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engineering that are addressed by mathematical modeling and computation;
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and encourages the education of the next generation of applied and computational
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Northwestern University
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Elementary
Calculus
of Financial
Mathematics
A. J. Roberts
University of Adelaide
Adelaide, South Australia, Australia
Society for Industrial and Applied Mathematics
Philadelphia
Copyright © 2009 by the Society for Industrial and Applied Mathematics.
10 9 8 7 6 5 4 3 2 1
All rights reserved. Printed in the United States of America. No part of this book may be
reproduced, stored, or transmitted in any manner without the written permission of the
publisher. For information, write to the Society for Industrial and Applied Mathematics,
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Trademarked names may be used in this book without the inclusion of a trademark
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MATLAB is a registered trademark of The MathWorks, Inc. For MATLAB product
information, please contact The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA
01760-2098 USA, 508-647-7000, Fax: [phone], [email protected],
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Library of Congress Cataloging-in-Publication Data
Roberts, A. J.
Elementary calculus of financial mathematics / A. J. Roberts.
p. cm. -- (Mathematical modeling and computation ; 15)
Includes bibliographical references and index.
ISBN 978-0-898716-67-2
1. Finance--Mathematical models. 2. Stochastic processes. 3. Investments--
Mathematics. 4. Calculus. I. Title.
HG106.R63 2009
332.01'51923--dc22 2008042349
is a registered trademark.
To Barbara, Sam, Ben, and Nicky
for their support over the years
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Contents
Preface ix
ListofAlgorithms xi
1 FinancialIndicesAppeartoBeStochasticProcesses 1
1.1 BrownianmotionisalsocalledaWienerprocess . . . . . . . . . . . . 3
1.2 Stochasticdriftandvolatilityareunique . . . . . . . . . . . . . . . . 9
1.3 BasicnumericssimulateanSDE . . . . . . . . . . . . . . . . . . . . 14
1.4 Abinomiallatticepricescalloption . . . . . . . . . . . . . . . . . . . 20
1.5 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 Ito’sStochasticCalculusIntroduced 39
2.1 Multiplicativenoisereducesexponentialgrowth . . . . . . . . . . . . 39
2.2 Ito’sformulasolvessomeSDEs . . . . . . . . . . . . . . . . . . . . . 43
2.3 TheBlack–Scholesequationpricesoptionsaccurately . . . . . . . . . 48
2.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 TheFokker–PlanckEquationDescribestheProbabilityDistribution 61
3.1 Theprobabilitydistributionevolvesforwardintime . . . . . . . . . . 65
3.2 Stochasticallysolvedeterministicdifferentialequations . . . . . . . . 76
3.3 TheKolmogorovbackwardequationcompletesthepicture . . . . . . 84
3.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4 StochasticIntegrationProvesIto’sFormula 93
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b
4.1 TheItointegral fdW . . . . . . . . . . . . . . . . . . . . . . . . . 95
a
4.2 TheItoformula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.3 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
AppendixA ExtraMATLAB/SCILABCode 115
AppendixB TwoAlternateProofs 119
B.1 Fokker–Planckequation . . . . . . . . . . . . . . . . . . . . . . . . . 119
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viii Contents
B.2 Kolmogorovbackwardequation. . . . . . . . . . . . . . . . . . . . . 121
Bibliography 125
Index 127
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Preface
Welcome! This book leads youon an introductioninto the fascinating realm of fi-
nancialmathematicsanditscalculus. Modernfinancialmathematicsreliesonadeepand
sophisticated theory of random processes in time. Such randomness reflects the erratic
fluctuationsinfinancialmarkets. Itakeonthechallengeofintroducingyoutothecrucial
conceptsneededtounderstandandvaluefinancialoptionsamongsuchfluctuations. This
booksupportsyourlearningwiththebareminimumofnecessaryprerequisitemathematics.
Todeliverunderstandingwithaminimumofanalysis,thebookstartswithagraph-
ical/numerical introduction to how to adapt random walks to describe the typical erratic
fluctuationsoffinancialmarkets.Thensimplenumericalsimulationsbothdemonstratethe
approachandsuggestthesymbologyofstochasticcalculus. Thefinitestepsofthenumeri-
calapproachunderlietheintroductionofthebinomiallatticemodelforevaluatingfinancial
options.
Fluctuationsinafinancialenvironmentmaybankruptbusinessesthatotherwisewould
grow.Discreteanalysisofthisproblemleadstothesurprisinglysimpleextensionofclassic
calculusneededto performstochastic calculus. The key is to replacesquared noise by a
meandrift: ineffect,dW2=dt. Thissimplebutpowerfulruleenablesustodifferentiate,
integrate, solve stochastic differential equations, and to triumphantly derive and use the
Black–Scholesequationtoaccuratelyvaluefinancialoptions.
The first two chapters deal with individualrealizations and simulations. However,
some applications require exploring the distribution of possibilities. The Fokker–Planck
andKolmogorovequationslinkevolvingprobabilitydistributionstostochasticdifferential
equations (SDEs). Such transformationsempower us not only to value financial options
but also to model the natural fluctuations in biology models and to approximately solve
differentialequationsusingstochasticsimulation.
Lastly,theformalrulesusedpreviouslyarejustifiedmorerigorouslybyanintroduc-
tion to a sound definition of stochastic integration. Integration in turn leads to a sound
interpretationofIto’sformulathatwefindsousefulinfinancialapplications.
Prerequisites
Basic algebra, calculus, data analysis, probability and Markov chains are prerequisites
for this course. There will be many times throughoutthis book when you will need the
concepts and techniquesof such courses. Be sure you are familiar with those, and have
appropriatereferencesonhand.
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x Preface
Computer simulations
IncorporatedintothisbookareMATLAB/SCILAB scriptstoenhanceyourabilitytoprobe
the problems and concepts presented and thus to improve learning. You can purchase
MATLABfromtheMathworkscompany,http://www.mathworks.com.SCILABisavailable
forfreeviahttp://www.scilab.org.
A.J.Roberts
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