Monte Carlo Simulation with Applications to Finance

291 Pages·2012·2.005 MB·English

by  Hui Wang

Mathematics
Hui Wang
Monte Carlo Simulation
with Applications to Finance
M
Developed from the author’s course on Monte Carlo simulation at
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Brown University, Monte Carlo Simulation with Applications to p n
Finance provides a self-contained introduction to Monte Carlo p t Monte Carlo
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methods in financial engineering. Only requiring some familiarity with i
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probability and statistics, the book keeps much of the mathematics
a a
at an informal level and avoids technical measure-theoretic jargon to t r Simulation with
i l
provide a practical understanding of the basics. It includes a large o o

number of examples as well as MATLAB® coding exercises that are n S
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designed in a progressive manner so that no prior experience with m Applications to
t
MATLAB is needed.
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The author first presents the necessary mathematical tools for F a
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simulation, arbitrary free option pricing, and the basic implementation in io Finance
of Monte Carlo schemes. He then describes variance reduction a n
techniques, including control variates, stratification, conditioning, n
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importance sampling, and cross-entropy. The text concludes with
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stochastic calculus and the simulation of diffusion processes. h

Features
• Presents common variance reduction techniques as well as the
cross-entropy method
• Covers the simulation of diffusion process models
• Requires minimal background in mathematics and finance
• Contains numerous examples of option pricing, risk analysis,
and sensitivity analysis
• Includes many hand-and-paper and MATLAB coding exercises
at the end of every chapter
W
a
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g
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Monte Carlo
Simulation with
Applications to
Finance
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CHAPMAN & HALL/CRC

Financial Mathematics Series
Aims and scope:
The field of financial mathematics forms an ever-expanding slice of the financial sector. This series
aims to capture new developments and summarize what is known over the whole spectrum of this
field. It will include a broad range of textbooks, reference works and handbooks that are meant to
appeal to both academics and practitioners. The inclusion of numerical code and concrete real-
world examples is highly encouraged.
Series Editors
M.A.H. Dempster Dilip B. Madan Rama Cont
Centre for Financial Research Robert H. Smith School Center for Financial
Department of Pure of Business Engineering
Mathematics and Statistics University of Maryland Columbia University
University of Cambridge New York
Published Titles
American-Style Derivatives; Valuation and Computation, Jerome Detemple
Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing,
 Pierre Henry-Labordère
An Introduction to Exotic Option Pricing, Peter Buchen
Credit Risk: Models, Derivatives, and Management, Niklas Wagner
Engineering BGM, Alan Brace
Financial Modelling with Jump Processes, Rama Cont and Peter Tankov
Interest Rate Modeling: Theory and Practice, Lixin Wu
Introduction to Credit Risk Modeling, Second Edition, Christian Bluhm, Ludger Overbeck, and
 Christoph Wagner
Introduction to Stochastic Calculus Applied to Finance, Second Edition,
 Damien Lamberton and Bernard Lapeyre
Monte Carlo Methods and Models in Finance and Insurance, Ralf Korn, Elke Korn,
 and Gerald Kroisandt
Monte Carlo Simulation with Applications to Finance, Hui Wang
Numerical Methods for Finance, John A. D. Appleby, David C. Edelman, and John J. H. Miller
Option Valuation: A First Course in Financial Mathematics, Hugo D. Junghenn
Portfolio Optimization and Performance Analysis, Jean-Luc Prigent
Quantitative Fund Management, M. A. H. Dempster, Georg Pflug, and Gautam Mitra
Risk Analysis in Finance and Insurance, Second Edition, Alexander Melnikov
Robust Libor Modelling and Pricing of Derivative Products, John Schoenmakers
Stochastic Finance: A Numeraire Approach, Jan Vecer
Stochastic Financial Models, Douglas Kennedy
Structured Credit Portfolio Analysis, Baskets & CDOs, Christian Bluhm and Ludger Overbeck
Understanding Risk: The Theory and Practice of Financial Risk Management, David Murphy
Unravelling the Credit Crunch, David Murphy
Proposals for the series should be submitted to one of the series editors above or directly to:
CRC Press, Taylor & Francis Group
4th, Floor, Albert House
1-4 Singer Street
London EC2A 4BQ
UK
K12713_FM.indd 2 4/26/12 5:56 PM
Monte Carlo
Simulation with
Applications to
Finance
Hui Wang
Brown University
Providence, Rhode Island, USA
K12713_FM.indd 3 4/26/12 5:56 PM
MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does
not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MAT-
LAB® software or related products does not constitute endorsement or sponsorship by The MathWorks
of a particular pedagogical approach or particular use of the MATLAB® software.
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Preface
This book can serve as the text for a one-semester course on Monte Carlo
simulation. Theintendedaudienceisadvancedundergraduatestudentsor
studentsinmaster’sprogramswhowishtolearnthebasicsofthisexciting
topicanditsapplications tofinance.
The book is largely self-contained. The only prerequisite is some expe-
rience with probability and statistics. Prior knowledge on option pricing
is helpful but not essential. As in any study of Monte Carlo simulation,
codingisanintegralpartandcannotbeignored.Thebookcontainsalarge
numberofMATLAB(cid:13)R codingexercises.Theyaredesignedinaprogressive
mannersothatnopriorexperiencewithMATLABisrequired.
Much of the mathematics in the book is informal. For example, ran-
domvariablesaresimplydefinedtobefunctionsonthesamplespace,even
though they should bemeasurablewithrespect toappropriateσ-algebras;
exchanging theorderofintegrations iscarriedoutliberally,eventhough it
should be justified by theTonelli–Fubini Theorem. The motivation for do-
ing so is to avoid the technical measure theoretic jargon, which is of little
concern in practice and does not help much to further the understanding
ofthetopic.
The book is an extension of the lecture notes thatI havedevelopedfor
an undergraduate course on MonteCarlo simulation at BrownUniversity.
Iwouldliketothankthestudentswhohavetakenthecourse,aswellasthe
Division ofAppliedMathematicsatBrown,fortheirsupport.
HuiWang
Providence,RhodeIsland
January, 2012
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vi
MATLAB(cid:13)R is a trademark of The MathWorks, Inc. For product informa-
tion,pleasecontact:
TheMathWorks,Inc.
3AppleHillDrive
Natick,MA01760-2098USA
Tel:[phone]
Fax: [phone]
E-mail:[email protected]
Web:www.mathworks.com
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Contents
1 ReviewofProbability 1
1.1 Probability Space . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 IndependenceandConditional Probability . . . . . . . . . . 2
1.3 RandomVariables . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 RandomVectors . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 ConditionalDistributions . . . . . . . . . . . . . . . . . . . . 18
1.6 ConditionalExpectation . . . . . . . . . . . . . . . . . . . . . 21
1.7 ClassicalLimitTheorems . . . . . . . . . . . . . . . . . . . . . 23
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 BrownianMotion 31
2.1 BrownianMotion . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Running MaximumofBrownianMotion. . . . . . . . . . . . 33
2.3 DerivativesandBlack–ScholesPrices . . . . . . . . . . . . . . 35
2.4 Multidimensional BrownianMotions . . . . . . . . . . . . . . 43
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 ArbitrageFreePricing 51
3.1 ArbitrageFreePrinciple . . . . . . . . . . . . . . . . . . . . . 51
3.2 AssetPricing withBinomialTrees . . . . . . . . . . . . . . . . 53
3.3 TheBlack–ScholesModel . . . . . . . . . . . . . . . . . . . . 61
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 MonteCarloSimulation 67
4.1 BasicsofMonteCarloSimulation . . . . . . . . . . . . . . . . 67
4.2 StandardErrorandConfidenceInterval . . . . . . . . . . . . 69
4.3 ExamplesofMonteCarloSimulation . . . . . . . . . . . . . . 72
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
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viii CONTENTS
5 GeneratingRandomVariables 87
5.1 InverseTransformMethod . . . . . . . . . . . . . . . . . . . . 87
5.2 Acceptance-RejectionMethod . . . . . . . . . . . . . . . . . . 90
5.3 SamplingMultivariateNormalDistributions . . . . . . . . . 93
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6 VarianceReductionTechniques 103
6.1 AntitheticSampling . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2 ControlVariates . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3 StratifiedSampling . . . . . . . . . . . . . . . . . . . . . . . . 115
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7 ImportanceSampling 133
7.1 BasicIdeasofImportanceSampling . . . . . . . . . . . . . . 133
7.2 TheCross-Entropy Method . . . . . . . . . . . . . . . . . . . 144
7.3 Applications toRiskAnalysis . . . . . . . . . . . . . . . . . . 164
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8 StochasticCalculus 183
8.1 StochasticIntegrals . . . . . . . . . . . . . . . . . . . . . . . . 184
8.2 Itoˆ Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8.3 StochasticDifferential Equations . . . . . . . . . . . . . . . . 194
8.4 Risk-NeutralPricing . . . . . . . . . . . . . . . . . . . . . . . 197
8.5 Black–ScholesEquation . . . . . . . . . . . . . . . . . . . . . 200
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
9 SimulationofDiffusions 205
9.1 EulerScheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
9.2 EliminatingDiscretizationError. . . . . . . . . . . . . . . . . 207
9.3 RefinementsofEulerScheme . . . . . . . . . . . . . . . . . . 208
9.4 TheLampertiTransform . . . . . . . . . . . . . . . . . . . . . 209
9.5 NumericalExamples . . . . . . . . . . . . . . . . . . . . . . . 211
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
10 SensitivityAnalysis 237
10.1 CommonlyUsedGreeks . . . . . . . . . . . . . . . . . . . . . 238
10.2 MonteCarloSimulationofGreeks . . . . . . . . . . . . . . . 239
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
A MultivariateNormalDistributions 257
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CONTENTS ix
B AmericanOptionPricing 259
B.1 TheValueofanAmerican Option . . . . . . . . . . . . . . . . 259
B.2 Dynamic ProgrammingandBinomialTrees . . . . . . . . . . 261
B.3 Diffusion Models:BinomialApproximation . . . . . . . . . . 264
C OptionPricingFormulas 269
Bibliography 277
Index 280
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