Monte Carlo Methods in Finance

235 Pages·2002·16.016 MB·English

by  Peter Jaeckel

Monte Carlo Methods in Finance
Peter Ja¨ckel
24th November 2001
ii
Contents
Preface ix
MathematicalNotation xiii
1 Introduction 1
2 ThemathematicsbehindMonteCarlomethods 5
2.1 Afewbasictermsinprobabilityandstatistics . . . . . . . . . . . . . . . . . . . . . . 5
2.2 MonteCarlosimulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 MonteCarlosupremacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Multidimensionalintegration . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Somecommondistributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Kolmogorov’sstronglaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Thecentrallimittheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Thecontinuousmappingtheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.7 ErrorestimationforMonteCarlomethods . . . . . . . . . . . . . . . . . . . . . . . 19
2.8 TheFeynman-Kactheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.9 TheMoore-Penrosepseudo-inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Stochasticdynamics 23
3.1 Brownianmotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Itoˆ’slemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Normalprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Lognormalprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.5 TheMarkovianWienerprocessembeddingdimension . . . . . . . . . . . . . . . . . 27
3.6 Besselprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.7 Constantelasticityofvarianceprocesses . . . . . . . . . . . . . . . . . . . . . . . . 28
3.8 Displaceddiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
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iv CONTENTS
4 Processdrivensampling 31
4.1 Strongversusweakconvergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Numericalsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2.1 TheEulerscheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2.2 TheMilsteinscheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.3 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.4 Predictor-Corrector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3 Spuriouspaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4 StrongconvergenceforEulerandMilstein . . . . . . . . . . . . . . . . . . . . . . . 37
5 Correlationandco-movement 41
5.1 Measuresforco-dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2 Copulæ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2.1 TheGaussiancopula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2.2 Thet-copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2.3 Archimedeancopulæ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6 Salvagingalinearcorrelationmatrix 57
6.1 Hyperspheredecomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.2 Spectraldecomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.3 Angulardecompositionoflowertriangularform . . . . . . . . . . . . . . . . . . . . 61
6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.5 Angularcoordinatesonahypersphereofunitradius . . . . . . . . . . . . . . . . . . 63
7 Pseudo-randomnumbers 65
7.1 Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.2 Themid-squaremethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.3 Congruentialgeneration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.4 Ran0toRan3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.5 TheMersennetwister . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.6 Whichonetouse? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
CONTENTS v
8 Low-discrepancynumbers 75
8.1 Discrepancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
8.2 Haltonnumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
8.3 Sobol’numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
8.3.1 Primitivepolynomialsmodulotwo . . . . . . . . . . . . . . . . . . . . . . . . 79
8.3.2 TheconstructionofSobol’numbers . . . . . . . . . . . . . . . . . . . . . . . 80
8.3.3 TheGraycode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
8.3.4 TheinitialisationofSobol’numbers . . . . . . . . . . . . . . . . . . . . . . . 83
8.4 Niederreiter(1988)numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
8.5 Pairwiseprojections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
8.6 Empiricaldiscrepancies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
8.7 Thenumberofiterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8.8.1 ExplicitformulafortheL -normdiscrepancyontheunithypercube . . . . . . 93
2
8.8.2 ExpectedL -normdiscrepancyoftrulyrandomnumbers . . . . . . . . . . . . 94
2
9 Non-uniformvariates 95
9.1 Inversionofthecumulativeprobabilityfunction . . . . . . . . . . . . . . . . . . . . 95
9.2 Usingasamplerdensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
9.2.1 Importancesampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
9.2.2 Rejectionsampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
9.3 Normalvariates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
9.3.1 TheBox-Mullermethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
9.3.2 TheNeaveeffect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
9.4 Simulatingmulti-variatecopuladraws . . . . . . . . . . . . . . . . . . . . . . . . . 104
10 Variancereductiontechniques 107
10.1 Antitheticsampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
10.2 Variaterecycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
10.3 Controlvariates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
10.4 Stratifiedsampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
10.5 Importancesampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
10.6 Momentmatching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.7 Latinhypercubesampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
vi CONTENTS
10.8 Pathconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
10.8.1 Incremental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
10.8.2 Spectral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
10.8.3 TheBrownianbridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
10.8.4 Acomparisonofpathconstructionmethods . . . . . . . . . . . . . . . . . . . 124
10.8.5 Multivariatepathconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 126
10.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
10.9.1 Eigenvaluesandeigenvectorsofadiscrete-timecovariancematrix . . . . . . . 129
10.9.2 TheconditionaldistributionoftheBrownianbridge . . . . . . . . . . . . . . . 132
11 Greeks 135
11.1 ImportanceofGreeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
11.2 AnUp-Out-Calloption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
11.3 Finitedifferencingwithpathrecycling . . . . . . . . . . . . . . . . . . . . . . . . . 137
11.4 Finitedifferencingwithimportancesampling . . . . . . . . . . . . . . . . . . . . . . 139
11.5 Pathwisedifferentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
11.6 Thelikelihoodratiomethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
11.7 Comparativefigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
11.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
11.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
11.9.1 ThelikelihoodratioformulaforVega . . . . . . . . . . . . . . . . . . . . . . 147
11.9.2 ThelikelihoodratioformulaforRho . . . . . . . . . . . . . . . . . . . . . . . 149
12 MonteCarlointheBGM/Jframework 151
12.1 TheBrace-Gatarek-Musiela/Jamshidianmarketmodel . . . . . . . . . . . . . . . . . 151
12.2 Factorisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
12.3 Bermudanswaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
12.4 CalibrationtoEuropeanswaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
12.5 ThePredictor-Correctorscheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
12.6 Heuristicsoftheexerciseboundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
12.7 Exerciseboundaryparametrisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
12.8 Thealgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
12.9 Numericalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
12.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
CONTENTS vii
13 Non-recombiningtrees 175
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
13.2 Evolvingtheforwardrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
13.3 Optimalsimplexalignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
13.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
13.5 Convergenceperformance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
13.6 Variancematching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
13.7 Exactmartingaleconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
13.8 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
13.9 Asimpleexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
13.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
14 Miscellanea 193
14.1 Interpolationofthetermstructureofimpliedvolatility . . . . . . . . . . . . . . . . . 193
14.2 Watchyourcpuusage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
14.3 Numericaloverflowandunderflow . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
14.4 Asinglenumberoraconvergencediagram? . . . . . . . . . . . . . . . . . . . . . . 198
14.5 Embeddedpathcreation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
14.6 Howslowisexp()? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
14.7 Parallelcomputingandmulti-threading . . . . . . . . . . . . . . . . . . . . . . . . . 201
Bibliography 205
Index 215
viii CONTENTS
Preface
This book is about Monte Carlo methods and close relatives thereof. It is about the application of
traditional and state-of-the-art sampling techniques to problems encountered in the world of modern
finance. TheapproachItakeistoexplainmethodsalongsideactualexamplesthatIencounteredinmy
professionalworkingday. Thisiswhytheremaybeabiastowardsapplicationstoinvestmentbanking
and derivatives pricing in particular. However, many of the methods presented here equally apply to
similarmathematicalproblemsthatariseinmarketriskmanagement,creditriskassessment,theinsur-
ancebusinesses,strategicmanagementconsultancy,andotherareaswheretheeffectofmanyunknown
variables (in the sense that we can only make assumptions about their probability distributions) is to
beevaluated.
ThepurposeofthisbookistobeanintroductiontothebasicMonteCarlotechniquesusednowadays
byexpertpractitionersinthefield. TherearesomanyareasofMonteCarlomethodsinfinancethatany
attempt to try and provide a book on the subject that is both introductory and comprehensive would
havemeantmanyyearsof(part-time)writing. Instead,inordertofilltheneedforanintroductorytext
more timely, I decided to rather focus on the issues most pressing to any novice to financial Monte
Carlo simulations and to omit many of the more advanced topics. The subjects not covered include
the whole family of Markov Chain techniques, and almost all of the recent advances in Monte Carlo
methods tailored specifically for the pricing of American, Bermudan, or any other derivative contract
whose ideal value is given by the maximal (discounted) expected payoff over all possible exercise
strategy, i.e. by finding the truly optimal exercise strategy. An exception to this is perhaps the identi-
fication of a suitable exercise boundary optimisation for the purpose of Bermudan swaption pricing in
the Brace-Gatarek-Musiela/Jamshidian framework presented in chapter 12. At the same time, though,
Ihavetriedtoincludemostofthepresentlyusedtechniquesthatenablethepractitionertocreaterather
powerfulMonteCarlosimulationapplicationsindeed.
WhilstIalwaysendeavourtoexplainthebasicprinciplesoftheparticularproblemtowhichatech-
niqueisapplied,thisbookisnotmeanttobeanintroductiontofinancialmathematics. Iassumethatthe
reader either has background knowledge in the relevant areas, or could follow up the given references
for a deeper understanding of the financial and/or economical reasoning behind specific mathematical
assumptions. After all, this is not a book about the reasoning behind option pricing. This is a book
about mathematical and numerical techniques that may be used for the solution of the mathematical
equations that were derived by experts in financial theory and economics. I do not attempt to give a
justification for the assumption of complete markets, market efficiency, specific stochastic differential
ix
x Preface
equations, etc.; I leave this up to the authors of the excellent books on those issues subject in the lit-
erature [Hul97, Mer90, Reb98, Wil98]. Instead I have focussed on the implementational aspects of
Monte Carlo methods. Any Monte Carlo method will invariable have to run on a computing device,
and this means that numerical issues can be of paramount importance. In order for this book to be
of some practical value to the practitioner having to implement Monte Carlo methods, I made the at-
tempt to link the fundamental concepts of any one technique directly to the algorithm that has to be
programmed, and often explicitly in terms of the C++ language, often taking into account aspects of
numericalanalysissuchasround-offerrorpropagationetc.
Thenatureofthesubjectofthisbookisstronglyentwinedwiththeconceptofconvergence. Ingen-
eral,MonteCarlomethodsgiveusatbestastatisticalerrorestimate. Thisisincontrasttovariousother
numerical methods. A Monte Carlo calculation is typically of the following structure: carry out the
sameproceduremanytimes,takeintoaccountalloftheindividualresults,andsummarisethemintoan
overall approximation to the problem in question. For most Monte Carlo methods (in particular those
providing serial decorrelation of the individual results), we can choose any subset of the individual
results and summarise them to obtain an estimate. The numerically exact solution will be approached
by the method only as we iterate the procedure more and more times, eventually converging at infin-
ity. Clearly, we are not just interested in a method to converge to the correct answer after an infinite
amountofcalculationtime,butratherwewishtohaveagoodapproximationquickly. Therefore,once
we are satisfied that a particular Monte Carlo method works in the limit, we are naturally interested in
its convergence behaviour, or, more specifically, its convergence speed. A good part of this book is
dedicated to various techniques and tricks to improve the convergence speed of Monte Carlo methods
and their relatives. In order to present the reader not just with a description of the algorithms, but also
to foster an intuitive grasp of the potential benefit from the implementation of a specific technique,
we have attempted to include many diagrams of typical convergence behaviour: frequently these are
used to highlight the differences between the performances of different methods. In particular where
such comparisons are made, we often display the convergence behaviour as a function of cpu time
used by the different methods since the human user’s utility is much more closely related to the time
elapseduntilacalculationofsufficientaccuracyhasbeencompletedratherthantothenumberofactual
iterationscarriedout.
You may wonder why there is no explicit chapter on option pricing, considering that that’s one of
the most immediate applications of Monte Carlo methods in finance. As it happens, there isn’t one
chapter on option pricing, but every chapter is written with option pricing in the mind. My foremost
use of Monte Carlo methods has been in the area of derivatives pricing. Since a lot of the examples I
give are directly with respect to option valuation, I considered it unnecessary to have a chapter on the
subject by itself, only to repeat what is written in other chapters already. I hope the reader will agree
withme.