Computational Finance Using C and C# (Quantitative Finance)

385 Pages·2008·4.31 MB·English

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Computational Finance Using C and C#
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Quantitative Finance Series
Aims and Objectives
• Booksbasedontheworkoffinancialmarketpractitionersandacademics
• Presentingcutting-edgeresearchtotheprofessional/practitionermarket
• Combiningintellectualrigourandpracticalapplication
• Coveringtheinteractionbetweenmathematicaltheoryandfinancialpractice
• Toimproveportfolioperformance,riskmanagementandtradingbookperformance
• Coveringquantitativetechniques
Market
Brokers/Traders; Actuaries; Consultants; Asset Managers; Fund Managers; Regula-
tors;CentralBankers;TreasuryOfficials;TechnicalAnalysis;andAcademicsforMas-
tersinFinanceandMBAmarket.
Series Titles
ComputationalFinanceUsingCandC#
TheAnalyticsofRiskModelValidation
ForecastingExpectedReturnsintheFinancialMarkets
CorporateGovernanceandRegulatoryImpactonMergersandAcquisitions
InternationalMergersandAcquisitionsActivitySince1990
ForecastingVolatilityintheFinancialMarkets,ThirdEdition
VentureCapitalinEurope
FundsofHedgeFunds
InitialPublicOfferings
LinearFactorModelsinFinance
ComputationalFinance
AdvancesinPortfolioConstructionandImplementation
AdvancedTradingRules,SecondEdition
RealR&DOptions
PerformanceMeasurementinFinance
EconomicsforFinancialMarkets
ManagingDownsideRiskinFinancialMarkets
DerivativeInstruments:Theory,Valuation,Analysis
ReturnDistributionsinFinance
Series Editor: Dr Stephen Satchell
Dr Satchell is Reader in Financial Econometrics at Trinity College, Cambridge;
Visiting Professor at Birkbeck College, City University Business School and Univer-
sity of Technology, Sydney. He also works in a consultative capacity to many firms,
andeditsthejournalDerivatives:use,tradingandregulationsandtheJournalofAsset
Management.
Computational Finance
Using C and C#
George Levy
AMSTERDAM•BOSTON•HEIDELBERG•LONDON•NEWYORK
OXFORD•PARIS•SANDIEGO•SANFRANCISCO•SINGAPORE
SYDNEY•TOKYO
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LibraryofCongressCataloging-in-PublicationData
Levy,George.
ComputationalFinanceUsingCandC#/GeorgeLevy.
p.cm.–(Quantitativefinance)
Includesbibliographicalreferencesandindex.
ISBN-13:978-0-7506-6919-1(alk.paper)1.Finance-Mathematicalmodels.I.Title.
HG106.L4842008
332.0285’5133-dc22
2008000470
BritishLibraryCataloguing-in-PublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
ForinformationonallAcademicPresspublications
visitourWebsiteatwww.books.elsevier.com
PrintedintheUnitedStatesofAmerica
08 09 10 11 9 8 7 6 5 4 3 2 1
To my parents Paul and Paula
Thispageintentionallyleftblank
Contents
Preface xi
1 Overviewoffinancialderivatives 1
2 Introductiontostochasticprocesses 5
2.1 Brownianmotion 5
2.2 ABrownianmodelofassetpricemovements 9
2.3 Ito’sformula(orlemma) 10
2.4 Girsanov’stheorem 12
2.5 Ito’slemmaformultiassetgeometricBrownianmotion 13
2.6 Itoproductandquotientrulesintwodimensions 15
2.7 Itoproductinndimensions 18
2.8 TheBrownianbridge 19
2.9 Time-transformedBrownianmotion 21
2.10 Ornstein–Uhlenbeckprocess 24
2.11 TheOrnstein–Uhlenbeckbridge 27
2.12 Otherusefulresults 31
2.13 Selectedproblems 33
3 Generationofrandomvariates 37
3.1 Introduction 37
3.2 Pseudo-randomandquasi-randomsequences 38
3.3 Generationofmultivariatedistributions:independentvariates 41
3.4 Generationofmultivariatedistributions:correlatedvariates 47
4 Europeanoptions 59
4.1 Introduction 59
4.2 Pricingderivativesusingamartingalemeasure 59
4.3 Putcallparity 60
4.4 VanillaoptionsandtheBlack–Scholesmodel 62
4.5 Barrieroptions 85
5 SingleassetAmericanoptions 97
5.1 Introduction 97
5.2 ApproximationsforvanillaAmericanoptions 97
5.3 Latticemethodsforvanillaoptions 114
viii ComputationalFinanceUsingCandC#
5.4 Gridmethodsforvanillaoptions 135
5.5 PricingAmericanoptionsusingastochasticlattice 172
6 Multiassetoptions 181
6.1 Introduction 181
6.2 ThemultiassetBlack–Scholesequation 181
6.3 MultidimensionalMonteCarlomethods 183
6.4 Introductiontomultidimensionallatticemethods 185
6.5 Twoassetoptions 190
6.6 Threeassetoptions 201
6.7 Fourassetoptions 205
7 Otherfinancialderivatives 209
7.1 Introduction 209
7.2 Interestratederivatives 209
7.3 Foreignexchangederivatives 228
7.4 Creditderivatives 232
7.5 Equityderivatives 237
8 C#portfoliopricingapplication 245
8.1 Introduction 245
8.2 Storingandretrievingthemarketdata 254
8.3 ThePricingUtilsclassandtheAnalytics_MathLib 262
8.4 Equitydealclasses 267
8.5 FXdealclasses 280
AppendixA: TheGreeksforvanillaEuropeanoptions 289
A.1 Introduction 289
A.2 Gamma 290
A.3 Delta 291
A.4 Theta 292
A.5 Rho 293
A.6 Vega 294
AppendixB: Barrieroptionintegrals 295
B.1 Thedownandoutcall 295
B.2 Theupandoutcall 298
AppendixC: Standardstatisticalresults 303
C.1 Thelawoflargenumbers 303
C.2 Thecentrallimittheorem 303
C.3 Thevarianceandcovarianceofrandomvariables 305
C.4 Conditionalmeanandcovarianceofnormaldistributions 310
C.5 Momentgeneratingfunctions 311
Contents ix
AppendixD: Statisticaldistributionfunctions 313
D.1 Thenormal(Gaussian)distribution 313
D.2 Thelognormaldistribution 315
D.3 TheStudent’st distribution 317
D.4 Thegeneralerrordistribution 319
AppendixE: Mathematicalreference 321
E.1 Standardintegrals 321
E.2 Gammafunction 321
E.3 Thecumulativenormaldistributionfunction 322
E.4 Arithmeticandgeometricprogressions 323
AppendixF: Black–Scholesfinite-differenceschemes 325
F.1 Thegeneralcase 325
F.2 Thelogtransformationandauniformgrid 325
AppendixG: TheBrownianbridge:alternativederivation 329
AppendixH: Brownianmotion:moreresults 333
H.1 SomeresultsconcerningBrownianmotion 333
H.2 ProofofEq.(H.1.2) 334
H.3 ProofofEq.(H.1.4) 335
H.4 ProofofEq.(H.1.5) 335
H.5 ProofofEq.(H.1.6) 335
H.6 ProofofEq.(H.1.7) 338
H.7 ProofofEq.(H.1.8) 338
H.8 ProofofEq.(H.1.9) 338
H.9 ProofofEq.(H.1.10) 339
AppendixI: TheFeynman–Kacformula 341
AppendixJ: Answerstoproblems 343
J.1 Problem1 343
J.2 Problem2 344
J.3 Problem3 345
J.4 Problem4 346
J.5 Problem5 346
J.6 Problem6 347
J.7 Problem7 348
J.8 Problem8 350
J.9 Problem9 350
J.10 Problem10 352
J.11 Problem11 354
References 355
Index 361