Quantitative finance

476 Pages·2020·3.622 MB·English

by  Florescu, Ionuţ, Mariani, Maria C

QuantitativeFinance
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Quantitative Finance
Maria C. Mariani
University of Texas at El Paso
Texas, United States
Ionut Florescu
Stevens Institute of Technology
Hoboken, United States
Thiseditionfirstpublished2020
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LibraryofCongressCataloging-in-PublicationData
Names:Mariani,MariaC.,author.|Florescu,Ionut,1973–author.
Title:Quantitativefinance/MariaC.Mariani,UniversityofTexasatElPaso,Texas,
UnitedStates,IonutFlorescu,StevensIntistuteofTechnology,Hoboken,UnitedStates.
Description:Hoboken,NJ:Wiley,2020.|Series:Wileyseriesin
statisticsinpractice|Includesindex.
Identifiers:LCCN2019035349(print)|LCCN2019035350(ebook)|ISBN
9781118629956(hardback)|ISBN9781118629963(adobepdf)|ISBN
9781118629888(epub)
Subjects:LCSH:Businessmathematics.|Finance–Mathematicalmodels.|
Finance–Econometricmodels.
Classification:LCCHF5691.M292020(print)|LCCHF5691(ebook)|DDC
332.01/5195–dc23
LCrecordavailableathttps://lccn.loc.gov/2019035349
LCebookrecordavailableathttps://lccn.loc.gov/2019035350
CoverDesign:Wiley
CoverImage:CourtesyofMariaC.Mariani
Setin9.5/12.5ptSTIXTwoTextbySPiGlobal,Pondicherry,India
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
v
Contents
ListofFigures xv
ListofTables xvii
PartI StochasticProcessesandFinance 1
1 StochasticProcesses 3
1.1 Introduction 3
1.2 GeneralCharacteristicsofStochasticProcesses 4
1.2.1 TheIndexSet 4
1.2.2 TheStateSpace 4
1.2.3 Adaptiveness,Filtration,andStandardFiltration 5
1.2.4 PathwiseRealizations 7
1.2.5 TheFiniteDimensionalDistributionofStochasticProcesses 8
1.2.6 IndependentComponents 9
1.2.7 StationaryProcess 9
1.2.8 StationaryandIndependentIncrements 10
1.3 VariationandQuadraticVariationofStochasticProcesses 11
1.4 OtherMoreSpecificProperties 13
1.5 ExamplesofStochasticProcesses 14
1.5.1 TheBernoulliProcess(SimpleRandomWalk) 14
1.5.2 TheBrownianMotion(WienerProcess) 17
1.6 Borel—CantelliLemmas 19
1.7 CentralLimitTheorem 20
1.8 StochasticDifferentialEquation 20
1.9 StochasticIntegral 21
1.9.1 PropertiesoftheStochasticIntegral 22
1.10 MaximizationandParameterCalibrationofStochasticProcesses 22
1.10.1 ApproximationoftheLikelihoodFunction(PseudoMaximum
LikelihoodEstimation) 24
1.10.2 OzakiMethod 24
vi Contents
1.10.3 Shoji-OzakiMethod 25
1.10.4 KesslerMethod 25
1.11 QuadratureMethods 26
1.11.1 RectangleRule:(n=1)(DarbouxSums) 27
1.11.2 MidpointRule 28
1.11.3 TrapezoidRule 28
1.11.4 Simpson’sRule 28
1.12 Problems 29
2 BasicsofFinance 33
2.1 Introduction 33
2.2 Arbitrage 33
2.3 Options 35
2.3.1 VanillaOptions 35
2.3.2 Put–CallParity 36
2.4 Hedging 39
2.5 ModelingReturnofStocks 40
2.6 ContinuousTimeModel 41
2.6.1 Itô’sLemma 42
2.7 Problems 45
PartII QuantitativeFinanceinPractice 47
3 SomeModelsUsedinQuantitativeFinance 49
3.1 Introduction 49
3.2 AssumptionsfortheBlack–Scholes–MertonDerivation 49
3.3 TheB-SModel 50
3.4 SomeRemarksontheB-SModel 58
3.4.1 Remark1 58
3.4.2 Remark2 58
3.5 HestonModel 60
3.5.1 HestonPDEDerivation 61
3.6 TheCox–Ingersoll–Ross(CIR)Model 63
3.7 Stochastic𝛼,𝛽,𝜌(SABR)Model 64
3.7.1 SABRImpliedVolatility 64
3.8 MethodsforFindingRootsofFunctions:ImpliedVolatility 65
3.8.1 Introduction 65
3.8.2 TheBisectionMethod 65
3.8.3 TheNewton’sMethod 66
3.8.4 SecantMethod 67
3.8.5 ComputationofImpliedVolatilityUsingtheNewton’sMethod 68
3.9 SomeRemarksofImpliedVolatility(Put–CallParity) 69
Contents vii
3.10 HedgingUsingVolatility 70
3.11 FunctionalApproximationMethods 73
3.11.1 LocalVolatilityModel 74
3.11.2 Dupire’sEquation 74
3.11.3 SplineApproximation 77
3.11.4 NumericalSolutionTechniques 78
3.11.5 PricingSurface 79
3.12 Problems 79
4 SolvingPartialDifferentialEquations 83
4.1 Introduction 83
4.2 UsefulDefinitionsandTypesofPDEs 83
4.2.1 TypesofPDEs(2-D) 83
4.2.2 BoundaryConditions(BC)forPDEs 84
4.3 FunctionalSpacesUsefulforPDEs 85
4.4 SeparationofVariables 88
4.5 Moment-GeneratingLaplaceTransform 91
4.5.1 NumericInversionforLaplaceTransform 92
4.5.2 FourierSeriesApproximationMethod 93
4.6 ApplicationoftheLaplaceTransformtotheBlack–ScholesPDE 96
4.7 Problems 99
5 WaveletsandFourierTransforms 101
5.1 Introduction 101
5.2 DynamicFourierAnalysis 101
5.2.1 Tapering 102
5.2.2 EstimationofSpectralDensitywithDaniellKernel 103
5.2.3 DiscreteFourierTransform 104
5.2.4 TheFastFourierTransform(FFT)Method 106
5.3 WaveletsTheory 109
5.3.1 Definition 109
5.3.2 WaveletsandTimeSeries 110
5.4 ExamplesofDiscreteWaveletsTransforms(DWT) 112
5.4.1 HaarWavelets 112
5.4.2 DaubechiesWavelets 115
5.5 ApplicationofWaveletsTransform 116
5.5.1 Finance 116
5.5.2 ModelingandForecasting 117
5.5.3 ImageCompression 117
5.5.4 SeismicSignals 117
5.5.5 DamageDetectioninFrameStructures 118
5.6 Problems 118
viii Contents
6 TreeMethods 121
6.1 Introduction 121
6.2 TreeMethods:theBinomialTree 122
6.2.1 One-StepBinomialTree 122
6.2.2 UsingtheTreetoPriceaEuropeanOption 125
6.2.3 UsingtheTreetoPriceanAmericanOption 126
6.2.4 UsingtheTreetoPriceAnyPath-DependentOption 127
6.2.5 UsingtheTreeforComputingHedgeSensitivities:theGreeks 128
6.2.6 FurtherDiscussionontheAmericanOptionPricing 128
6.2.7 AParenthesis:theBrownianMotionasaLimitofSimpleRandom
Walk 132
6.3 TreeMethodsforDividend-PayingAssets 135
6.3.1 OptionsonAssetsPayingaContinuousDividend 135
6.3.2 OptionsonAssetsPayingaKnownDiscreteProportional
Dividend 136
6.3.3 OptionsonAssetsPayingaKnownDiscreteCashDividend 136
6.3.4 TreeforKnown(Deterministic)Time-VaryingVolatility 137
6.4 PricingPath-DependentOptions:BarrierOptions 139
6.5 TrinomialTreeMethodandOtherConsiderations 140
6.6 MarkovProcess 143
6.6.1 TransitionFunction 143
6.7 BasicElementsofOperatorsandSemigroupTheory 146
6.7.1 InfinitesimalOperatorofSemigroup 150
6.7.2 FellerSemigroup 151
6.8 GeneralDiffusionProcess 152
6.8.1 Example:DerivationofOptionPricingPDE 155
6.9 AGeneralDiffusionApproximationMethod 156
6.10 ParticleFilterConstruction 159
6.11 QuadrinomialTreeApproximation 163
6.11.1 ConstructionoftheOne-PeriodModel 164
6.11.2 ConstructionoftheMultiperiodModel:OptionValuation 170
6.12 Problems 173
7 ApproximatingPDEs 177
7.1 Introduction 177
7.2 TheExplicitFiniteDifferenceMethod 179
7.2.1 StabilityandConvergence 180
7.3 TheImplicitFiniteDifferenceMethod 180
7.3.1 StabilityandConvergence 182
7.4 TheCrank–NicolsonFiniteDifferenceMethod 183
7.4.1 StabilityandConvergence 183
Contents ix
7.5 ADiscussionAbouttheNecessaryNumberofNodesinthe
Schemes 184
7.5.1 ExplicitFiniteDifferenceMethod 184
7.5.2 ImplicitFiniteDifferenceMethod 185
7.5.3 Crank–NicolsonFiniteDifferenceMethod 185
7.6 SolutionofaTridiagonalSystem 186
7.6.1 InvertingtheTridiagonalMatrix 186
7.6.2 AlgorithmforSolvingaTridiagonalSystem 187
7.7 HestonPDE 188
7.7.1 BoundaryConditions 189
7.7.2 DerivativeApproximationforNonuniformGrid 190
7.8 MethodsforFreeBoundaryProblems 191
7.8.1 AmericanOptionValuations 192
7.8.2 FreeBoundaryProblem 192
7.8.3 LinearComplementarityProblem(LCP) 193
7.8.4 TheObstacleProblem 196
7.9 MethodsforPricingAmericanOptions 199
7.10 Problems 201
8 ApproximatingStochasticProcesses 203
8.1 Introduction 203
8.2 PlainVanillaMonteCarloMethod 203
8.3 ApproximationofIntegralsUsingtheMonteCarloMethod 205
8.4 VarianceReduction 205
8.4.1 AntitheticVariates 205
8.4.2 ControlVariates 206
8.5 AmericanOptionPricingwithMonteCarloSimulation 208
8.5.1 Introduction 209
8.5.2 MartingaleOptimization 210
8.5.3 LeastSquaresMonteCarlo(LSM) 210
8.6 NonstandardMonteCarloMethods 216
8.6.1 SequentialMonteCarlo(SMC)Method 216
8.6.2 MarkovChainMonteCarlo(MCMC)Method 217
8.7 GeneratingOne-DimensionalRandomVariablesbyInvertingthe
cdf 218
8.8 GeneratingOne-DimensionalNormalRandomVariables 220
8.8.1 TheBox–MullerMethod 221
8.8.2 ThePolarRejectionMethod 222
8.9 GeneratingRandomVariables:RejectionSamplingMethod 224
8.9.1 Marsaglia’sZigguratMethod 226
8.10 GeneratingRandomVariables:ImportanceSampling 236
8.10.1 SamplingImportanceResampling 240