Implementing models in quantitative finance: methods and cases

618 Pages·2008·10.274 MB·English

by  Gianluca Fusai, Andrea Roncoroni

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BarucciE.,FinancialMarketsTheory.Equilibrium,EfficiencyandInformation(2003)
BieleckiT.R.andRutkowskiM.,CreditRisk:Modeling,ValuationandHedging(2002)
BinghamN.H.andKieselR.,Risk-NeutralValuation:PricingandHedgingofFinancial
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BrigoD.andMercurioF.,InterestRateModels:TheoryandPractice(2001,2nded.2006)
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Gianluca Fusai · Andrea Roncoroni
Implementing Models in
Quantitative Finance:
Methods and Cases
GianlucaFusai AndreaRoncoroni
DipartimentodiScienzeEconomiche FinanceDepartment
eMetodiQuantitativi ESSECGraduateBusinessSchool
FacoltàdiEconomia AvenueBernardHirschBP50105
UniversitàdelPiemonte CergyPontoiseCedex
Orientale“A.Avogadro” France
ViaPerrone,18 E-mails:[email protected]
28100Novara [email protected]
Italy
E-mail:[email protected]
MathematicsSubjectClassification(2000):35-01,65-01,65C05,65C10,65C20,
65C30,91B28
JELClassification:G11,G13,C15,C22,C63
LibraryofCongressControlNumber:2007931341
ISBN978-3-540-22348-1SpringerBerlinHeidelbergNewYork
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Contents
Introduction....................................................... xv
PartI Methods
1 StaticMonteCarlo ............................................. 3
1.1 MotivationandIssues ....................................... 3
1.1.1 Issue1:MonteCarloEstimation........................ 5
1.1.2 Issue2:EfficiencyandSampleSize..................... 7
1.1.3 Issue3:HowtoSimulateSamples ...................... 8
1.1.4 Issue4:HowtoEvaluateFinancialDerivatives ........... 9
1.1.5 TheMonteCarloSimulationAlgorithm ................. 11
1.2 SimulationofRandomVariables.............................. 11
1.2.1 UniformNumbersGeneration.......................... 12
1.2.2 TransformationMethods .............................. 14
1.2.3 Acceptance–RejectionMethods ........................ 20
1.2.4 HazardRateFunctionMethod ......................... 23
1.2.5 SpecialMethods..................................... 24
1.3 VarianceReduction......................................... 31
1.3.1 AntitheticVariables .................................. 31
1.3.2 ControlVariables .................................... 33
1.3.3 ImportanceSampling................................. 35
1.4 Comments ................................................ 39
2 DynamicMonteCarlo .......................................... 41
2.1 MainIssues ............................................... 41
2.2 ContinuousDiffusions ...................................... 45
2.2.1 MethodI:ExactTransition ............................ 45
2.2.2 MethodII:ExactSolution............................. 46
2.2.3 MethodIII:ApproximateDynamics .................... 46
viii
2.2.4 Example:OptionValuationunderAlternativeSimulation
Schemes............................................ 48
2.3 JumpProcesses ............................................ 49
2.3.1 CompoundJumpProcesses............................ 49
2.3.2 ModellingviaJumpIntensity .......................... 51
2.3.3 SimulationwithConstantIntensity ..................... 53
2.3.4 SimulationwithDeterministicIntensity ................. 54
2.4 Mixed-JumpDiffusions ..................................... 56
2.4.1 StatementoftheProblem ............................. 56
2.4.2 MethodI:TransitionProbability........................ 58
2.4.3 MethodII:ExactSolution............................. 58
2.4.4 MethodIII.A:ApproximateDynamicswithDeterministic
Intensity............................................ 59
2.4.5 MethodIII.B:ApproximateDynamicswithRandomIntensity 60
2.5 GaussianProcesses ......................................... 62
2.6 Comments ................................................ 66
3 DynamicProgrammingforStochasticOptimization................ 69
3.1 ControlledDynamicalSystems ............................... 69
3.2 TheOptimalControlProblem ................................ 71
3.3 TheBellmanPrincipleofOptimality .......................... 73
3.4 DynamicProgramming...................................... 74
3.5 StochasticDynamicProgramming ............................ 76
3.6 Applications............................................... 77
3.6.1 AmericanOptionPricing.............................. 77
3.6.2 OptimalInvestmentProblem........................... 79
3.7 Comments ................................................ 81
4 FiniteDifferenceMethods....................................... 83
4.1 Introduction ............................................... 83
4.1.1 SecurityPricingandPartialDifferentialEquations ........ 83
4.1.2 ClassificationofPDEs................................ 84
4.2 FromBlack–ScholestotheHeatEquation...................... 87
4.2.1 ChangingtheTimeOrigin............................. 88
4.2.2 UndiscountedPrices.................................. 88
4.2.3 FromPricestoReturns ............................... 89
4.2.4 HeatEquation....................................... 89
4.2.5 ExtendingTransformationstoOtherProcesses............ 90
4.3 DiscretizationSetting ....................................... 91
4.3.1 Finite-DifferenceApproximations ...................... 91
4.3.2 Grid ............................................... 93
4.3.3 ExplicitScheme ..................................... 94
4.3.4 ImplicitScheme ..................................... 101
4.3.5 Crank–NicolsonScheme .............................. 103
4.3.6 ComputingtheGreeks................................ 109
ix
4.4 Consistency,ConvergenceandStability ........................ 110
4.5 GeneralLinearParabolicPDEs............................... 115
4.5.1 ExplicitScheme ..................................... 116
4.5.2 ImplicitScheme ..................................... 117
4.5.3 Crank–NicolsonScheme .............................. 118
4.6 AVBACodeforSolvingGeneralLinearParabolicPDEs......... 119
4.7 Comments ................................................ 119
5 NumericalSolutionofLinearSystems ............................ 121
5.1 DirectMethods:TheLUDecomposition ....................... 122
5.2 IterativeMethods........................................... 127
5.2.1 JacobiIteration:SimultaneousDisplacements ............ 128
5.2.2 Gauss–SeidelIteration(SuccessiveDisplacements)........ 130
5.2.3 SOR(SuccessiveOver-RelaxationMethod) .............. 131
5.2.4 ConjugateGradientMethod(CGM)..................... 133
5.2.5 ConvergenceofIterativeMethods ...................... 135
5.3 CodefortheSolutionofLinearSystems ....................... 140
5.3.1 VBACode.......................................... 140
5.3.2 MATLABCode ..................................... 141
5.4 IllustrativeExamples........................................ 143
5.4.1 PricingaPlainVanillaCallintheBlack–ScholesModel
(VBA) ............................................. 144
5.4.2 PricingaPlainVanillaCallintheSquare-RootModel(VBA)145
5.4.3 PricingAmericanOptionswiththeCNScheme(VBA) .... 147
5.4.4 PricingaDoubleBarrierCallintheBSModel(MATLAB
andVBA) .......................................... 149
5.4.5 PricinganOptiononaCouponBondintheCox–Ingersoll–
RossModel(MATLAB) .............................. 152
5.5 Comments ................................................ 155
6 QuadratureMethods ........................................... 157
6.1 QuadratureRules........................................... 158
6.2 Newton–CotesFormulae .................................... 159
6.2.1 CompositeNewton–CotesFormula ..................... 162
6.3 GaussianQuadratureFormulae ............................... 173
6.4 MatlabCode............................................... 180
6.4.1 TrapezoidalRule .................................... 180
6.4.2 SimpsonRule ....................................... 180
6.4.3 RombergExtrapolation ............................... 181
6.5 VBACode ................................................ 181
6.6 AdaptiveQuadrature........................................ 182
6.7 Examples ................................................. 185
6.7.1 VanillaOptionsintheBlack–ScholesModel ............. 186
6.7.2 VanillaOptionsintheSquare-RootModel ............... 188
6.7.3 BondOptionsintheCox–Ingersoll–RossModel .......... 190
x
6.7.4 DiscretelyMonitoredBarrierOptions ................... 193
6.8 PricingUsingCharacteristicFunctions......................... 197
6.8.1 MATLABandVBAAlgorithms........................ 202
6.8.2 OptionsPricingwithLévyProcesses.................... 206
6.9 Comments ................................................ 211
7 TheLaplaceTransform......................................... 213
7.1 DefinitionandProperties .................................... 213
7.2 NumericalInversion ........................................ 216
7.3 TheFourierSeriesMethod................................... 218
7.4 ApplicationstoQuantitativeFinance .......................... 219
7.4.1 Example............................................ 219
7.4.2 Example............................................ 225
7.5 Comments ................................................ 228
8 StructuringDependenceusingCopulaFunctions .................. 231
8.1 CopulaFunctions .......................................... 231
8.2 ConcordanceandDependence................................ 233
8.2.1 Fréchet–HoeffdingBounds ............................ 233
8.2.2 MeasuresofConcordance ............................. 234
8.2.3 MeasuresofDependence.............................. 235
8.2.4 ComparisonwiththeLinearCorrelation ................. 236
8.2.5 OtherNotionsofDependence.......................... 238
8.3 EllipticalCopulaFunctions .................................. 240
8.4 ArchimedeanCopulas....................................... 245
8.5 StatisticalInferenceforCopulas .............................. 251
8.5.1 ExactMaximumLikelihood ........................... 253
8.5.2 InferenceFunctionsforMargins........................ 254
8.5.3 Kernel-basedNonparametricEstimation................. 255
8.6 MonteCarloSimulation..................................... 257
8.6.1 DistributionalMethod ................................ 257
8.6.2 ConditionalSampling ................................ 259
8.6.3 CompoundCopulaSimulation ......................... 263
8.7 Comments ................................................ 265
PartII Problems
PortfolioManagementandTrading .................................. 271
9 PortfolioSelection:“Optimizing”anError........................ 273
9.1 ProblemStatement ......................................... 274
9.2 ModelandSolutionMethodology............................. 276
9.3 ImplementationandAlgorithm ............................... 278
9.4 ResultsandComments ...................................... 280
9.4.1 In-sampleAnalysis................................... 281
xi
9.4.2 Out-of-sampleSimulation............................. 285
10 Alpha,BetaandBeyond ........................................ 289
10.1 ProblemStatement ......................................... 290
10.2 SolutionMethodology ...................................... 291
10.2.1 ConstantBeta:OLSEstimation ........................ 292
10.2.2 ConstantBeta:RobustEstimation ...................... 293
10.2.3 ConstantBeta:ShrinkageEstimation.................... 295
10.2.4 ConstantBeta:BayesianEstimation..................... 296
10.2.5 Time-VaryingBeta:ExponentialSmoothing.............. 299
10.2.6 Time-VaryingBeta:TheKalmanFilter .................. 300
10.2.7 Comparingthemodels................................ 304
10.3 ImplementationandAlgorithm ............................... 306
10.4 ResultsandComments ...................................... 309
11 AutomaticTrading:WinningorLosinginakBit .................. 311
11.1 ProblemStatement ......................................... 312
11.2 ModelandSolutionMethodology............................. 314
11.2.1 MeasuringTradingSystemPerformance................. 314
11.2.2 StatisticalTesting .................................... 315
11.3 Code ..................................................... 317
11.4 ResultsandComments ...................................... 322
VanillaOptions .................................................... 329
12 EstimatingtheRisk-NeutralDensity ............................. 331
12.1 ProblemStatement ......................................... 332
12.2 SolutionMethodology ...................................... 332
12.3 ImplementationandAlgorithm ............................... 335
12.4 ResultsandComments ...................................... 338
13 An“American”MonteCarlo.................................... 345
13.1 ProblemStatement ......................................... 346
13.2 ModelandSolutionMethodology............................. 347
13.3 ImplementationandAlgorithm ............................... 348
13.4 ResultsandComments ...................................... 349
14 FixingVolatileVolatility ........................................ 353
14.1 ProblemStatement ......................................... 354
14.2 ModelandSolutionMethodology............................. 356
14.2.1 AnalyticalTransforms ................................ 356
14.2.2 ModelCalibration ................................... 358
14.3 ImplementationandAlgorithm ............................... 360
14.3.1 CodeDescription .................................... 361
14.4 ResultsandComments ...................................... 362