Introduction .......................................................xv
PartIMethods
1StaticMonteCarlo .............................................3
1.1MotivationandIssues.......................................3
1.1.1Issue1:MonteCarloEstimation........................5
1.1.2Issue2:Ef?ciencyandSampleSize.....................7
1.1.3Issue3:HowtoSimulateSamples......................8
1.1.4Issue4:HowtoEvaluateFinancialDerivatives...........9
1.1.5TheMonteCarloSimulationAlgorithm.................11
1.2SimulationofRandomVariables..............................11
1.2.1UniformNumbersGeneration..........................12
1.2.2TransformationMethods..............................14
1.2.3Acceptance(R)CRejectionMethods........................2
1.2.4HazardRateFunctionMethod.........................23
1.2.5SpecialMethods.....................................24
1.3VarianceReduction.........................................31
1.3.1AntitheticVariables..................................31
1.3.2ControlVariables....................................33
1.3.3ImportanceSampling.................................35
1.4Comments................................................39
2DynamicMonteCarlo ..........................................41
2.1MainIssues...............................................41
2.2ContinuousDiffusions......................................45
2.2.1MethodI:ExactTransition............................45
2.2.2MethodII:ExactSolution.............................46
2.2.3MethodIII:ApproximateDynamics....................46

viii
2.2.4Example:OptionValuationunderAlternativeSimulation
Schemes............................................48
2.3JumpProcesses............................................49
2.3.1CompoundJumpProcesses............................49
2.3.2ModellingviaJumpIntensity..........................51
2.3.3SimulationwithConstantIntensity.....................53
2.3.4SimulationwithDeterministicIntensity.................54
2.4Mixed-JumpDiffusions.....................................56
2.4.1StatementoftheProblem.............................56
2.4.2MethodI:TransitionProbability........................58
2.4.3MethodII:ExactSolution.............................58
2.4.4MethodIII.A:ApproximateDynamicswithDeterministic
Intensity............................................59
2.4.5MethodIII.B:ApproximateDynamicswithRandomIntensity60
2.5GaussianProcesses.........................................62
2.6Comments................................................66
3DynamicProgrammingforStochasticOptimization................69
3.1ControlledDynamicalSystems...............................69
3.2TheOptimalControlProblem................................71
3.3TheBellmanPrincipleofOptimality..........................73
3.4DynamicProgramming......................................74
3.5StochasticDynamicProgramming............................76
3.6Applications...............................................77
3.6.1AmericanOptionPricing..............................77
3.6.2OptimalInvestmentProblem...........................79
3.7Comments................................................81
4FiniteDifferenceMethods.......................................83
4.1Introduction...............................................83
4.1.1SecurityPricingandPartialDifferentialEquations........83
4.1.2Classi?cationofPDEs................................84
4.2FromBlack(R)CScholestotheHeatEquatio ......................87
4.2.1ChangingtheTimeOrigin.............................88
4.2.2UndiscountedPrices..................................88
4.2.3FromPricestoReturns...............................89
4.2.4HeatEquation.......................................89
4.2.5ExtendingTransformationstoOtherProcesses............90
4.3DiscretizationSetting.......................................91
4.3.1Finite-DifferenceApproximations......................91
4.3.2Grid...............................................93
4.3.3ExplicitScheme.....................................94
4.3.4ImplicitScheme.....................................101
4.3.5Crank(R)CNicolsonScheme..............................10
4.3.6ComputingtheGreeks................................109

4.4Consistency,ConvergenceandStability........................110
4.5GeneralLinearParabolicPDEs...............................115
4.5.1ExplicitScheme.....................................116
4.5.2ImplicitScheme.....................................117
4.5.3Crank(R)CNicolsonScheme..............................11
4.6AVBACodeforSolvingGeneralLinearParabolicPDEs.........119
4.7Comments................................................119
5NumericalSolutionofLinearSystems ............................121
5.1DirectMethods:TheLUDecomposition.......................122
5.2IterativeMethods...........................................127
5.2.1JacobiIteration:SimultaneousDisplacements............128
5.2.2Gauss(R)CSeidelIteration(SuccessiveDisplacements)........13
5.2.3SOR(SuccessiveOver-RelaxationMethod)..............131
5.2.4ConjugateGradientMethod(CGM).....................133
5.2.5ConvergenceofIterativeMethods......................135
5.3CodefortheSolutionofLinearSystems.......................140
5.3.1VBACode..........................................140
5.3.2MATLABCode.....................................141
5.4IllustrativeExamples........................................143
5.4.1PricingaPlainVanillaCallintheBlack(R)CScholesMode
(VBA).............................................144
5.4.2PricingaPlainVanillaCallintheSquare-RootModel(VBA)145
5.4.3PricingAmericanOptionswiththeCNScheme(VBA)....147
5.4.4PricingaDoubleBarrierCallintheBSModel(MATLAB
andVBA)..........................................149
5.4.5PricinganOptiononaCouponBondintheCox(R)CIngersoll
RossModel(MATLAB)..............................152
5.5Comments................................................155
6QuadratureMethods ...........................................157
6.1QuadratureRules...........................................158
6.2Newton(R)CCotesFormulae....................................15
6.2.1CompositeNewton(R)CCotesFormula.....................16
6.3GaussianQuadratureFormulae...............................173
6.4MatlabCode...............................................180
6.4.1TrapezoidalRule....................................180
6.4.2SimpsonRule.......................................180
6.4.3RombergExtrapolation...............................181
6.5VBACode................................................181
6.6AdaptiveQuadrature........................................182
6.7Examples.................................................185
6.7.1VanillaOptionsintheBlack(R)CScholesModel.............18
6.7.2VanillaOptionsintheSquare-RootModel...............188
6.7.3BondOptionsintheCox(R)CIngersoll(R)CRossModel..........1

x
6.7.4DiscretelyMonitoredBarrierOptions...................193
6.8PricingUsingCharacteristicFunctions.........................197
6.8.1MATLABandVBAAlgorithms........................202
6.8.2OptionsPricingwithL(R)PvyProcesses....................20
6.9Comments................................................211
7TheLaplaceTransform .........................................213
7.1De?nitionandProperties....................................213
7.2NumericalInversion........................................216
7.3TheFourierSeriesMethod...................................218
7.4ApplicationstoQuantitativeFinance..........................219
7.4.1Example............................................219
7.4.2Example............................................225
7.5Comments................................................228
8StructuringDependenceusingCopulaFunctions ..................231
8.1CopulaFunctions..........................................231
8.2ConcordanceandDependence................................233
8.2.1Fr(R)Pchet(R)CHoeffdingBounds............................2
8.2.2MeasuresofConcordance.............................234
8.2.3MeasuresofDependence..............................235
8.2.4ComparisonwiththeLinearCorrelation.................236
8.2.5OtherNotionsofDependence..........................238
8.3EllipticalCopulaFunctions..................................240
8.4ArchimedeanCopulas.......................................245
8.5StatisticalInferenceforCopulas..............................251
8.5.1ExactMaximumLikelihood...........................253
8.5.2InferenceFunctionsforMargins........................254
8.5.3Kernel-basedNonparametricEstimation.................255
8.6MonteCarloSimulation.....................................257
8.6.1DistributionalMethod................................257
8.6.2ConditionalSampling................................259
8.6.3CompoundCopulaSimulation.........................263
8.7Comments................................................265
PartIIProblems
PortfolioManagementandTrading ..................................271
9PortfolioSelection:!°Optimizing!±anErr........................273
9.1ProblemStatement.........................................274
9.2ModelandSolutionMethodology.............................276
9.3ImplementationandAlgorithm...............................278
9.4ResultsandComments......................................280
9.4.1In-sampleAnalysis...................................281

xi
9.4.2Out-of-sampleSimulation.............................285
10Alpha,BetaandBeyond ........................................289
10.1ProblemStatement.........................................290
10.2SolutionMethodology......................................291
10.2.1ConstantBeta:OLSEstimation........................292
10.2.2ConstantBeta:RobustEstimation......................293
10.2.3ConstantBeta:ShrinkageEstimation....................295
10.2.4ConstantBeta:BayesianEstimation.....................296
10.2.5Time-VaryingBeta:ExponentialSmoothing..............299
10.2.6Time-VaryingBeta:TheKalmanFilter..................300
10.2.7Comparingthemodels................................304
10.3ImplementationandAlgorithm...............................306
10.4ResultsandComments......................................309
11AutomaticTrading:WinningorLosinginakBit ..................311
11.1ProblemStatement.........................................312
11.2ModelandSolutionMethodology.............................314
11.2.1MeasuringTradingSystemPerformance.................314
11.2.2StatisticalTesting....................................315
11.3Code.....................................................317
11.4ResultsandComments......................................322
VanillaOptions ....................................................329
12EstimatingtheRisk-NeutralDensity .............................331
12.1ProblemStatement.........................................332
12.2SolutionMethodology......................................332
12.3ImplementationandAlgorithm...............................335
12.4ResultsandComments......................................338
13An!°American!±MonteCar ....................................345
13.1ProblemStatement.........................................346
13.2ModelandSolutionMethodology.............................347
13.3ImplementationandAlgorithm...............................348
13.4ResultsandComments......................................349
14FixingVolatileVolatility ........................................353
14.1ProblemStatement.........................................354
14.2ModelandSolutionMethodology.............................356
14.2.1AnalyticalTransforms................................356
14.2.2ModelCalibration...................................358
14.3ImplementationandAlgorithm...............................360
14.3.1CodeDescription....................................361
14.4ResultsandComments......................................362

xii
ExoticDerivatives ..................................................371
15AnAverageProblem ...........................................373
15.1ProblemStatement.........................................374
15.2ModelandSolutionMethodology.............................374
15.2.1MomentMatching...................................375
15.2.2UpperandLowerPriceBounds........................378
15.2.3NumericalSolutionofthePricingPDE..................379
15.2.4TransformApproach.................................382
15.3ImplementationandAlgorithm...............................386
15.4ResultsandComments......................................390
16Quasi-MonteCarlo:AnAsianBet ...............................395
16.1ProblemStatement.........................................396
16.2SolutionMetodology.......................................398
16.2.1Strati?cationandLatinHypercubeSampling.............399
16.2.2LowDiscrepancySequences...........................401
16.2.3DigitalNets.........................................402
16.2.4TheSobol°OSequence.................................40
16.2.5ScramblingTechniques...............................404
16.3ImplementationandAlgorithm...............................406
16.4ResultsandComments......................................407
17LookbackOptions:ADiscreteProblem...........................411
17.1ProblemStatement.........................................412
17.2ModelandSolutionMethodology.............................414
17.2.1AnalyticalApproach.................................414
17.2.2FiniteDifferenceMethod.............................417
17.2.3MonteCarloSimulation..............................419
17.2.4ContinuousMonitoringFormula.......................419
17.3ImplementationandAlgorithm...............................420
17.4ResultsandComments......................................421
18ElectrifyingthePriceofPower ..................................427
18.1ProblemStatement.........................................429
18.1.1TheDemandSide....................................429
18.1.2TheBidSide........................................429
18.1.3TheBidCostFunction................................430
18.1.4TheBidStrategy.....................................432
18.1.5AMulti-PeriodExtension.............................433
18.2SolutionMethodology......................................433
18.3ImplementationandExperimentalResults......................435
19ASparklingOption ............................................441
19.1ProblemStatement.........................................441
19.2ModelandSolutionMethodology.............................444

xiii
19.3ImplementationandAlgorithm...............................450
19.4ResultsandComments......................................453
20SwingingonaTree .............................................457
20.1ProblemStatement.........................................458
20.2ModelandSolutionMethodology.............................460
20.3ImplementationandAlgorithm...............................461
20.3.1GasPriceTree.......................................461
20.3.2BackwardRecursion.................................463
20.3.3Code...............................................464
20.4ResultsandComments......................................464
Interest-RateandCreditDerivatives .................................469
21FloatingMortgages.............................................471
21.1ProblemStatementandSolutionMethod.......................473
21.1.1Fixed-RateMortgage.................................473
21.1.2Flexible-RateMortgage...............................474
21.2ImplementationandAlgorithm...............................476
21.2.1MarkovControlPolicies..............................476
21.2.2DynamicProgrammingAlgorithm......................477
21.2.3TransactionCosts....................................480
21.2.4Code...............................................480
21.3ResultsandComments......................................482
22BasketDefaultSwaps...........................................487
22.1ProblemStatement.........................................487
22.2ModelsandSolutionMethodologies...........................489
22.2.1Pricingnth-to-defaultHomogeneousBasketSwaps........489
22.2.2ModellingDefaultTimes..............................490
22.2.3MonteCarloMethod.................................491
22.2.4AOne-FactorGaussianModel.........................491
22.2.5Convolutions,CharacteristicFunctionsandFourier
Transforms.........................................493
22.2.6TheHullandWhiteRecursion.........................495
22.3ImplementationandAlgorithm...............................495
22.3.1MonteCarloMethod.................................496
22.3.2FastFourierTransform...............................496
22.3.3Hull(R)CWhiteRecursion................................49
22.3.4Code...............................................497
22.4ResultsandComments......................................497
23ScenarioSimulationUsingPrincipalComponents .................505
23.1ProblemStatementandSolutionMethodology..................506
23.2ImplementationandAlgorithm...............................508
23.2.1PrincipalComponentsAnalysis........................508

xiv
23.2.2Code...............................................511
23.3ResultsandComments......................................511
FinancialEconometrics .............................................515
24ParametricEstimationofJump-Diffusions........................519
24.1ProblemStatement.........................................520
24.2SolutionMethodology......................................520
24.3ImplementationandAlgorithm...............................522
24.3.1TheContinuousSquare-RootModel....................523
24.3.2TheMixed-JumpSquare-RootModel...................525
24.4ResultsandComments......................................528
24.4.1EstimatingaContinuousSquare-RootModel.............528
24.4.2EstimatingaMixed-JumpSquare-RootModel............530
25NonparametricEstimationofJump-Diffusions ....................531
25.1ProblemStatement.........................................532
25.2SolutionMethodology......................................533
25.3ImplementationandAlgorithm...............................535
25.4ResultsandComments......................................537
26ASmilingGARCH .............................................543
26.1ProblemStatement.........................................543
26.2ModelandSolutionMethodology.............................545
26.3ImplementationandAlgorithm...............................547
26.3.1CodeDescription....................................551
26.4ResultsandComments......................................554
AAppendix:ProofoftheThinningAlgorithm.......................557
BAppendix:SampleProblemsforMonteCarlo .....................559
CAppendix:TheMatlabSolver ...................................563
DAppendix:OptimalControl .....................................569
D.1SettinguptheOptimalStoppingProblem......................569
D.2ProofoftheBellmanPrincipleofOptimality....................570
D.3ProofoftheDynamicProgrammingAlgorithm..................570
Bibliography ......................................................573
Index .............................................................599

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