随机波动金融市场衍生品

Thisbookaddressesproblemsinfinancialmathematicsofpricingandhedgingderivativesecuritiesinanenvironmentofuncertainandchangingmarketvolatility.Theseproblemsareimportanttoinvestorsrangingfromlargetradinginstitutionstopensionfunds.Theauthorspresentmathematicalandstatisticaltoolsthatexploitthe"bursty"natureofmarketvolatility.Themathematicsisintroducedthroughexamplesandillustratedwithsimulations,andtheapproachdescribedisvalidatedandtestedonmarketdata.

Thematerialissuitableforaone-semestercourseforgraduatestudentswhohavebeenexposedtomethodsofstochasticmodelingandarbitragepricingtheoryinfinance.Itiseasilyaccessibletoderivativespractitionersintheinancialengineeringindustry.

Introduction
1TheBlack-ScholesTheoryofDerivativePricing
1.1MarketModel
1.1.1BrownianMotion
1.1.2StochasticIntegrals
1.1.3RiskyAssetPriceModel
1.1.4ItosFormula
1.1.5LognormalRiskyAssetPrice
1.2DerivativeContracts
1.2.1EuropeanCallandPutOptions
1.2.2AmericanOptions
1.2.3OtherExoticOptions
1.3ReplicatingStrategies
1.3.1ReplicatingSelf-FinancingPortfolios
1.3.2TheBlack-ScholesPartialDifferentialEquation
1.3.3PricingtoHedge
1.3.4TheBlack-ScholesFormula
1.4Risk-NeutralPricing
1.4.1EquivalentMartingaleMeasure
1.4.2Self-FinancingPortfolios
1.4.3Risk-NeutralValuation
1.4.4UsingtheMarkovProperty
1.5Risk-NeutralExpectationsandPartialDifferentialEquations
1.5.1InfinitesimalGeneratorsandAssociatedMartingales
1.5.2ConditionalExpectationsandParabolicPartialDifferentialEquations
1.5.3ApplicationtotheBlack-ScholesPartialDifferentialEquation
1.5.4AmericanOptionsandFreeBoundaryProblems
1.5.5Path-DependentDerivatives
1.6CompleteMarket

2IntroductiontoStochasticVolatilityModels
2.1ImpliedVolatilityandtheSmileCurve
2.1.1InterpretationoftheSmileCurve
2.1.2WhatDatatoUse
2.2ImpliedDeterministicVolatility
2.2.1Time-DependentVolatility
2.2.2Level-DependentVolatility
2.2.3Short-TimeTightFitversusLong-TimeRoughFit
2.3StochasticVolatilityModels
2.3.1Mean-RevertingStochasticVolatilityModels
2.3.2Stock-PriceDistributionunderStochasticVolatility
2.4DerivativePricing
2.5PricingwithEquivalentMartingaleMeasures
2.6ImpliedVolatilityasaFunctionofMoneyness
2.7MarketPriceofVolatilityRiskandData
2.8SpecialCase：UncorrelatedVolatility
2.8.1Hull-WhiteFormula
2.8.2StochasticVolatilityImpliesSmile
2.8.3RemarkonCorrelatedVolatility
2.9SummaryandConclusions

3ScalesinMean-RevertingStochasticVolatility
3.1ScalinginSimpleModels
3.2ModelsofClustering
3.2.1Example：MarkovChain
3.2.2Example：AnotherJumpProcess
3.2.3Example：Ornstein-UhlenbeckProcess
3.2.4Summary
3.3ConvergencetoBlack-ScholesunderFastMean-RevertingVolatility
3.4ScalesintheReturnsProcess
3.4.1TheReturnsProcess
3.4.2ReturnsProcesswithJumpVolatility
3.4.3ReturnsProcesswithOUVolatility
3.4.4S&P500ReturnsProcess

4ToolsforEstimatingtheRateofMeanReversion
4.1ModelandData
4.1.1Mean-RevertingStochasticVolatility
4.1.2DiscreteData
4.2VariogramAnalysis
4.2.1ComputationoftheVariogram
4.2.2ComparisonandSensitivityAnalysiswithSimulatedData
4.2.3TheDayEffect
4.3SpectralAnalysis

5AsymptoticsforPricingEuropeanDerivatives
5.1Preliminaries
5.1.lTheRescaledStochasticVolatilityModel
5.1.2TheRescaledPricingEquation
5.1.3TheOperatorNotation
5.2TheFormalExpansion
5.2.1TheDivergingTerms
5.2.2PoissonEquations
5.2.3TheZero-OrderTerm
5.2.4TheFirstCorrection
5.2.5UniversalMarketGroupParameters
5.2.6ProbabilisticInterpretationoftheSourceTerm
5.2.7Put-CallParity
5.2.8TheSkewEffect
5.3ImpliedVolatilitiesandCalibration
5.4AccuracyoftheApproximation
5.5RegionofValidity

6ImplementationandStability
6.1Step-by-StepProcedure
6.2CommentsabouttheMethod
6.3Dividends
6.4TheSecondCorrection

7HedgingStrategies
7.1Black-ScholesDeltaHedging
7.1.1TheStrategyandItsCost
7.1.2AveragingEffect
7.2MeanSelf-FinancingHedgingStrategy
7.3StayingClosetothePrice

8ApplicationtoExoticDerivatives
8.lEuropeanBinaryOptions
8.2BarrierOptions
8.3AsianOptions

9ApplicationtoAmericanDerivatives
9.1AmericanProblemunderStochasticVolatility
9.2StochasticVolatilityCorrectionforanAmericanPut
9.2.1Expansions
9.2.2FirstApproximation
9.2.3TheStochasticVolatilityCorrection
9.2.4UncorrelatedVolatility
9.2.5ProbabilisticRepresentation
9.3NumericalComputation
9.3.1SolutionoftheBlack-ScholesProblem
9.3.2ComputationoftheCorrection

10Generalizations
10.1PortfolioOptimizationunderStochasticVolatility
10.1.1ConstantVolatilityMertonProblem
10.1.2StochasticVolatilityMertonProblem
10.1.3APracticalSolution
10.2PeriodicDayEffect
10.3OtherMarkovianVolatilityModels
10.3.1MarkovianJumpVolatilityModels
10.3.2PricingandAsymptotics
10.4MartingaleApproach
10.4.1MainArgument
10.4.2DecompositionResult
10.4.3ComparisonwiththePDEApproach
10.5Non-MarkovianModelsofVolatility
10.5.1Setting：AnExample
10.5.2AsymptoticsintheNon-MarkovianCase
10.6MultidimensionalModels

11ApplicationstoInterest-RateModels
11.1BondPricingintheVasicekModel
11.1.1ReviewoftheConstantVolatilityVasicekModel
11.1.2StochasticVolatilityVasicekModels
11.2BondOptionPricing
11.2.1TheConstantVolatilityCase
11.2.2CorrectionforStochasticVolatility
11.2.3Implications
11.3AsymptoticsaroundtheCIRModel
11.4IllustrationfromData
11.4.1VariogramAnalysis
11.4.2YieldCurveFitting
Bibliography
Index