A.Mijatovic - Stochastic Volatility Models with Jumps [Warwick 2011]

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1、StochasticVolatilityModelswithJumpsExoticDerivativesinFinancialMarketsAleksandarMijatovic´DepartmentofStatistics,UniversityofWarwickMASDOC:StatisticalFrontiersST912January2011StochasticVolatilityModelswithJumps–p.1OverviewoftheCoursePart0:WhatgoeswrongintheBlack-Scholesworld?PartI:TheModel

2、s(SVJ)PartII:ExoticDerivatives(volatilityderivatives,forward-startingoptions,asymptoticsoftheimpliedvolatilitysmile)PartIII:FluctuationTheoryandBarrierOptionsStochasticVolatilityModelswithJumps–p.2Black-Scholesmodelσ2St=S0·expµ−t+σWt,2Dailylog-returnYt+1=log[St+1/St]isnormal:2Yt=µ−σ/2+

3、σZwhereZ∼N(0,1)meanµ−σ2/2varianceσ2skewness0kurtosis3Financialdataexhibits:•asymmetriclog-returns•excesskurtosisoflog-returns(wellabove3)StochasticVolatilityModelswithJumps–p.3FTSE100dailylog-returnsfrom1984to20091ááææææá0.1áææáá0.01áææáááá0.001áááææááá-0.03-0.02-0.010.000.010.020.03Dailyr

4、eturns•empiricaldistribution:mean0.01%,volatility7.83%,skewness-0.401,kurtosis:9.085.•normaldistribution(maximumlikehoodestimate)StochasticVolatilityModelswithJumps–p.4ProblemswithBalck-Scholesmodel(a)Log-returnshavethintailsandsymmetricdistribution.(b)Noauto-correlationinlog-returns.(c)Ca

5、librationacrossstrikes/maturitiesimpossible.(d)Exoticoptionpricesarewidelyoffthemarket.StochasticVolatilityModelswithJumps–p.5Whatcanwedoabouttheseproblems?Twoessentialingrediantsofthesolution:(i)Replaceσbyastochasticprocess(stochasticvolatility).(ii)Introducejumpsintheevoluationofthestock

6、price.Keyconsiderationsinfinancialapplications:•AnalyticaltractabilityforpricesandGreeks.•Numericalaccuracyofpricingalgorithms.StochasticVolatilityModelswithJumps–p.6PartITheModelsStochasticVolatilityModelswithJumps–p.7TheModelsWhataremodelsusedfor?Understandingtheriskofportfoliosofderivati

7、vesecurities:•Pricing•Hedging•RiskManagementFeaturestheymustpossess:•Jumps(GammaRegime)•StochasticityofVolatility(VegaRegime,VolatilityClustering)•AnalyticalTractability(Calibration,HedgingandRiskManagemement)StochasticVolatilityModelswithJumps–p.8Regimeswitch