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1、MULTISCALEMODEL.SIMUL.c2003SocietyforIndustrialandAppliedMathematicsVol.2,No.1,pp.22–42MULTISCALESTOCHASTICVOLATILITYASYMPTOTICS∗JEAN-PIERREFOUQUE†,GEORGEPAPANICOLAOU‡,RONNIESIRCAR§,ANDKNUTSOLNA¶Abstract.Inthispaperweproposetouseacombinationofregularandsingularperturbationstoanalyzepara
2、bolicPDEsthatariseinthecontextofpricingoptionswhenthevolatilityisastochasticprocessthatvariesonseveralcharacteristictimescales.TheclassicalBlack–ScholesformulagivesthepriceofcalloptionswhentheunderlyingisageometricBrownianmotionwithaconstantvolatility.Theunderlyingmightbethepriceofastock
3、oranindex,say,andaconstantvolatilitycorrespondstoafixedstandarddeviationfortherandomfluctuationsinthereturnsoftheunderlying.Modernmarketphenomenamakeitimportanttoanalyzethesituationwhenthisvolatilityisnotfixedbutratherisheterogeneousandvarieswithtime.Inpreviouswork(see,forinstance,[J.P.Fouq
4、ue,G.Papanicolaou,andK.R.Sircar,DerivativesinFinancialMarketswithStochasticVolatility,CambridgeUniversityPress,Cambridge,UK,2000]),weconsideredthesituationwhenthevolatilityisfastmeanreverting.Usingasingularperturbationexpansionwederivedanapproximationforoptionprices.Wealsoprovidedacalibr
5、ationmethodusingobservedoptionpricesasrepresentedbytheso-calledtermstructureofimpliedvolatility.Ouranalysisofmarketdata,however,showstheneedforintroducingalsoaslowlyvaryingfactorinthemodelforthestochasticvolatility.Thecombinationofregularandsingularperturbationsapproachthatwesetforthinth
6、ispaperdealswiththiscase.Theresultingapproximationisstillindependentoftheparticulardetailsofthevolatilitymodelandgivesmoreflexibilityintheparametrizationoftheimpliedvolatilitysurface.Inparticular,theintroductionoftheslowfactorgivesamuchbetterfitforoptionswithlongermaturities.Weuseoptiondat
7、atoillustrateourresultsandshowhowexoticoptionpricesalsocanbeapproximatedusingourmultiscaleperturbationapproach.Keywords.stochasticvolatility,timescales,singularregularperturbations,optionpricing,impliedvolatilityAMSsubjectclassifications.34E10,34E13,35K20,60H15,60H30,60J60
