清华大学金融学lessonstochasticvolatility

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1、个人收集整理勿做商业用途封面作者：ZHANGJIAN仅供个人学习，勿做商业用途5/5个人收集整理勿做商业用途1.Impliedvolatility&volatilitysmilePayoffOutofmoneyputsoutofmoneycallsk3k2k1St=k0k4k5underlyingstockpriceThevolatilityimpliedbyoptionpriceobservedinthemkt.-InB-S,isassumedtobeconstant:-Inreality,-Someempiricalfacts:(1)Optionsoneqityindices:o

2、ne-sided“smile”or“skew”精品文档收集整理汇总ImpliedvolatilitySt=equityindexStkn…k2k1k0精品文档收集整理汇总(2)OptionsonFX:moreorlesssymmetric“smile”Impliedvolatilityet=exchangerateetkn…k2k1k0k1knlumpiervolatility精品文档收集整理汇总(3)Optionsoninterestrates:moremonotonousone-sidedskew(Caps/Floors/Swaptions)精品文档收集整理汇总2.Arbitra

3、gePricingandHedgingwithSmile-Thegoal:Tobuildaspotprocessthat:a)iscompatiblewiththeobservedSmilesatallmaturity.精品文档收集整理汇总b)keepsthemodelcompleteGivenEuropeancallsofallstrikesKandmaturityT:C(K,T)toconstructarisk-neutralprocessforthespotprice:精品文档收集整理汇总where(S,t)isadeterministicfunctionofspotprice

4、sandtimet.精品文档收集整理汇总-Theproblem:Assumer=0,fT(S)istherisk-neutraldensityfunctionofthespotpriceattimeT.精品文档收集整理汇总Fokker-Planckequation:IntegratetwiceinSforconstt:Assume:3.ArbitragepricingwithstochasticvolatilityAssume:acontinuumofallCK,Taretraded.3.1LogcontractForsimplicityassumer=05/5个人收集整理勿做商业用

5、途Def:ALogcontractisacontingentclaimonS,withpayoffattimeTequalslog(ST).(Logcontractcanbesynthesized)LetitspricebedenotedLT(t).精品文档收集整理汇总3.2ForwardvariancebyIto’slemma:CumulativeinstantaneousvarianceorFwdvariancebetweenT1andT2Self-financingstrategyTwologcontract精品文档收集整理汇总3.3ArbitragefreepriceofFw

6、dvariance:2(LT1(t)-LT2(t))精品文档收集整理汇总Def:FwdtimeTimeT,instantaneousvariance(observedatt):精品文档收集整理汇总3.4Stochasticassumptionsonfwdvarianceandrisk-neutralprocess精品文档收集整理汇总3.5risk-neutralprocessesforinstantaneousvarianceandvolatility精品文档收集整理汇总Defineinstantaneousvarianceattimet3.6Contingentclaimprici

7、ngRisk-neutralprocessforjointLetAbeacontingentdaimthatdeliversinTapayoffdependentonthepathsfollowedbyspotSandvariance,i.e.,asquare-integrable-measurablerandomvariableon.精品文档收集整理汇总5/5个人收集整理勿做商业用途版权申明本文部分内容，包括文字、图片、以及设计等在网上搜集整理。版权为张