asymptotic analysis of volatility

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1、CHAPTER55asymptoticanalysisofvolatilityInthisChapter...•asymptoticanalysisandsmallorlargeparameters•aseriessolutionforimpliedvolatilityunderarbitrarystochasticvolatility55.1INTRODUCTIONAsymptoticanalysisisasystematicmethodforexploitingthelargenessorsmallnessofaparameterinsomeequation.Inourcon

2、texttheequationisthatforanoptionunderstochasticvolatilityandtheparametersmeasurethespeedofmeanreversionofvolatilityandthevolatilityofvolatility.Themeanreversionisfast(largeparameter)andthevolatilityofvolatilityislarge(anotherlargeparameter).Sincewe’llbelookingforasymptoticsolutionswe’llseehow

3、theprecisespecificationofthemodelisirrelevantasfarasfindingclosed-formsolutionsisconcerned.Inotherwords,wedon’thavetosacrificeaccuracyfortractabilityanymore.ThischapterisheavilybasedonapaperbymyselfandHenrikRasmussen,Rasmussen&Wilmott(2002).55.2FASTMEANREVERSIONANDHIGHVOLATILITYOFVOLATILITYWecon

4、siderthepricingofoptionswhentheunderlyingassetvalueSanditsvolatilityσaredescribedbythestochasticdifferentialequations,dS/S=rdt+σdXdσ=Adt+BdYdX·dY=ρdtwhereXandYareBrownianmotions,ristheshortrate,andthecoefficientsAandBarefunctionsofonlyσ.Whencalibratingsuchastochasticvolatilitymodeltomarketpric

5、es,one902PartFiveadvancedtopicsusuallyfindsthatthevolatilityofvolatilityB/σisgreaterthanthevolatilityσoftheunderlying.Forinstance(Wiggins,1987),σ∝0.2B∝0.2,inwhichcasetheratiobetweenvolatilityandthevolatilityofvolatilitybecomesB∝5.σ2Accordingly,weintroduceasmallparameterβB=√,suchthat(fortypic

6、alvaluesofσ)β∝1.0σ2whichimplies∝0.04.TheparametercanbeinterpretedasaratiooftimescalesforfluctuationsinvolatilityσandintheassetpriceS.IfTisacharacteristictimescaleforS,thenTisacharacteristictimescaleforthevolatilityprocessσ.Notethatwearewritinglargeparametersasinversesofasmallparameter.That’

7、sbecausewewillshortlybelookingforaseriesexpansioninpowersof.Sincewewouldliketothinkofasaratioofcharacteristictimescales,weletthedrifttermAscalelikeA=α/,andtherebyarriveattheequationsdS/S=rdt+σdXαβdσ=dt+√dYdX·dY=ρdt.LetVdenotethevalueofaE