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1、TheMomentFormulaforImpliedVolatilityatExtremeStrikesRogerW.Lee¤DepartmentofMathematics,StanfordUniversityCourantInstituteofMathematicalSciences,NYUFirstversion:March27,2002Thisversion:August11,2003ForthcominginMathematicalFinanceAbstractConsideroptionsonanonn
2、egativeunderlyingrandomvariablewitharbitrarydistribution.Intheabsenceofarbitrage,weshowthatatanymaturityT,thelarge-striketailoftheBlack-Scholesimpliedvolatilityskewisboundedbythesquarerootof2jxj=T,wherexislog-moneyness.Thesmallestcoeffi-cientthatcanreplacethe2
3、dependsonlyonthenumberoffinitemomentsintheunderlyingdistribution.Weprovethemomentformula,whichexpressesexplicitlythismodel-independentrelationship.Weprovealsothereciprocalmomentformulaforthesmall-striketail,andweexhibitthesymmetrybetweentheformulas.Themomentfo
4、rmula,whichevaluatesreadilyinmanycasesofpracticalinterest,hasapplicationstoskewextrapolationandmodelcalibration.Keywords:Impliedvolatility,moment.¤ThisworkwaspartiallysupportedbyanNSFMathematicalSciencesPostdoctoralFellowship.IthankMarcoAvellaneda,PeterCarr,A
5、mirDembo,DarrellDuffie,JimGatheral,AlanLewis,GeorgePapanicolaou,andseminarparticipantsatBancofAmericaSecurities,theCourantInstitute,StanfordUniversity,andtheSIAM50conference.Email:rogerlee@math.stanford.edu11IntroductionThissectionoutlinesbrieflythecontribution
6、ofthispaper,deferringtosections2and3theexplicitdefini-tionsofsometerminology.LetuswritethesquaredBlack-ScholesimpliedvolatilityI2asacoefficienttimesjxj=T,theratioofabsolute-log-moneynesstomaturity.InSection3,weshowthatasx!¥,thelimsupofthiscoefficientisanumberbR2
7、[0;2],whichcanbetermedtheright-handorOTM-callorlarge-striketailslope.Similarly,weshowthatasx!¡¥,thelimsupisanumberbL2[0;2],whichcanbetermedtheleft-handorOTM-putorsmall-striketailslope.Thenweestablishtheexplicitone-to-onecorrespondencelarge-striketailslopeÃ!nu
8、mberoffinitemomentsoftheunderlyingST:Inparticular,bR=2ifandonlyiftheunderlyinghasnofinitemomentsofordergreaterthan1;attheoppositeextreme,bR=0ifandonlyiftheunderlyinghasfinitemomentsofallposi
