Modeling the Implied Volatility Surface

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1、ModelingtheImpliedVolatilitySurfaceJimGatheralStanfordFinancialMathematicsSeminarFebruary28,2003ThispresentationrepresentsonlythepersonalopinionsoftheauthorandnotthoseofMerrillLynch,itssubsidiariesoraffiliates.JimGatheral,MerrillLynch,February-2003Outlineofthistal

2、knAcompoundPoissonmodelofstocktradingnTherelationshipbetweenvolatilityandvolumenClusteringnCorrelationbetweenvolatilitychangesandlogreturnsnStochasticvolatilitynDynamicsofthevolatilityskewnSimilaritiesbetweenstochasticvolatilitymodelsnDostochasticvolatilitymodelsf

3、itoptionprices?nJumpsnTheimpactoflargeoptiontradesJimGatheral,MerrillLynch,February-2003StocktradingasacompoundPoissonprocessnConsiderarandomtimechangefromconventionalcalendartimetotradingtimesuchthattherateofarrivalofstocktradesinagiven(transformed)timeintervalis

4、aconstant.lnIntuitively,relativetorealtime,tradingtimeflowsfasterwhenthereismoreactivityinthestockandmoreslowlywhenthereislessactivity.nSupposethatthe(random)sizeofatradeisindependentofthelevelofnactivityinthestock.nAssumefurtherthateachtradeimpactsthemid-log-pric

5、eofthestockbyanamountproportionalto.n•ThisisastandardassumptioninthemarketmicrostructureliteraturenThenthechangeinlog-mid-priceoversometimeintervalisgivenbyND=xsgn()nnåiii=1nNotethatboththenumberoftradesandthesizeofeachtradeinaNnigiventimeintervalarerandom.DtJimGa

6、theral,MerrillLynch,February-2003Volatilityandvolume:arelationshipnThevarianceofthisrandomsumofrandomvariablesisgivenby2Var[Dx]=+EN[]VaréëaaniiùûVarN[]Enéùëû2=DaltEn[i]nRewritingthisintermsofvolatility,weobtain22Var[Dx]=sDt=DaltEn[i]nButisjusttheexpectationofthevo

7、lumeoverthetimelDtEn[i]interval.DtnThefactorcancelsandtransformingbacktorealtime,weseethatDtvarianceisdirectlyproportionaltovolumeinthissimplemodel.nMoreover,thedistributionofreturnsintradingtimeisapproximatelyGaussianforlarge.DtJimGatheral,MerrillLynch,February-2

8、003TherelationshipnnAkeyassumptioninoursimplemodelisthatmarketimpactisproportionaltothesquarerootofthetradesize.Thefollowingnargumentshowswhythisisplaus