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ID:40351813
大小:4.42 MB
页数:49页
时间:2019-07-31
《Modeling the Implied Volatility Surface》由会员上传分享,免费在线阅读,更多相关内容在学术论文-天天文库。
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
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