LONG MEMORY STOCHASTIC VOLATILITY IN OPTION PRICING

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1、May3,200514:8WSPC-104-IJTAFSPI-J07100301InternationalJournalofTheoreticalandAppliedFinanceVol.8,No.3(2005)381–392
cWorldScientificPublishingCompanyLONGMEMORYSTOCHASTICVOLATILITYINOPTIONPRICINGSERGEIFEDOTOV∗andABBYTANSchoolofMathematics,TheUniversityofManchesterM601QD,

2、UK∗sergei.fedotov@manchester.ac.ukReceived16March2004Accepted21September2004Theaimofthispaperistopresentastochasticmodelthataccountsfortheeffectsofalong-memoryinvolatilityonoptionpricing.ThestartingpointisthestochasticBlack–Scholesequationinvolvingvolatilitywithlong-r

3、angedependence.WedefinethestochasticoptionpriceasasumofclassicalBlack–Scholespriceandrandomdeviationdescribingtheriskfromtherandomvolatility.Byusingthefactthattheoptionpriceandrandomvolatilitychangeondifferenttimescales,wederivetheasymptoticequationforthisdeviationinvo

4、lvingfractionalBrownianmotion.Thesolutiontothisequationallowsustofindthepricingbandsforoptions.Keywords:Longmemory;stochasticvolatility;optionpricing.1.IntroductionOverthelastfewyears,self-similarityandlong-rangedependencehavebecomeimportantconceptsinanalyzingthefinanc

5、ialtimeseries[24,26].Thereisstrongevidencethatthereturn,rt,haslittleornoautocorrelation,whereasitssquare,r2,orabsolutereturn,

6、r

7、,exhibitnoticeableautocorrelation[6].ThisphenomenonttcanbedescribedbytheARCH(p)model[14]oritsGARCH(p,q)extension[4].However,theexponentiald

8、ecayforλ=cov(r2,r2)isbelievedtobetoofasttostt+sdescribecorrectlythepersistentdependencebetweentheseriesobservationsasthetimelagincreases.Itturnsout[3,27]thatthemodelswithhyperbolicdecaywhichhaveslowlydecayingcovariancesprovidebetterfittingtofinancialtimeseries.Thechara

9、cteristicfeatureofthesemodelsisthattheircovarianceλshasthepowerlawdecays2d−1(0

10、ay3,200514:8WSPC-104-IJTAFSPI-J07100301382S.Fedotov&A.Tanthecorrelationsdecayveryslowlytozero.Letusnotethattheseriesissaidtohavesho