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1、Multi-scaleJumpandVolatilityAnalysisforHigh-FrequencyFinancialData¤JianqingFanandYazhenWangVersionofMay2007AbstractThewideavailabilityofhigh-frequencydataformany¯nancialinstrumentsstimulatesanupsurgeinterestinstatisticalresearchontheestimationofvolatil-ity.Jump-di®us
2、ionprocessesobservedwithmarketmicrostructurenoisearefrequentlyusedtomodelhigh-frequency¯nancialdata.Yet,existingmethodsaredevelopedforeithernoisydatafromacontinuousdi®usionpricemodelordatafromajump-di®usionpricemodelwithoutnoise.Weproposemethodstocopewithbothjumpsint
3、hepriceandmarketmicrostructurenoiseintheobserveddata.Theyallowustoestimatebothintegratedvolatilityandjumpvariationfromthedatasampledfromjump-di®usionpriceprocesses,contam-inatedwiththemarketmicrostructurenoise.Ourapproachisto¯rstremoveJianqingFanisFrederickMoore'18Pr
4、ofessorofFinance,DepartmentofOperationResearchandFinancialEngineering,PrincetonUniversity,Princeton,NJ08544.YazhenWangisprofessor,DepartmentofStatistics,UniversityofConnecticut,Storrs,CT06269.Fan'sresearchwaspartiallysupportedbysupportedbytheNSFgrantDMS-0532370andWan
5、g'sresearchwaspartiallysup-portedbytheNSFgrantDMS-0504323.Theauthorsthanktheeditor,associateeditor,andtwoanonymousrefereesforstimulatingcommentsandsuggestions,whichledtosigni¯cantimprovementsinbothsubstanceandthepresentationofthepaper.1jumpsfromthedataandthenapplynoi
6、se-resistantmethodstoestimatetheintegratedvolatility.Theasymptoticanalysisandthesimulationstudyrevealthattheproposedwaveletmethodscansuccessfullyremovethejumpsinthepriceprocessesandtheintegratedvolatilitycanbeestimatedasaccuratelyasinthecasewithnopresenceofjumpsinthe
7、priceprocesses.Inaddition,theyhaveoutstandingstatisticale±ciency.Themethodsareillustratedbyapplicationstotwohigh-frequencyexchangeratedatasets.1IntroductionDi®usionbasedstochasticmodelsareoftenemployedtodescribecomplexdynamicsystemswhereitisessentialtoincorporateinte
8、rnallyorexternallyoriginatingran-dom°uctuationsinthesystem.Theyhavebeenwidelyappliedtoproblemsin¯eldssuchasbiology,engineering,¯nan
