Dependence Structures for Multivariate High--Frequ

DependenceStructuresforMultivariateHigh–FrequencyDatainFinanceWolfgangBreymann∗,AlexandraDias†,PaulEmbrechtsDepartmentofMathematicsETHZCH–8092Z¨urichSwitzerland–frequencydatainfinancearewell–known.Theyincludescalingbehaviour,volatilityclustering,heavytails,andseasonalities.Themultivariateproblem,however,hasscarcelybeenaddresseduptonow.Inthispaper,bivariateseriesofhigh–frequencyFXspotdataformajorFXmarketsareinves-tigated.First,asanindispensableprerequisiteforfurtheranalysis,theproblemofsimultaneousdeseasonalisationofhigh–frequencydataisaddressed.Inthebulkofthepaperweanalyseinde-tailthedependencestructureasafunctionofthetimescale.Particularemphasisisputonthetailbehaviour,whichisinvestigatedbymeansofcopulasandspectralmeasures.1IntroductionNumerouspapershavestudiedstatisticalpropertiesofone–dimensionalreturndatainfinance.Resultslikeleptokurtosis,stochasticvolatilityeffects,occurrenceofextremes,seasonalities,andscalingbehav-iorarenowreferencedtoasstylisedfactsofempiricalfinance.TheworkbyOlsen&Associateshasextendedthesefactsacrosssamplingfrequenciesreachingfromminutestomonths;seeforinstanceDa-corognaetal.(2001).Similarresultsformore–dimensionalreturndataarehoweverscarce.InEmbrechtsetal.(2002)someofthebasictechniquesfortheanalysisofdependencebeyondlinearcorrelationwereintroducedthroughthenotionofcopula.Inthispaper,thelattertechniqueswillbeappliedtoatwo–dimensionalhigh–frequence(hourly)datasetofFXreturns.Assuch,thechangeindependenceasafunctionofthesamplingfrequencywillbeestablished.Also,foreachseparatefrequency,ellipticalitywillbetested.Finally,severalstatisticaltechniquesforthestudyofextremalclusteringinhigherdimen-sionswillbeapplied.Asanecessaryprerequisiteforthisanalysis,amethodfordeseasonalisingbivariatereturnsfortimehorizonsuptoonedaywillbepresented.Theoutlineofthepaperisasfollows.Section2presentsthetransformationfromtherawhigh–frequency(tick–by–tick)datatoproperlydeseasonaliseddata.InSection3,severalfamiliesofcopulaswillbefittedtodeseasonalisedtwo–dimensionalFXdata,andthisatseveralfrequencies(fromhourlytodaily).Goodness–of–fittests,includingtestsforellipticality,arepresentedinSection4.InSection5,∗ResearchsupportedbyCreditSuisseGroup,SwissRe,andUBSAGthroughRiskLab,Switzerland.†SupportfromFundac¸˜aoparaaCiˆenciaeaTecnologia-FCT/POCTIandFaculdadedeCiˆenciaseTecnologia,UniversidadeNovadeLisboa,Portugal,isgratefullyacknowledged.1theproblemofclusteringofextremesintwodimensionaldataisdiscussed.Finally,Section6givesanoutlookforfurtherresearch.2ThedataWeinvestigateahigh-frequencybivariatetimeseriesconsistingofUSD/DEMandUSD/JPYspotrates.Beforetheyaretobeusedfordependencyanalysis,thefollowingpreliminarystepshavebeenperformed:•collectionandfiltering,•regularisationandtransformationtologarithmicmiddleprices,and•deseasonalisation.Theyaredescribedinturn.2.1CollectionandfilteringThedatasetconsistsoftick–by–tickdataoriginatingmainlyfromReutersandcollectedandfilteredbyOlsenData.Itconsistsofalargepartbutnotallofthequotesemittedinthemarketbecausethemarketcoverageofthedataprovidersitnotcompleteanddependsontheregionoftheworld.Thehigh-frequencyseriesareirregularlyspaced;theystartFebruary1986andendJune30,2001.Sincewearein-terestedinUSD/DEM,whichendsDecember31,1998,wediscardlaterdataalsoforothercurrencies.Asinglequoteattimetconsistsofabidprice,pBidα,t,andanaskprice,pAskα,t,α∈{USD/DEM,USD/JPY}.Forbothseries,middlepricesaredisplayedinFigure1.Inafirststepthedataarecleanedbymeansofaspecialfilter,describedinDacorognaetal.(2001),thattakespeculiaritiesofthefinancialmarketintoaccount.Amongothersitcorrectsfordecimalerrorscausedbythetransmissionlineandremovesautomaticallygeneratedfakequotesduringinactiveperiodsusedbymarketparticipantstotestthetrans-missionchannel.Sinceonlyasmallfractionofquotesisremovedbythefilterthefilteredtimeseriesisstillirregularlyspaced,andthenumberofdatapointsisveryhigh(about10millionfortheUSD/DEMseries).2.2RegularisationandtransformationtologarithmicmiddlepricesToreducethedataweregularisethetimeseriestoaregularserieswithstepsizeδ=5minutesbylinearinterpolation.Sincewearenotinterestedineffectsrelatedtothebid–askspread,wewillworkwithlogarithmicmiddlepricesξα,tdefinedasξα,t=logpBidα,t·pAskα,t2.(1)ReturnswithrespecttoatimehorizonΔTarethendefinedasthedifferenceoflogarithmicmiddleprices:rα,t[ΔT]=ξα,t−ξα,t−ΔT.(2)HourlyUSD/DEMreturnsaredisplayedinFigure2.Theadvantageoftakingthelogarithmisthatreturnsoftheinvertedrate(e.g.DEM/USD)arejustthenegativeofthecorrespondingreturnsoftheoriginalrateUSD/DEM,asoneintuitivelyexpects.2Figure1:FXmiddlepricesforUSD/DEM(top)andUSD/JPY(bottom).3Figure2:HourlyUSD/DEMreturnsofthewhole11-yearperiod(top)andfor1year(bottom).Noticetheweeklyseasonalities.42.3DeseasonalisationoffinancialdataPracticallyallfinancialtimeseriesexhibitseasonalities.Themoststrikingoneistheabsenceofanyactivityduringweekends,whichcausesaweeklyseasonalityintheautocorrelationfunctionoflaggedabsolutereturns.Withhigh–frequencydatatheproblemofseasonalitybecomesmuchmoreimportantandmorediffic