A summary about realized volatility in high freqeuncy financial data.docx
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A summary about realized volatility in high freqeuncy financial data.docx
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Asummaryaboutrealizedvolatilityinhighfreqeuncyfinancialdata
Asummaryaboutrealizedvolatilityinhighfreqeuncyfinancialdata
Abundanttradingdataiscreatedeverydayinnowaday'sfinancialworld,anditoffersplentyofinformation.AfterTorbenG.AndersenandTimBollerslev(1998)proposedinterdailyvolatilitymodel,moreandmoreresearchersfocusonstochasticvolatilityresearchabouthighfrequencydata.SuchasCduToitWJConradie(2006)studiedrealizedvolatilitymeasurementwithmicrostructureeffects,OleE.Barndorff-Nielsen(2002),whowritemanypapersfrom2001to2011inthisfield,userealizedvarianceandrealizedvolatilitytoestimatequadraticvariationandstochasticvolatilityrespectively,in2001,OleE.Barndorff-NielsenandNeilShephardstudiedhigherordervariationandstochasticvolatilitymodels,theyproposedanotation“higherordervariations”thatdefinedas
whenr>2.InastudybyBarndor-NielsenandShephard(2001d)ofthepropertiesofrealisedvolatility,thatisthesumofsquaresofintra-dayreturnsonspeculativeassets,itbecamenecessaryinadditiontoquadraticvariationofthestochasticprocessestoconsideralsoaspectsofhigherordervariation.Therequisitemathematicalresultsonhigherordervariationseemofsomeindependentinterest.
Nowthefollowingissomeintruductionof“realizedvolatility”.
Atriple(
;F;P)iscalledaprobabilityspace,if
isagivenset,Fisa
-algebraon
andPisaprobabilitymeasureon(
;F).
Asequence{Xn}ofrandomvariablesdefinedonaprobabilityspace(
;F;P)issaidtoconvergetoXinprobability,denotedby
if
{Xn}issaidtoconvergetoXinmeansquareif
Markov'sInequalityguaranteesthatconvergenceinmeansquareisstrongerthanconvergenceinprobability.It'swellknowthatconvergenceinprobabilityimpliesconvergenceindistribution.
Astochasticprocess
issaidtobeastandardBrownianmotionif
(1)B0=0;
(2)
hasstationaryindependentincrements;
(3)foreveryt>0,Btisnormallydistributedwithmean0andvariancet.
DenotebyFtthe
-algebrageneratedbytherandomvariable
andbyFthe
-algebrageneratedby
.Aprocessg(t;w):
iscalledFt-adaptedifforeacht
0thefunction
isFt-measurable.
SupposethatXtisareal-valuedFt-adaptedstochasticprocessdefinedonaprobabilityspace(
;F;P).Thequantity
iscalledtherealizedvolatilityofthestochasticprocessXt,where
isanypartitionoftheinterval[0,t].ThequadraticvariationofXt,ifexists,isdefinedbythefollowing
where
isthelengthofthelongestsubintervalinthepartition
FortheItoprocess
whereBtisastandardBrownianmotion,
isthedriftcoefficientand
istheinstantaneousvarianceofthereturnprocessXt,thefollowingresultiswellknown.
TheoremA.Let
betheBrownianmotion,and
beFt-adaptedstochasticprocess.ThenthequadraticvariationoftheItoprocessis
.
Let
betheclassoffunctions
suchthat
(1)
is
-measurable,whereBdenotestheBorel
-algebraon
.
(2)
isFt-adapted;
(3)
.
Theneweconometricsismotivatedbytheadventofcompleterecordsofquotesortransactionpricesformanyfinancialassets.Althoughmarketmicrostructureeffects(e.g.discretenessofprices,bid/askbounce,irregulartradingetc.)meansthatthereisamismatchbetweenassetpricingtheorybasedonsemimartingalesandthedataatveryfinetimeintervals(see,forexample,Bai,Russell,andTiao(2000))itdoessuggestthedesirabilityofestablishinganasymptoticdistributiontheoryforestimatorsasweusemoreandmorehighlyfrequentobservations.Realizedvariance,beingthesummationofsquaredintra-dayreturns,hasquicklygainedpopularityasameasureofdailyvolatility.FollowingParkinson(1980)wereplaceeachsquaredintra-dayreturnbythehigh-lowrangeforthatperiodtocreateanovelandmoreefficientestimatorcalledtherealizedrange.Intheory,therealizedvarianceisanunbiasedandhighlyefficientestimator,asillustratedinAndersenetal.(2001b),andconvergestothetrueunderlyingintegratedvariancewhenthelengthoftheintra-dayintervalsgoestozero,seeBarndorff-NielsenandShephard(2002).Inpractice,marketmicrostructureeffectssuchasbid-askbounceposelimitationstothechoiceofsamplingfrequency.Returnsatveryhighfrequenciesaredistortedsuchthattherealizedvariancebecomesbiasedandinconsistent,seeBandiandRussell(2005a,b),andHansenandLunde(2006b).Popularchoicesinempiricalapplicationsarethefive-andthirty-minuteintervals,whicharebelievedtostrikeabalancebetweentheincreasingaccuracyofhigherfrequenciesandtheadverseeffectsofmarketmicrostructurefrictions,seee.g.AndersenandBollerslev(1998),Andersenetal.(2001a),Andersenetal.(2003),andFlemingetal.(2003).Analternativewayofmeasuringvolatilityisbasedonthedifferencebetweenthemaximumandminimumpricesobservedduringacertainperiod.Parkinson(1980)showsthatthedaily(log)high-lowrange,properlyscaled,notonlyisanunbiasedestimatorofdailyvolatilitybutisfivetimesmoreefficientthanthesquareddailyclose-to-closereturn.Correspondingly,AndersenandBollerslev(1998)andBrandtandDiebold(2006)findthattheefficiencyofthedailyhigh-lowrangeisbetweenthatoftherealizedvariancecomputedusing3-hourand6-hourreturns.
Inthepaper“Estimatingquadraticvariationusingrealisedvariance”writtenbyOleE.Barndorff-Nielsen(2002),heproposedtwoquestionsabout“realisedvariance”,thatisthesumofMsquaredreturns.Andthemainresultofthispaperis
ThisisamixedGaussianlimittheory,thatisthedenominatorisitselfrandom.Ofcoursethistheorycanbeusedtoprovideapproximationsforrealisedvolatilityaswellasrealisedvariance.Thedistributionofrealisedvolatiliescanalsobeapproximatedindirectlyvia
usingthedeltamethodwhichgives
Thelog-basedapproximation
islikelytobepreferredinpracticewhenweconstructconfidenceintervalsforrealisedvolatility.
Inastudy“Arealisedvolatilitymeasurementusingquadraticvariationanddealingwithmicrostructureeffects”byCduToit∗andWJConradie†,theyaddmicrostructureeffectsintorealisedvolatilityresearch.InAnderson,etal.(2001a,2001b),Barndorff-NielsenandShepard(2001,2002a,2002b,2002c)andComteandRenault(1998),amodelfree(non-parametric)volatilitymeasurementisspecifiedandstudied.
Themainresultsofthispaperis
And
Buildingonrealizedvarianceandbipowervariationmeasuresconstructedfromhigh-frequencyfinancialprices,TorbenG.Andersena,TimBollerslev,XinHuang(2011)proposeasimplereducedformframeworkforeffectivelyincorporatingintradaydataintothemodelingofdailyreturnvolatility.Theydecomposethetotaldailyreturnvariabilityintothecontinuoussamplepathvariance,thevariationarisingfromdiscontinuousjumpsthatoccurduringthetradingday,aswellastheovernightreturnvariance.Theirempiricalresults,basedonlongsamplesofhigh-frequencyequityandbondfuturesreturns,suggestthatthedynamicdependenciesinthedailycontinuoussamplepathvariabilityarewelldescribedbyanapproximatelong-memoryHAR–GARCHmodel,whiletheovernightreturnsmaybemodeledbyanaugmentedGARCHtypestructure.Thedynamicdependenciesinthenon-parametricallyidentifiedsignificantjumpsappeartobewelldescribedbythecombinationofanACHmodelforthetime-varyingjumpintensitiescoupledwitharelativelysimplelog-linearstructureforthejumpsizes.
Analysisofhighfrequencyfinacialdataposesinteresttoeconometricmodelingandstatisticalanalysis.Modelingandmeasuringfinancialvolatilityarekeystepsforderivativepricing,portfolioallocationandriskmanagement.Howcanwemodelandmeasurehighfrequencydataeffectively?
Intheory,thesumofsquaresoflogreturnssampledathighfrequencyestimatestheirvariance(see,forexampleinthepaper“Andersen,T.G,Bollerslev,T.,Diebold,F.X.andLabys,P,Modelingandforecastingrealizedvolatility,Econometrica,71,579-625,2003”and“Barndorff-Nielsen,OleE.andShephard,N,Econometricanalysisofrealizedvolatilityanditsuseinestimatingstochasticvolatilitymodels,Journalofroyalstatisticalsociety,seriesB,64,253-280,2002”).Forexample,forlogpricedatafollowingadiffusionprocesswithoutnoise,therealizedvolatilityconvergestoitsquadraticvariation(Barndorff-Nielsen,OleE.andShephard,N,Econometricanalysisofrealizedvolatilityanditsuseinestimatingstochasticvolatilitymodels,Journalofroyalstatisticalsociety,seriesB,64,253-280,2002).Muchresearchhasbeendonerecently.
Somuchforthereviewaboutriealisedvolatility,mymajorstudydirectionisthehigherorderrealisedvolatilityinhighfreqeuncyfinanacialdata.Now,Ihaveareadyhadaresultinthisfield,andthemainideaisalloftheriskfunctioncanberepresentedbyapolynomial.Inmyworkingpaper,wecalculatetheexpectationandthevarianceofthepolynomialandwefindsomeinterestingresults.Themainresultofourworkingpaperistheconstant-matrixinthevariancematrix.Thatistosayanyriskvariancecanberepresentsbyavectormultipletheconstant-matrixandmultiplethetransposefothevectorabove.Thisisaveryinterestingresultwhichshockedmeatfirst.Idonotknowwhetherthisisasurprisingresult,andIdonotknowwhethereffectintherealfinancialworld,butIknowitsaninterestingresultinmathematicsatleast,especiallytheconstand-matrix.Formyownreason,theworkingpaperhavenotdone,thepaperisnotexcellentnow,therestillmuchwordformetodo.IhopeIcanfinishmypaperinsixmonthandhaveaverygoodresultintheend.Istillhopet
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