A summary about realized volatility in high freqeuncy financial data.docx

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