Modeling Financial Volatilities ARCH, GARCH, CARR and Other Models.ppt

modellingfinancialvolatilities arch garch carrandothermodels rayy chou周雨田academiasinica nationalchiao tunguniversitypresentedat南開大學經濟學院4 11 12 2007 2 planofthetalks 1 overviewofarchmodeling2 frontiersofarchmodeling3 carrandacarrmodels4 carrdccandcovarianceforecasting5 carrindynamichedgeratio 3 2003nobelprizewinnerrobertengle 4 2003nobellaureate engle snobelcitationwasexplicitly formethodsofanalyzingeconomictimeserieswithtime varyingvolatility arch 5 25yearsofarchmodeling archsurveypapers engle 1982 econometricabollerslev chouandkroner 1992 journalofeconometrics bollerslev engleandnelson 1994 handbooksofeconometricsengle 2002 journalofappliedeconometricsengle 2004 nobellecture americaneconomicreview 6 thefirstarchmodel rollingvolatilityor historical volatilityestimatorweightsareequalforjnwhatisn 7 1982archpaper weightscanbeestimatedarch p 8 whatisarch autoregressiveconditionalheteroskedasticitypredictive conditional uncertainty heteroskedasticity thatfluctuatesovertime autoregressive 9 thesimplestproblem whatisvolatilitynow oneansweristhestandarddeviationoverthelast5yearsbutthiswillincludelotsofoldinformationthatmaynotberelevantforshorttermforecastinganotheransweristhestandarddeviationoverthelast5daysbutthiswillbehighlyvariablebecausethereissolittleinformation 10 thearchanswer useaweightedaverageofthevolatilityoveralongperiodwithhigherweightsontherecentpastandsmallbutnon zeroweightsonthedistantpast choosetheseweightsbylookingatthepastdata whatforecastingmodelwouldhavebeenbesthistorically thisisastatisticalestimationproblem 11 rollingwindowvolatilitiesnumberofdays 5 260 1300 12 arch garchvolatilities 13 confidenceintervals 14 valueatrisk futurelossesareuncertain findalossthatyouare99 sureisworsethanwhateverwilloccur thisisthevalueatrisk onedayinadvancemanydaysinadvancethissinglenumber aquantile isusedtorepresentafulldistribution itcanbemisleading 15 calculatingvar forecasttheonedaystandarddeviation garchstylemodelsarewidelyused then assumingnormality multiplyby2 33withoutassumingnormality multiplybythequantileofthestandardizedresiduals fortheexample multiplier 2 65 16 multi dayhorizons ifvolatilitywereconstant thenthemulti dayvolatilitywouldsimplyrequiremultiplyingbythesquarerootofthedays becausevolatilityisdynamicandasymmetric thelowertailismoreextremeandthevarshouldbegreater 17 twoperiodreturns twoperiodreturnisthesumoftwooneperiodcontinuouslycompoundedreturnslookatbinomialtreeversionasymmetrygivesnegativeskewness highvariance lowvariance 18 multiplierfor10days fora10day99 valueatrisk conventionalpracticemultipliesthedailystandarddeviationby7 36forthesamemultiplierwithasymmetricgarchitissimulatedfromtheexampletobe7 88bootstrappingfromtheresidualsthemultiplierbecomes8 52 19 options tradedoptionsalwayshavemultipledaystoexpiration hencethedistributionoffuturepricelevelsisnegativelyskewed thustheblackscholesimpliedvolatilityshoulddependonstrikeifoptionsarepricedbygarch askewinimpliedvolatilitywillresultfromasymmetricgarch atleastforshortmaturities 20 impliedvolatilityskewfor10dayoption fromsimulated riskneutral finalvalues findaverageputoptionpayoffforeachstrike calculateblackscholesimpliedvolatilitiesandplotagainststrike noticethecleardownwardslope thiswouldbezeroforconstantvolatility 21 whataboutheteroskedasticity 22 exponentialsmoother anothersimplemodelweightsaredecliningnofinitecutoffwhatislambda riskmetrics 06 23 thegarchmodel thevarianceofrtisaweightedaverageofthreecomponentsaconstantorunconditionalvarianceyesterday sforecastyesterday snews 24 25 forecastingwithgarch garch 1 1 canbewrittenasarma 1 1 theautoregressivecoefficientisthemovingaveragecoefficientis 26 garch 1 1 forecasts 27 forecastingaveragevolatility annualizedvol squarerootof252timestheaveragedailystandarddeviationassumethatreturnsareuncorrelated 28 variancetargeting rewritingthegarchmodelwhereiseasilyseentobetheunconditionalorlongrunvariancethisparametercanbeconstrainedtobeequaltosomenumbersuchasthesamplevariance mleonlyestimatesthedynamics 29 thecomponentmodel engleandlee 1999 qislongruncomponentand h q istransitoryvolatilitymeanrevertstoaslowlymovinglongruncomponent 30 taylor schwert standarddeviationmodel 31 asymmetricmodels theleverageeffect engleandng 1993 followingnelson 1989 newsimpactcurverelatestoday sreturnstotomorrowsvolatilitydefinedasadummyvariablewhichis1fordowndays 32 newsimpactcurve 33 otherasymmetricmodels 34 newarchmodels gjr garchtarchstarchaarchnarchmarchswarchsnparchaparchtaylor schwert figarchfiegarchcomponentasymmetriccomponentsqgarchcesgarchstudenttgedsparch 35 financialeconometrics thismayalsobethebirthoffinancialeconometricsstatisticalmodelsdevelopedspecificallyforfinancialapplicationstodaythisisaverypopularandactiveresearchareawithmanyapplications 36 exogenousvariablesinagarchmodel includepredeterminedvariablesintothevarianceequationeasytoestimateandforecastonestepmulti stepforecastingisdifficult 37 examples non lineareffectsdeterministiceffectsnewsfromothermarketsheatwavesvs meteorshowersotherassetsimpliedvolatilitiesindexvolatilitymacrovariablesorevents 38 stochasticvolatilitymodels easytosimulatemodelseasytocalculaterealizedvolatilitydifficulttosummarizepastinformationsethowtodefineinnovation 39 svmodels taylor 1982 40 41 whathavebeenaccomplishedin25years stochasticpropertiesofarchmodelsthealphabetsoupofarchtypemodelsculminatinginfiglewski syaarchcomparisonwithlatentorstochasticvolatilitymacroapplicationsinflation policysimpleoptionstradingstrategiesefficiencyofoptionsmarketsmodelingtheriskreturntrade offassetpricing capmmeasuringrisk var 42 newfrontiers highfrequencyvolatilityclocktimeticktimeusehighfrequencydatatoimprovedailyvolatilityestimates 43 anotherfrontier multivariatearchhowgeneralshouldamultivariategarchmodelbe thedynamicconditionalcorrelationmodelengle 2002 engleandsheppard 2002 highlyrestrictedparameterizationseparatesthevolatilityandcorrelationparametershowtomeasurecorrelationswithhighfrequencydataepps 1972 zebedee 2001 44 themultivariateproblem assetallocationandriskmanagementproblemsrequirelargecovariancematricescreditrisknowalsorequiresbigcorrelationmatricestoaccuratelymodellossordefaultcorrelationsmultivariategarchhasneverbeenwidelyused itistoodifficulttospecifyandestimate 45 dynamicconditionalcorrelation dccisanewtypeofmultivariategarchmodelthatisparticularlyconvenientforbigsystems seeengle 2002 orengle 2004 46 dcc estimatevolatilitiesforeachassetandcomputethestandardizedresidualsorvolatilityadjustedreturns estimatethetimevaryingcovariancesbetweentheseusingamaximumlikelihoodcriterionandoneofseveralmodelsforthecorrelations formthecorrelationmatrixandcovariancematrix theyareguaranteedtobepositivedefinite 47 howitworks whentwoassetsmoveinthesamedirection thecorrelationisincreasedslightly whentheymoveintheoppositedirectionitisdecreased thiseffectmaybestrongerindownmarkets thecorrelationsoftenareassumedtoonlytemporarilydeviatefromalongrunmean 48 stillmorefrontiers optionspricingandhedgingsimulationmethods englemustafa 1992 treemethods trevorandrichken 1999 riskneutralizationbylocalquadraticapproximation duan 1995 empiricalpricingkernel rosenbergandengle 2001 pathdependentoptions 49 twomorefrontiers modelingnon negativeprocessesusingarch garchmodelsforawiderrangeoftimeseriesproblemssimulationmethodsformodelanalysis seeengle spaperfordetails 50 modelingnon negativeprocesses supposeisanon negativeprocesswhichhasnon zeroprobabilityofbeingzeroatanytimeamodelsuchasmightcommonlybeemployedwheretheconditionalmeanandvariancedependonpredeterminedandweaklyexogenousvariables howeverdcannothaveaconstantrangedisunlikelytohaveconstantvarianceleastsquaresisconsistentbutinefficient 51 multiplicativeerrormodel instead considerthemodelnowtheerrordistributioncanbei i d withoutviolatingtheassumptionsofthemodel ofcourseitstillmightnotbe forecastsofxdonotdependupontheerrordensity 52 estimationofmem specifythemean forexampleassumeittobelinearinlaggeddependentvariables laggedmeansandotherpredeterminedandweaklyexogenousvariableszspecifytheerrordensityandanyheteroskedasticityitmayhave forexample assumeitisunitexponential 53 loglikelihood inthiscase nowifthisissimplygarch p q withgaussianlikelihoodfunctionandexogenousvariablesz 54 estimationwithgarch thislikelihoodcanbemaximizedwithagaussiangarchprogramsuchaseviews simplyconsidersquarerootofxasthedependen