課程-金融計(jì)量學(xué)lecture8garch

GARCHandOtherVolatilityModelsOutline1Introduction2ARCH3GARCH4OtherGARCHfamilymodels5Application:VaR6Realizedvolatilitymodel7Impliedvolatility閱讀： Enders，Ch3Tsay，Ch31IntroductiontovolatilitymodelWhatisvolatility?Volatilityissimplyastatisticalmeasureofdispersionaroundthemean,i.e.,standarddeviationWhyisvolatilitysoimportant?OptionpricingandhedgingDynamichedgeratiosVaRTimeseriesmodelingforforecastandhypothesistestWhyisvolatilitysodifficulttomodel?Haveyoueverobservedvolatility?Thenhowabouttoestimatestdusinghistoricdata?Thenextfigureisthestd’sofS&P500,Russell2000andastockusingtwo-monthdailyreturns.(candoitforyou)ItisalsocalledhistoricvolatilityItisobviousthatvolatilityistime-varying.Anunobservabletime-varyingobject!Nowonderitisdifficulttomodel.Fromthishistoricvolatilityfigure,wecanseethattimevaryingthereareperiodsoftimewherethevolatilityishigh(orlow)anditstaysthatway.ThisiscalledvolatilityclusteringLargeshocksareusuallyfollowedbyanotherlargeshock.volatilitygoesupandthencomesbackdownagain.Youcanclearlyseethereisarangeandthatiscalledmeanreversion.clusteringandmeanrevertingtogetherindicatethatvolatilityislikeastationaryAR(1)processwithpositivecoefficient.Canwejustusethishistoricvolatility?Inallapplications(optionpricing,riskmanagement,etc.),whatweneedisthevolatilityoffuturereturn,ortheexpectedvolatility!Ifvolatilityisconstant,wecanusehistoricvolatilityastheforecastoffuturevolatility.Unfortunately,volatilityistime-varying.Henceweneedmodelsforthefuturevolatilityforecastproblem.Wehavetwoapproaches:Two-stepsapproach:estimatevolatilityinthehistoryfirstandthenmodelitsmovementThemovementmodelisoftenARMAtypesimplemodel.Inindustry,randomwalkmodeliswidelyused,i.e.,today’svolatilityisusedastheforecastoftomorrow’sPros:simpleandintuitive,forexample,tradingoptionisoftencalledtradingvolatilityCons:notheoreticalfoundation.Volatilityestimatingmodelandmovementmodelarenotconsistent.SimilartooutsidemodelhedgingOne-stepapproach:estimateandmodelitsmovementinonemodelThemodelinputisobservableinformationsuchasreturns,themodelistofitthereturns.Volatilityanditsmovementispartofthemodel.Pros:hassolidtheoreticalfoundationCons:notflexible,difficulttoestimate.Beforegotomodeldetails,pleasenotethat:Inmostapplications,volatilityisannualized.Wecangetitbyadjustingthedatafrequency.Butitisn’tthevolatilityfortheyearlyreturn!Forecastvolatility(standarddeviation)orvariance?SinceE(s)=/sqrt(E(s^2)),itisbettertomodelwhateveryouwillneedtoforecastThevolatilityistime-varying,thenthereturnprocess(S&P500,Russell2000)isnotstationary?Notnecessarybecausethevolatilityreportedaboveisanestimatoroftheabsolutevolatility,nottheabsolutevolatilityitself.Wewillseethatconditionallyheteroscedasticity(time-varying)volatilitycangeneratetime-varyingvolatilityestimatorasabove.Infact,theheteroscedasticityincross-sectionaldataisalsoconditional.Wewilldiscusstheone-stepapproachfirst.One-stepapproachestimatesandmodelsvolatility’smovementinonemodel.Themodelinputisobservableinformationsuchasreturns,volatilityispartofthemodel.Hencevolatility,intheory,isafunctionoftheavailableinformation.Stochasticfunctionoftheavailableinformation:SVfamilymodelsMelinoandTurnbull(1990),Harvey,RuizandShephard(1994)Jacquier,PolsonandRossi(1994).“Fixedfunction"oftheavailableinformation:GARCHfamilymodelsAllarebasedonARCHmodelofEngle(1982)RobertF.EnglesharedtheNobelprize(2003)“formethodsofanalyzingeconomictimeserieswithtime-varyingvolatility(ARCH)withCliveW.J.Grangerwhoreceivedtheprize“formethodsofanalyzingeconomictimeserieswithcommontrends(cointegration).”“Fixedfunction"oftheavailableinformation:GARCHfamilymodelsARCHmodelofEngle(1982),GeneralizedARCH(GARCH)modelofBollerslev(1986),GARCH-Mmodels,IGARCHmodelofNelson(1990),ExponentialGARCH(EGARCH)modelofNelson(1991),ThresholdGARCHmodelofZakoian(1994)orGJRmodelofGlosten,Jagannathan,andRunkle(1993),AsymmetricparametricARCH(APARCH)modelsofDing,GrangerandEngle(1994),[TGARCHandGJRmodelsarespecialcasesofAPARCH]2ARCHARCH(AutoregressiveConditionalHeteroscedastic)modelisproposedbyEngle(1982).Itsbasicidea:explicitlymodelthemovementofvolatilityinthemodelofreturnprocess回顧在橫截面模型中，我們?nèi)绾翁幚恚l件）異方差的方法（GLS）我們將用這種方法處理時(shí)間系列模型中的（條件）異方差2ARCH與GLS不同的是，這里方差不是可觀察變量的函數(shù)，而是擬合誤差的函數(shù)Ifnosigmapart,wehavethestandardAR(p)processEconomicimplication:ShocksofassetreturnsareNOTseriallycorrelated,butdependent.Thatis,theserialdependenceisnonlinear.Consistentwithempiricalfindingthatstockreturnisnotpredictable,butvolatilityismean-revertingThemeanpartcanbeanyotherprocess(evenwithexogeneousvariable).Ourdiscussionbelowwillfocusonvolatilitypartonly.UnconditionalpropertiesofARCHIfa1>=1,thevolatilityprocesswillbenon-stationaryForARCH(m),thepropertiesabovehold,butformulasarecomplicatedConditionalvolatilityinARCHLikeaMAprocesswhenm>1ItisobviousthatvolatilityistimevaryingFurthermore,wehavevolatilityclusteringSinceerrortermainaboveequationcanberewrittenasproductsofvolatilityandinnovationThispropertycanbeusedastoestimate“historic”volatilityItcanalsobeusedtoforecastfuturevolatilitysincetheexpectedvolatilityoftimetisknownattimet-1Wewilldiscussthevolatilityforecastindetaillater.ButweneedtoknowtheparameterfirstEstimation:MLEForexample:AR(1)-ARCH(1)當(dāng)樣本比較大時(shí)，ConditionalMLE沒有什么誤差特別是對(duì)高階ARCH，MLE更復(fù)雜，CMLE更常用。其它的ARCH，包括不同的均值過程，都有類似的likelihood方程對(duì)比較標(biāo)準(zhǔn)的ARCH模型，SAS等軟件有現(xiàn)成的命令ProcedureautoreginSASisaproceduretodealwithautocorrelatedand/orheteroscadasticerrordataone;v=0;lret=0;dotime=-10to500;u=rannor(12346);sigma=0.1+0.3*v*v;v=sqrt(sigma)*u;dreteq=1.2+0.5*lret+v;iftime>0thenoutput;lret=dreteq;end;run;模擬的模型為：dreteq=1.2+0.5lret+vv(t)=se

s2=0.1+0.3v(t-1)^2procautoregdata=one;modeldreteq=lret/garch=(q=2);quit;估計(jì)為：dreteq=1.2348+0.4811lret+vv(t)=se

s2=0.089+0.371v(t-1)^2+0.0v(t-2)^2ARCH模型的均值部分理論上可以是ARMA，即誤差除了有異方差，還可以是自相關(guān)的但procautoreg不能處理。兩種辦法處理自相關(guān)：用ARMA-ARCH模型，但用procmodel命令用procautoreg，即均值部分用AR，誤差部分用AR+ARCH這里如果沒有sigma，就是前面初級(jí)計(jì)量經(jīng)濟(jì)學(xué)講的誤差系列相關(guān)模型dataone;v1=0;lret=0;dotime=-10to1000;u=rannor(12346);sigma=0.1+0.3*v1*v1;v=0.5*v1+sqrt(sigma)*u;dreteq=1.2+0.5*lret+v;iftime>0thenoutput;lret=dreteq;end;run;procautoregdata=one;modeldreteq=lret/nlag=1garch=(q=2);quit;估計(jì)的模型應(yīng)該為：dreteq=1.2+0.5lret+vv(t)=0.5v(t-1）+

se

s2=0.1+0.3v(t-1)^2估計(jì)為：dreteq=1.153+0.516lret+vv(t)=0.3984v(t-1）+

se

s2=0.1004+0.3578v(t-1)^2+0.00928v(t-2)^2注意誤差A(yù)R（1）的系數(shù)在SAS中是相反數(shù)BuildinganARCHModelARCHmodelitselfisnotwidelyusedanymore,ageneratedARCH(GARCH)ismuchbetter.WewillleaveittoGARCHpartSimilarly,ForecastingvolatilityusingARCHmodelisalsodiscussedinGARCH3GARCHBasicideaInpracticeweoftentoneedhighorderARCHmodeltofitdataEngle(1982)findsthatUKCPIisARCH(4)ButARCHislikeMA,AhighorderMAcanbeapproximatedbyasimpleARBasicstructure沒有beta是就是ARCH，但加上beta，可以大幅減小sGARCH（1,1）Re-parameterization:ThisisanARMAformforthesquarederrorseriesItisusefultounderstandpropertiesofGARCHmodels,e.g.momentequations,forecasting,etc.UnconditionalpropertiesofGARCHFocusonGARCH(1,1)Unconditionalmeanofvolatility:alpha0/(1-alpha1-beta1)可見其平穩(wěn)性條件，Weaklystationarycondition:FattailsConditionalpropertiesofGARCHTime-varyingvolatilityVolatilityclustersMuchclearthaninARCHmodelbecauseoftheARcomponentsInfact,autocorrelationofsquaredresidualsissimilartotheAR(1)withparameter(alpha1+beta1)Bothalphaandbetageneratevolatilityclustering,whichoneismoreimportant?Alphacapturesthepreviousshock’simpact,andbetacapturesthepreviousvolatility’simpactLargerbetaindicatesthatshockstovolatilitytakealongtimetodieout,sovolatilityismorepersistentLargeralphameansthatvolatilityreactsquiteintenselytothepreviousmarketmovement,sovolatilitytendtobemorespiky.NextfigureshowsthispatternItiscommontoseethatbeta>0.8andalpha<0.2infinancialmarketGARCHmodelestimation:SimilarARCH,MLENeedinitialconditionforbotherrorandvolatilityUnconditionalmeanofvolatility:alpha0/(1-alpha1-beta1)dataone;v=0;lret=0;sigma1=0;dotime=-10to1000;u=rannor(12346);sigma=0.1+0.4*sigma1+0.3*v*v;v=sqrt(sigma)*u;dreteq=1.2+0.5*lret+v;iftime>0thenoutput;lret=dreteq;sigma1=sigma;end;run;procautoregdata=one;modeldreteq=lret/garch=(p=1,q=2);quit;估計(jì)的模型應(yīng)該為：dreteq=1.2+0.5lret+vv(t)=se

s2=0.1+0.3v(t-1)^2+0.4s2(t-1)估計(jì)為：dreteq=1.2037+0.4952lret+vv(t)=se

s2=0.0609+0.3578v(t-1)^2-0.1719v(t-2)^2+0.6248s2(t-1)VolatilityestimateandforecastinGARCHRecursiveDependsoninitialconditionTheerrortermattime0isassumedtobe0orato-be-estimatedparameterthevolatilityattime0isassumedtobesamplevariance，orlong-termvariance,orato-be-estimatedparameterprocautoregdata=one;modeldreteq=lret/garch=(p=1,q=2);Outputout=outcev=vhatr=residual;quit;這里的vhat事實(shí)上是sigma的擬合值比較擬合值與原始值(因?yàn)槭悄M數(shù)據(jù)）：vhat--sigma；residual–vVolatilityestimateandforecastinGARCHOut-sample預(yù)測(cè)？SASprocautoregcannotautomaticallygiveitButitiseasytocalculateone-dayaheadvolatilitybytherecursiveformulaafterweobtainparametersestimatedandvolatilityestimateuptodateHowaboutj-day(j>1)aheadvolatilityforecastsincenoobservationsonfutureinnovationa?VolatilityestimateandforecastinGARCHThevolatilityforecastsofarforone-periodreturn’sconditionalvolatility,butinapplications,weoftenneedtheforecastforthenextn-periodreturn’svolatility.Forexample,topriceancalloptionwithn-daystoexpire由這個(gè)公式，我們會(huì)發(fā)現(xiàn)，實(shí)證上，不同期回報(bào)的volatility(年化后）會(huì)不一樣。這個(gè)特征也叫波動(dòng)率的期限結(jié)構(gòu)（在BS模型中，它是常數(shù)）前面討論的模型估計(jì)/波動(dòng)率預(yù)測(cè)是建立在給定的GARCH模型基礎(chǔ)上，如何BuildinganGARCHModel？我們通過模擬一個(gè)AR(1)-GARCH（1,1）過程的建模來說明dataone;retainseed14500;v1=0;lret=0;sigma1=0;dotime=-10to1000;callrannor(seed1,u);sigma=0.1+0.4*sigma1+0.3*v1*v1;v=sqrt(sigma)*u;dreteq=1.2+0.5*lret+v;iftime>0thenoutput;lret=dreteq;sigma1=sigma;v1=v;end;run;Modelingthemeaneffect一般由經(jīng)濟(jì)學(xué)理論來，預(yù)測(cè)時(shí)則用AR過程procarimadata=one;identifyvar=dreteq;run;選AR（1）BuildinganGARCHModel2. TestingforARCHeffects（存在異方差？）UseQ-statisticsofsquaredresidualsprocautoregdata=one;modeldreteq=lret/archtest;quit;Q-test表明有強(qiáng)烈的ARCH效應(yīng)也可能有誤差自相關(guān)，procautoregdata=one;modeldreteq=lret/dwprob;quit;誤差自相關(guān)不強(qiáng)，可以先不管3.UsingAIC/BICtoselectmodelorder先試ARCH（1）procautoregdata=one;modeldreteq=lret/garch=(q=1);quit;再試GARCH（1,1）procautoregdata=one;modeldreteq=lret/garch=(p=1,q=1);quit;再試GARCH（1,2）procautoregdata=one;modeldreteq=lret/garch=(p=1,q=2);quit;再試GARCH（2,1）procautoregdata=one;modeldreteq=lret/garch=(p=2,q=1);quit;由AIC/BIC我們選GARCH（1,1）3.UsingAIC/BICtoselectmodelorder可能的誤差自相關(guān)，試AR(1)-GARCH（1,1）procautoregdata=one;modeldreteq=lret/nlag=1garch=(p=1,q=1);quit;AR1系數(shù)不顯著AIC更大AR(1)-GARCH（1,1）還是最優(yōu)選擇。正是用來模擬數(shù)據(jù)的模型4.Modelchecking:If

themodeliscorrectlyspecified,thestandardizedresidualsepsilon(anditssquaredterm)shouldbeunpredictableQ-statofstandardizedresidualsandsquaredstandardizedresidualsprocautoregdata=one;modeldreteq=lret/garch=(p=1,q=1);outputout=outr=residualcev=vhat;quit;datatwo;setout;rs=residual/sqrt(vhat);rs2=rs*rs;r2=residual*residual;run;procarimadata=two;identifyvar=rs;run;procarimadata=two;identifyvar=rs2;Run;4.Modelchecking:Q-statofstandardizedresidualsandsquaredstandardizedresiduals為比較，我們看看誤差a本身（及其平方項(xiàng)）的可預(yù)測(cè)性procarimadata=two;identifyvar=residual;run;有自相關(guān)的方差的誤差系列有微弱的自相關(guān)procarimadata=two;identifyvar=r2;run;本質(zhì)上就是ARCHtest故AR（1）-GARCH（1,1）合適選對(duì)了！GARCH模型選擇的其它問題：均值部分可以是任何符合經(jīng)濟(jì)學(xué)原理的模型，比如想要估計(jì)股票回報(bào)的特質(zhì)風(fēng)險(xiǎn)，均值部分就是市場(chǎng)模型，甚至再加上市場(chǎng)回報(bào)的滯后。波動(dòng)率部分，一般來說，很少有超過GARCH(2,2)的。因?yàn)镚ARCH模型的MLE估計(jì)收斂大都不太好，參數(shù)太多時(shí)問題更大還有一種導(dǎo)致不收斂的原因是outliers的存在。實(shí)證發(fā)現(xiàn)，美國市場(chǎng)上幾個(gè)主要股票（Ford，BankOne，P&G)的1996年到2000年的日度數(shù)據(jù)的GARCH模型估計(jì)收斂困難。而在去掉它們數(shù)據(jù)中的幾個(gè)奇異值后，收斂就好了怎么知道有奇異值？Skewness和Kurtosis非常大！我們可以看見，事實(shí)上這些股票回報(bào)的ARCH檢驗(yàn)值也都很小，原因也是outliers的存在因此，估計(jì)模型前先看看數(shù)據(jù)總是一個(gè)好習(xí)慣那如何處理outliers？如果是數(shù)據(jù)錯(cuò)誤，簡(jiǎn)單地去掉如果不是數(shù)據(jù)錯(cuò)誤，比如1987年的黑色星期五，1998年LTCM的幾乎破產(chǎn)等導(dǎo)致的市場(chǎng)激烈波動(dòng)等在用GARCH估計(jì)S&P500指數(shù)時(shí)，是否包括1987年的黑色星期五，估計(jì)的volatility的長(zhǎng)期平均值會(huì)有幾個(gè)百分點(diǎn)的差別是否去掉依賴你的直觀，即這樣的激烈波動(dòng)是否對(duì)將來的波動(dòng)有影響，甚至是否會(huì)有可能重演如果不能去掉outlier，如何解決模型的收斂問題（包括其他不能收斂的情況）改變參數(shù)估計(jì)的初始值改變最大化似然函數(shù)的數(shù)值方法改變模型，用簡(jiǎn)單些的模型GARCH模型選擇的其它問題：一般來說，大多金融市場(chǎng)的GARCH模型都是用日度或高頻數(shù)據(jù)，很少有用月度或更低頻的數(shù)據(jù)。這是因?yàn)榈皖l數(shù)據(jù)中的GARCHeffect比較弱做GARCH模型，用多長(zhǎng)時(shí)間的數(shù)據(jù)？這個(gè)問題與前面的outlier問題有關(guān)，我們現(xiàn)在是否應(yīng)該從1985年開始從而包括87年的黑色星期五？這樣的某些特殊事件點(diǎn)可以是作為樣本選擇的考慮，如果你認(rèn)為那個(gè)事件還有影響，樣本區(qū)間就應(yīng)該包括它大概沒有多少人認(rèn)為1929年的大蕭條會(huì)對(duì)今天的股市有影響因此，不是越長(zhǎng)越好太長(zhǎng)了，會(huì)把已經(jīng)沒有影響的“outlier”包括進(jìn)來。太長(zhǎng)了也會(huì)有模型的結(jié)果變化問題但太短了，會(huì)影響模型的估計(jì)精度一般原則是：應(yīng)該用幾年（日度數(shù)據(jù)）的數(shù)據(jù)，樣本大到使得當(dāng)進(jìn)行樣本窗口滾動(dòng)前移時(shí)，參數(shù)估計(jì)相對(duì)穩(wěn)定（肯定是越多越穩(wěn)定），但又沒有多到包括直觀認(rèn)為以前沒有影響的“outliers”。這個(gè)原則也適用于其它的模型數(shù)據(jù)選擇中GARCH模型選擇的其它問題：如果數(shù)據(jù)中有明顯的skew，應(yīng)該考慮student-t分布的GARCH。GARCHmodelwithotherfactorsprocautoregdata=wang.aindex;modeldreteq=lretgarch=(q=1,p=1);heterox1x2;outputout=outcev=vhat;Quit;這個(gè)模型可以用來研究波動(dòng)率在某一時(shí)點(diǎn)是否有跳躍（比如期貨的引入）Comments

onGARCHAdvantagesSimplicityGeneratesvolatilityclusteringHeavytails(highkurtosis)WeaknessesProvidesnoexplanation,apurestatisticalmodelNotsufficientlyadaptiveinpredictionSymmetricbtwpositive&negativepriorreturnsRestrictiveonparameters4OtherGARCHmodels有很多改進(jìn)的GARCH模型EGARCHIGARCHGARCH-in-meanThresholdGARCHEGARCHGARCHmodelhastwoproblemsPositiveconstraintforparametersCausedifficultyformodelestimationsymmetryinresponsestopastpositiveandnegativereturns,butweoftenobserveasymmetricimpactleverageeffectisonemechanismtocauseasymmetricresponseNelson(1991)proposesEGARCHmodelEGARCHAnEGARCH(m;s)model用對(duì)數(shù)形式，可以避免方差為負(fù)數(shù)的問題，從而不需要限制參數(shù)。這一點(diǎn)和cross-sectional中的異方差模型一樣估計(jì)、預(yù)測(cè)等與GARCH一樣procautoregdata=one;modeldreteq=lret/garch=(p=1,q=1，type=exp);quit;估計(jì)是dreteq=1.2152+0.4941lret+vv(t)=se

s2=-0.3854+0.6799s2(t-1)+0.5939g(e)g(e)=-0.004243e+|e|-E(|e|)q為負(fù),非對(duì)稱反應(yīng)，但不顯著因?yàn)閿?shù)據(jù)是前面基于GARCH（1,1）模擬的，沒有非對(duì)稱性Example:monthlylogreturnsofIBMstockfromJanuary1926toDecember1997for864observations（textbook）EGARCHisveryclosetothestochasticvolatilitymodelA(simple)SVmodelisDifferencefromtheSVmodelsFilteringisrequiredinSVmodeltoestimatevolatilityModelestimationmethod:Simulation-basedmethod(MCMC,EMM)GARCHorEGARCHEventhoughEGARCHhasbetterpropertiesthanGARCH,butitismoredifficulttoestimateanEGARCHmodelIGARCH金融時(shí)間系列數(shù)據(jù)中，波動(dòng)率一般有非常強(qiáng)的持續(xù)性。在GARCH框架下（alpha+beta）

接近1，特別是在外匯市場(chǎng)中。比如日度sterling-USdollar匯率（1988年1月5日到2000年10月6日）的GARCH模型估計(jì)中，alpha+beta=0.991這樣，波動(dòng)率幾乎是一個(gè)單位根過程為此，Nelson(1990)提出IGARCH(1,1)modelNelson（1990）showsthateventhoughthevolatilityprocessisnotstationary,thisprocessisn’tlikethenon-stationaryARMAprocess,itcanbeestimatedasstationaryGARCHmodelbutthereisnolong-termmeanforthevolatilitybecauseofthenon-stationarity.HencethevolatilitytermstructuredonotconvergeThismodelhasoneparameterlessthantheGARCH,whichmakesthemodelbetterbecauseoftheestimationproblemofGARCHmodelThespecialcaseofalpha0=0(itisactuallytheEWMAmodeldiscussedlater)isusedinRiskMetrics（DevelopedbyJ.P.Morgan）forVaRcalculation.Themodelestimation,modelbuilding,andvolatilityforecastareallsimilartoGARCHGARCH,EGARCHorIGARCH,whichonetouse?ThefollowingmodelbuildingexampleshowstheselectionprocessExample:DailyNYSEcompositeindexSectionof3.10ofEndersusesittoshowhowtobuildaGARCHmodelStep1:meanprocess

Whichonetouse?Becausetheyareobtainedunderconstantvolatility,wewilltryGARCHvolatilityanyway,sokeepthembothatthemoment.Step2:testGARCHeffectStep3:modelselectionIGARCH?HowaboutEGARCH?LeverageeffectIGARCHorEGARCH?Step4:modelcheckingWewilldiscusstwo-stepsapproach

volatilitymodelsnext.Sincetwo-stepsapproachestimatevolatilityinthehistoryfirstandthenuseARMAtypesimplemodeltoforecast,hencewewilldiscussvolatilityestimateonly.RealizedvolatilityImpliedvolatilityRangemodel5Application:VaRGARCH模型有很多應(yīng)用，這里講在風(fēng)險(xiǎn)管理上的應(yīng)用風(fēng)險(xiǎn)管理是任何一家金融機(jī)構(gòu)的核心問題之一前臺(tái)業(yè)務(wù)：掙錢風(fēng)險(xiǎn)管理：避免倒閉在險(xiǎn)價(jià)值(Value-at-risk,VaR)是一個(gè)被廣泛使用的風(fēng)險(xiǎn)度量定義：在險(xiǎn)價(jià)值(Value-at-risk,VaR)是一個(gè)資產(chǎn)，在一定時(shí)期內(nèi)（h天），損失大于等于V的概率小于a，那么我們說這個(gè)資產(chǎn)的對(duì)應(yīng)h和a的VaR等于V。由于這個(gè)定義依賴于時(shí)間h和概率a，故我們經(jīng)常標(biāo)注這個(gè)VaR為VaRa,h如何理解VaRa,h的經(jīng)濟(jì)含義？或者說為什么VaRa,h是一個(gè)好的風(fēng)險(xiǎn)度量？假如一金融機(jī)構(gòu)目前的總資產(chǎn)（asset）為Pt，自有資產(chǎn)（equity）為V，負(fù)債為Pt–V，且在時(shí)間（t+h)之前沒有可能有新的equity加入。要使得該機(jī)構(gòu)在時(shí)間（t+h)那一天不破產(chǎn)，就要求在（t+h)時(shí)的總資產(chǎn)Pt+h必須大于等于負(fù)債（Pt–V），即Pt+h>Pt–V或者說，如果這期間的損失Pt+h–Pt<(-V)，就破產(chǎn)了。LehmanBrothers但絕對(duì)不破產(chǎn)，幾乎是不可能的事。要革命總會(huì)有犧牲，連美國國債投資都會(huì)有違約風(fēng)險(xiǎn)，而LTCM的破產(chǎn)就是由于俄國國債違約。我們能做的是把這種破產(chǎn)的概率a控制在一定范圍內(nèi)，比如0.1%。這就是說，風(fēng)險(xiǎn)管理的最低要求是VaR小于自有資產(chǎn)！實(shí)際應(yīng)用VaRa,h有三個(gè)問題：設(shè)定一定時(shí)期（h天），設(shè)定概率a，計(jì)算先看給定a和h時(shí)的計(jì)算即是回報(bào)的VaR乘上當(dāng)期總資產(chǎn)如果mu=0，a=2.5%，x=-sigma*2x可以不是正態(tài)分布下面LTCM的例子表示了a和h的設(shè)定及VaR的簡(jiǎn)單計(jì)算過程上面LTCM例子中，計(jì)算VaR時(shí)用簡(jiǎn)單的歷史std，均值為零，正態(tài)分布（t分布）進(jìn)行但這是有問題的，VaR的計(jì)算的核心是回報(bào)的分布問題一般用t分布或正態(tài)分布要考慮均值部分，盡管比較簡(jiǎn)單問題在方差的預(yù)測(cè)前面介紹的波動(dòng)率模型可以用在這里下面用DailyreturnsofIBMstock（July3,62toDec.31,98，9190observations）Position:longon$10millions.問題：10天的回報(bào)均值，沒有調(diào)整10天的回報(bào)方差，做了簡(jiǎn)單調(diào)整。GARCH中有更合理的估計(jì)前面演示的是單個(gè)資產(chǎn)的VaR，如何計(jì)算一個(gè)具體的投資組合的VaR，比如LTCM？直接考察這個(gè)組合的回報(bào)數(shù)據(jù)優(yōu)點(diǎn)：簡(jiǎn)單缺點(diǎn)：有時(shí)沒有足夠的數(shù)據(jù)投資組合的結(jié)構(gòu)一般會(huì)變化，導(dǎo)致歷史回報(bào)數(shù)據(jù)不能很好反應(yīng)將來考察組合內(nèi)的每個(gè)資產(chǎn)的風(fēng)險(xiǎn)，根據(jù)當(dāng)前的組合結(jié)構(gòu)加總問題在于需要估計(jì)資產(chǎn)之間的相關(guān)性多元GARCH模型是解決這個(gè)相關(guān)性問題的一個(gè)有效模型上面演示了用正態(tài)分布的GARCH模型計(jì)算VaR，理論上還可以用其它模型Student-t分布的GARCHfamily模型，這時(shí)波動(dòng)率的預(yù)測(cè)和VaR中要同時(shí)用student-t分布6RealizedVolatilityUsingstandardmethodtocalculatethehistoricvolatilityisthemostintuitivemethod.ItisusedinLTCMcase.Itisalsocalledrealizedvolatility。ItisusuallycalculatedbyusinghigherfrequentdataSupposeweliketoestimatethemonthlyvolatilityofastockreturnforamonth,wecanusedailyreturndatawithinthemonthAdvantage:SimpleWeaknesses:隱含假設(shè)為結(jié)構(gòu)變化，即在一個(gè)月之內(nèi)的方差相同，而在月末相鄰兩天（分別屬于兩個(gè)月）卻不一樣。這個(gè)假設(shè)沒有理論基礎(chǔ)。對(duì)月度方差，一般只有21/22個(gè)數(shù)據(jù)，這樣的樣本量對(duì)方差估計(jì)來說偏少如果要計(jì)算周度、日度回報(bào)的方差，問題更大Whatifuseintradayreturntocalculatedaily/monthlyvolatilitytosolvethesmallsampleproblem?effectsofmarketmicrostructure(noises)Missing(oroverestimate)overnightvolatilities.Movingwindow(usingthepreviousndays’dailyreturns)isanothersolutiontothesmallsampleproblemButthereisaproblemknownas'ghosting'whereanextremereturnobservationcontinuestoaffectourvolatilityforecastandthensuddenlydropsoutofsampleTosolvethe“Ghosting”andimplicitstructurechangeproblem,EWMA(exponentiallyweightedmovingaverage)modelisproposedEWMAaddressestheseproblemsbyexponentiallyweightingthedata(inhistoricalvolatilitytheobservationsareequal-weighted)somorerecentreturnshavealargerimpactonthevolatilityestimation即同樣的邏輯，我們有l(wèi)越大，最近的return在波動(dòng)率的估計(jì)中越重要?但l=1并且n有限時(shí)，就是通常的樣本方差（也叫等權(quán)方差）。最近的波動(dòng)在波動(dòng)率的估計(jì)中不是更重要為更好地理解l，我們把EWMA模型改寫成l越大，當(dāng)期的方差越依賴于上期的方差，而較少依賴于上期的波動(dòng),即更強(qiáng)的persistence當(dāng)l等于1，方差為常數(shù)！這個(gè)模型有些像IGARCH(1,1)即alpha0等于0時(shí)的IGARCH（1，1）就是EWMA模型但參數(shù)l的選取不一樣。在IGARCH中，beta是用歷史數(shù)據(jù)估計(jì)出來的，而在EWMA模型中，它一般是使用者主觀選取的如何主觀地選擇l？Asaruleofthumb,forfinancialmarkets,lischosenfrom0.75to0.98Lowerlvalueforshort-termforecast,andhigher