在险风险值估计方法的比较研究(英文)

第27卷第4期2010年12月经济数学MATHEMATICSINECONOMICSVol.27,No.4Dec.2010ComparisonResearchofValueatRiskEstimationMethods*OUHui(CollegeofMathematics&ComputerScience,HunanNormalUniversity,Changsha,Hunan410081,China)AbstractItiswellknownthatfinancedatatendstoheavytailed.Onthebasisofanexponentialregressionmodel,thispaperproposedanextremequantileestimatormethodofheavytaileddistributionandobtainedtheestimationformulaofvalueatrisk(VaR).Then,theestimationsofvalueatrisk(VaR)ofthesynthesizedindexofShanghai,treasuryandcorporatebondindexesofChinawereobtained,andtheirextremalriskwascompared.Keywordsheavytaileddistribution;VaR;extremequantileF224;F830.9:A1IntroductionTheendofthisdecadehasbeencharacterizedbysignificantinstabilitiesinfinancialmarketsworldwide.Thishasledtonumerouscriticismsabouttheexistingriskmanagementsystemsandmotivatedthesearchformoreappropriatemethodologiesabletocopewithfinancialdisasters.TheValueatRisk(VaR)wasdevelopedinresponsetothesefinancialdisastersandobtainedanincreasingimportantroleinmarketriskmanagement.TheVaRsummarizestheworstlossoveratargethorizonwithagivenlevelofconfidence.Itisapopularapproachbecauseitprovidesasinglequantitythatsummarizestheoverallmarketriskfacedbyaninstitution.InaVaRcontext,precisepredictionoftheprobabilityofanextrememovementinthevalueofaportfolioisessentialforbothriskmanagementandregulatorypurposes.Bytheirverynature,extrememovementsarerelatedtothetailsofthedistributionoftheunderlyingdatageneratingprocess.SeveraltailstudiesafterthepioneeringworkbyMandelbort[1]indicatethatmostfinancialtimeseriesareheavytailed,andfindtheefficiencyisverylowwhileusingtheclassicalmethod,suchasVarianceCovariancemethod,historicalsimulationandMonteCarlomethodtocalculateVaR.Moreover,infinancialriskmanagement,peoplefocusonthecaseofhighvolatilityoffinancialassetreturnsratherthanthatofnormal.AsPhilippe[2]said:AGaussianmodelforthepricefluctuationsisneverjustifiedfortheextremee*收稿日期:20100805基金项目:国家自然科学基金资助项目(10871064);湖南省普通高校计算与随机数学及其应用!重点实验室,开放基金资助项目(09K026);湖南师范大学青年基金资助项目(71001)作者简介:欧辉(1978∀),女,湖南宁乡人,讲师,博士研究生Email:[e-mail]经济数学第27卷vents,sincetheCLTonlyappliesinthecenterofthedistributions.Now,itispreciselytheseextremerisksthatareofmostconcernforallfinancial,andthusthosewhichneedtobecontrolledinpriority.Inrecentlyyears,internationalregulatorshavetriedtoimposesomerulestolimittheexposureofbankstotheseextremerisks.Aboldsolutiontothisproblemissimplytoremovethecontributionofthesesocalledaberranteventsandthisisratherabsurd#.TosolvethisproblemandattainmorepreciseVaR,extremevaluetheory(EVT)isintroducedtoresearchfieldofthisproblem.Atpresent,twomainlyapproachesinEVTareusedtomeasurefinancialrisk.Oneisblockmaximamodelmethod(BMM),whichisusedtomodeltheblockmaxima,theotheroneisgeneralizationParetodistributionmodel(GPD),whichmodelstheexceedancesofaparticularthreshold.Thetwoapproacheshavebeenanumberofstudiesinrecentyears,suchasBali,T.G.[3],Embrechts,P.etal.[4],OuyangZisheng[5]andsoon.Inriskmanagement,thetypicalquestiononewouldliketoansweris:Ifthingsgowrong,howwrongcantheygo?#theproblemisthenhowcanwemodeltheserarephenomenawhichmainlylieoutsidetherangeofavailableobservations.Orinotherword,howcanwemodeltheseextremequantile?BecauseVaRisonlyaextremequantileoflossdistributioninessence,andifweknowtheextremequantilefromthedistributionofreturnsoffinancialasset,wealsoobtaintheVaR.Thispaperdealswiththebehaviorofthetailsoffinancialseries.Morespecifically,thefocusisontheuseofextremevaluetheorytoassesstailrelatedrisk;Itthusaimsatprovidingamodelingtoolforriskmanagement.2StatisticalModeling21DefinitionofHeavytailedDistributionCommonexamplesofheavytaileddistributionsarethePareto,Burr,Student∃st,loggammaandFrchetdistributions.Theyareusedinawidevarietymodelphenomenaanddatainreallife.Ininsurance,forinstance,theyprovidemodelsforclaimsizesandlosses,andinfinancetheycapturetheheavytailsofquantitiessuchasexchangeratesorlogreturnsonstocks.Inalltheseexamplesoneisofteninterestedinacertainhighlevel,whichwillbeexceededwithonlya(very)smallprobabilityp,i.e.anextremequantilexpoftheunderlyingdistribution.WecalladistributionF(x)heavytailedifitstailfunctionF=1-F,whichgivestheexceedanceprobabilities,decaysessentiallyasapowerfunction:1-F(x)=x-1/lF(x)0,(1)where,Ifwelet=1/,theniscalledthetailindexandlFisaslowlyvaryingfunction,i.e.forall0,lF(x)/lF(x)%1.Equivalently,thequantilefunctionQ(r)=inf{x:F(x)&r}(thegeneralizedinverseofF)mustsatisfyQ(1-p)=p-l(1/p),(2)lisaslowlyvaryingfunction.ForeaseofnotationwewilloftenwritexpforthequantileQ(1-p)with(usuallysmall)tailprobabilitypsuchthat∀16∀第4期欧辉:在险风险值估计方法的比较研究F(xp)=p,(3)ifF(x)isacontinuousdistribution.22DetectionoftheHeavytailDistributionAlthoughfinancialassertreturnsdatausuallypresentheavytails,asweanalyzethedataroughly,weneedtoexploreitstailcharacteristic.Twosimpleandpowerfulgraphicaltoolscanbeusedtodetecttailbehavior:1)Plotofthemeanexcessfunction(MEF).LetrandomvariableXhavefinitemean,i.e.E[X]+¥,themeanexcessfunction(MEF)associatedwithXisdefinedase(u)=E(X-u|Xu).Itiseasytoprovethat,ifXisexponentiallydistributed,itsmeane