风险度量方法研究

VaRESValueatriskVaRConditionValueatrisk(CVaR)Expectedshortfall(ES)DistortionRisk-Measure1VaRValueatRisk1994aXX()Fxa()min{|()}VaRxFxaa=-≥X2001VaRVaRVaRVaR2(Coherentmeasureofrisk)Artzneretal.(1999)1,0()0XVXXr∈≥⇒≤2,,()()()XVYVXYVXYXYrrr∈∈+∈⇒+≤+3,0,()()XVhhXVhXhhrr∈∈⇒=f4,()()XVaRXaXarr∈∈⇒+=-VaR100AB98.9ABA+BProb1701001703%2901001902%3100701703%4100901902%510010020090%ABAB5VaRABAB98.998.9197.85VaR8.98.927.8AB5VaRAB5VaR3CVaR(ConditionValueatRisk)CVaRaVaRaX()Fx{|()}CVaRExFxaa=-≤CVaRVaRVaRVaRVaRCVaRCVaRCVaR4ESExpectshortfall:ESCVaRXESCVaRXX()[]FxPXx=≤1()inf{|()}FpxFxp-=≥()()ESXa1()01()()ESXFpdpaaa-=-∫ESXESABABABAB98.998.9197.85ES20.920.927.8ABAB4DRMDistortionRisk-Measure:DRMg:[0,1][0,1]g→(0)0g=(1)1g=*()(())FxgFx=g*FFDRM()Xr0*0()()(())[1(())]XEXgFxdxgFxdxr+∞-∞==-+-∫∫DRMVaRCVaR{0()1uguuaa=≥pDRMVaR0()1uguuuaaaa⎧⎪=-⎨≥⎪-⎩pDRMCVaRDRM()XrgDRMES1ir1,2,,in=Lia1,2,,in=L11niia==∑0niiiarr==∑ir4r4ir,,()()()iiiXVYVXYVXYXYrrr∈∈+∈⇒+≤+1111()()[()()]()()()()nniiiiiiinniiiiiiXYaXYaXYaXaYXYrrrrrrrr====+=+≤+=+=+∑∑∑∑r3irarESaar=ar[,]aba∈()1badua=∫()0dua≥()dua()baduarar=∫ar4r4ar,,XVYVXYV∈∈+∈()()()iiiXYXYrrr+≤+()dua()1badua=∫()0dua≥()()()()(()())()()()()()()bbaabbaaXYduXYduXYduXduYXYaaaaarararrararrr+=+≤+=+=+∫∫∫∫ar3ESa10()1duaa=∫()0dua≥[0,1]a∈10()duESaraa=∫11100011001()()()()()duESduFpdpduFpdpaaaraaaaaa--==-=-∫∫∫∫∫11111000()()()()pduFpdpFpdpduaraa--=-=-∫∫∫∫1()()pfpdua=∫()0dua≥1()()0pfpdua=≥∫()fpp1111000010()()()()1pfpdpdpdududpduaaaaa====∫∫∫∫∫∫()fp()fppp()fp110()()Fpfpdpr-=-∫X()[]FxPXx=≤1()inf{|()}FpxFxp-=≥110()()Fpfpdpr-=-∫()fp()0fp≥10()1fpdp=∫()fpp110()()Fpfpdpr-=-∫11(0)1()pfpIaa≤≤=1()fp1110()()Fpfpdpr-=-∫11111(0)00101()()()1()pFpfpdpFpIdpFpdpESaaaraa--≤≤-=-=-=-=∫∫∫ESa220()()pfpppadaa≠⎧=-=⎨+∞=⎩2()fp2()fp1120()()Fpfpdpr-=-∫11112001()()()()()FpfpdpFppdpFVaRardaa---=-=--=-=∫∫VaRa()fp()fpESa[1]GiorgioSzego,“Measuresofrisk”,JournalofBanking&Finance,26(2002),1253-1272.[2]CarloAcerbi,DirkTasche,“ExpectedShortfall:anaturalcoherentalternativetoValueatRisk”,Workingpaper.[3]ShaunS.Wang,“ARiskMeasureThatGoesBeyondCoherence”,Workingpaper.[4]H.Yang,T.K.Siu,“CoherentRiskMeasuresForDerivativesUnderBlack-ScholesEconomy”,InternationalJournalofTheoreticalandAppliedFinance,Vol.4,No.5(2001),819-835.[5]CarloAcerbi,ClaudioNordio,CarloSirtori,“ExpectedShortfallasaToolforFinancialRiskManagement”,Workingpaper.[6]R.TyrrellRockafellar,StanislavUryasev,“Conditionalvalue-at-riskforgenerallossdistribution”,JournalofBanking&Finance26(2002),1443-1471.[7]()nESNo.4.2003.[8]20012