VaR数学模型及其计算方法-刘红波

2008,17(4):334-338ActaAgriculturaeBoreali-occidentalisSinicaVaR数学模型及其计算方法*刘红波,边宽江*,程波,袁志发(,712100):VaR(ValueatRisk),,,,VaR,,(VaR)VaR,,:VaR;;;:F224.9:A:1004-1389(2008)04-0334-05VaRMathematicalModelandItsComputingMethodsLIUHong-bo,BIANKuan-jiang*,CHENGBoandYUANZh-ifa(CollegeofScience,NorthwestA&FUniversity,YanglingShaanxi712100,China)Abstract:VaR(ValueatRisk),isanewcriteriontomeasurethemarketriskbyastandardstatisticaltechnology,anditiswidelyusedinfinancialmathematicsatpresent.Itisamethodtoanticipatethemostheavylossunderthenormalmarketconditionwiththegivenconfidentlevelandtimehorizon.ThetraditionalVaRcomputationalmethodisusedincalculatingtheopenstylefund,itmayoverest-imatetherisk.Thevalueofriskweobtainedunderthelogormaldistributionsuppositionmustbemoreapproachtheactualvaluecomparedtounderthenormaldistributionsupposition.Thispaperes-peciallystudythemathematicaltheoryandthecomputingmethodsofVaRmodel.Usingthelogormaldistributionsuppositiontoappraisetheriskoftheopenstylefund,toconfirmwhethertheresultise-venmoreapproachestherealvalueofrisk.Keywords:VaR;Confidentlevel;Timeseries;LogarithmnormaldistributionVaR,,,,,VaRVaR,[1],VaRUmbertoCheru-biniElisaLuciano(2001)VaR(2003)VaR,,/)0,VaR,,VaR*:2007-12-04:2008-04-05:(1982-),,,,*:(1963-),,,,,1VaR模型的数学原理VaR,,(,):Prob(vWVaR)=1-c=A,,vWvt;VaRA;c,,,99%,,VaR150,,1501%,,99%150W0,W,R,W=W0(1+R),,W*=W0(1+R*),R*VaR,:VaR()=E(W)-W*(1)0,:VaR()=W0-W*(2):W=W0(1+R)(3)W*=W0(1+R*)(4),(1)(2)(3)(4):VaR()=E[W0(1+R)]-W0(1+R*)=W0[E(R)-R*]=W0(L-R*)(5)VaR()=W0-W0(1+R*)=-W0R*(6),RI{R1,R2,,,Rt},,L,(1)~(6),VaRW*R*2VaR模型中变量的确定2.1,,{Wt},W0,W,R,W=W0(1+R)R,{Rt},Rt=Wt-Wt-1t-(t-1)/Wt-1t-(t-1),(t-1),Wt-1,{Rt}N(L,R2),VaR,,[5],2.2XH,X1,X2,,,XnH1(X1,X2,,,Xn)H2(X1,X2,,,Xn),A(0A1):P{H1(X1,X2,,,Xn)}[H[H2(X1,X2,,,Xn)}=1-A,100(1-A)%,100A%,,,95%~99%2.3(,vt),,VaR[7];,1~3dVaR;,3VaR计算方法3.1ARCH,Engle1982ARCH,NelsonDarielB1990,,EGARCH(p,q),:lnR2t=X+Eqi=1Ai[Hzt-i+C(|zt-i|-E|zt-i|)]+Eqi=1BilnR2(t-1)#335#4:VaR3.2RiskMetrics,,,H=(L1,L2;R1,R2;p),VaR:rtRt=p#f1(x;L1,R1)+(1-p)#f2(x;L2,R2)R2t+1|t=(1-K)r2t+KR2t|t-1fj(x;Lj,Rj)Lj,Rj,VaR=zARtk3.3,:¹,;ºEi(i=1,2,n),s(t+1),s(t+2),,s(t+n);»Pt+n=PTvpt+n=vpT,º,»,vpT1,vpT2,,VaR3.4MCMCMCMCMonteCarloMCMC,,MCMC,,4VaR计算方法的改进W,,,VaR,,(VaR)[6],lnWN(L,R2),W,:f(W)=12PRWe-(lnW-L)22R2W0WLW,R2W,LW=E(W)=Q+]-]W#f(W)dW=Q+]-]1R2Pe-(lnW-L)22R2dWlnW=L+R#t,dW=e(L+Rt)Rdt=1R2PQ+]-]e(-t22)e(L+Rt)dt=12PeLQ+]-]e(-t22+ot)dt=12PeLQ+]-]e(-(t-R)22)eR22dt=12Pe(L+R22)Q+]-]e-(t-R)22dtm=t-R=12Pe(L+R22)Q+]-]e-m22dmQ+]-]e-m22=2P,=e(L+R22)R2W=D(W)=E(W2)-(EW)2=Q+]-]W2#f(W)dW-(e(L+R22))2=Q+]-]W21RW2Pe-(lnW-L)22R2dW-e(2L+R2)=Q+]-]1RW2Pe-(lnW-L)22R2dW-e(2L+R2)lnW=L+R#t,W=e(L+Rt),dW=e(L+Rt)Rdt=1R2PQ+]-]e(L+Rt)e-t22e(L+Rt)Rdt-e(2L+R2)=12PQ+]-]e-(t-2R)22e(2L+2R2)dt-e(2L+R2)x=t-2R=12Pe(2L+2R2)Q+]-]e-x22dx-e(2L+R2)=e(2L+2R2)-e(2L+R2)=(eR2-1)e(2L+R2):1-c=QW*-]f(W)dW=QR*-]f(R)dR=QLa-]U(E)dE,La,U(E)P(RR*)=P(R-LWRWR*-LWRW)=1-c:R*-LWRW=LA]R*=LW+RWLAVaR:VaR=E(W)-W*=E[W0(1+R)]-W0(1+R*)=W0(LW-R*)#336#17=W0(LW-LW-RWLA)=-W0RWLA=-W0[e2L+R2(eR2-1)]LAVaR=W0-W*=W0-W0(1+R*)=-W0R*=-W0(LW+RwLA)=-W0(L+R22+[e2L+R2(eR2-1)]LA)2,,n,Ri(t)(i=1,2,,,n),R(t)=(R1(t),R2(t),,Rn(t)T),R(t),F=(Qi,j)n@nn,x=(x1,x2,,,xn,),En1xi=1,RW(t),RW(t)=x1R1(t)+x2R2(t)+,+xnRn(t),RW(t),,(VaR)VaRw=-W0RwLA(7)Rw,x=[x1x2,xn]xT=[x1x2,xn]T,U=R10,00R2,0,,,,00,RnF=1Q1,2,Q1,nQ2,11,Q2,n,,,,Qn,1Qn,2,1,,RwR:Rw2=xRFRxT,(7):VaRW=-W0(xRFRxT)1/2LA5计算实例VaR,8,,2006762007070941(:)30VaR,11VaR,895%99%,:L1=0.0246,L2=0.01,L3=0.0146,L4=0.01324L5=0.014,L6=0.01228,L7=0.0112,L8=0.0352R12=0.001659,R22=0.000127,R32=0.000462,R42=0.000199R52=0.000165,R62=0.000166,R72=0.000296,R82=0.003042VaR()=-W0[e2L+R2(eR2-1)]LA,L0.05=1.645,L0.01=2.33,:VaR1(95%)=1.775@[e2@0.0246+0.001659(e0.001659-1)]@1.645=3.033VaR1(99%)=1.775@[e2@0.0246+0.001659(e0.001659-1)]@2.33=4.295,,VaR8(95%)=3.246@[e2@0.0352+0.003042(e0.003042-1)]@1.645=5.729VaR8(99%)=3.246@[e2@0.0352+0.003042(e0.003042-1)]@2.33=8.11511VaRTable1ThevalueofVaRunderlogormaldistributionNameoffundsVarianceofsampleVaR(95%)VaRvalue(95%)/%VaR(99%)VaRvalue(99%)/%0.0016593.0334.2950.0001272.0562.912100ETF0.0004622.3953.3920.0001992.6413.7410.0001652.4283.4390.0001661.8942.68350ETF0.0002961.9192.719180ETF0.0030425.7298.115#337#4:VaRVaR,(1996)1122Table2TheareaofBaselruleTheareaNumberofexceptionGreenlight0~1Yellowlight2~3Redlight4VaR1,,95%,VaR,VaR6结论VaR,VaR:,;;,VaR,VaR,VaR,,VaR,,,VaR,,:[1],.VaR[J].,2003,(4):74-76.[2],.VaR[J].:,2005,3(3):142-144.[3],.VaR[J].,2003(2):6-10.[4],.[M].,2002.[5],,.VaR[J].,2006,(3):152-155.[6],.VaR[J].,2006,(2):34-39.[7].VaR[J].,2001,9.[8]SmonsKaterna.TheUseofValueatRiskbyInstitutionalInvestor[J].NewEnglandEconomicReview,2000,(6):21-31.[9]BerkowitzJ,BrienJ.Howaccuratearethevalue-at-riskmod-elsatcommercialbanks?[J].JournalofFinance,2002,(6):1093-1112.#338#17