贷款保证组合与联合保证之研究

______________________________________________________________()Merton()Lai&Yu(guarantor)()1CIRCIR(meanreversion)()CIRMerton(1977)MertonMerton()()Stulz&Johnson(1985)(secureddebt)(juniorsecureddebt)(seniorsecureddebt)()Lai(1992)Lai&Gendron(1994)CIR(MonteCarlo)Jones&Mason(1980)Merton(1973)Chen&Mazumdar(1996)(subordinatedebtclaim)(closed-formsolution)Merton(1977)Merton(1977)MertonBV(1)V≥BBV-B(2)VBVMin[V,B]Max[0,V-B]B(1)V≥B(2)VB(B–V)G(t)tG(0)=Max[0,B–V]Black&Scholes(1973))X(NV)X(NBe)t(G12rt-=-(1)tXXtt)2/r()B/Vln(X1221σ+=σ×σ+-=VB2σtrLai&Yu(1999)Lai&Yu(1999)WV(seniordebt)HDBF(subordinatedebt))W,V(f⎩⎨⎧++++++++≥+=FHDWV(2)Stulz&Johnson(1985)[][]{}F,0,H)T(WMax0,D)T(VMaxMin)W,V(f-+-=L(V,W)(risk-neutralizedbivariatedensityfunction)VW(jointlognormaldistribution)V2vσW2wσVW(covariance)VWσWVVWVWσσρ=σVWρVW(riskneutralvaluationrelationship,RNVR)(guaranteedloan)Bg∫∫∞∞-=001fgdVdW)W,V(L)W,V(fRB(3)2∫∫∞-++∞++-⋅=VF)HD(F)HD(1fgdVdW)W,V(LFRB∫∫-++-++++-+-++VF)HD(V)HD(F)HD()HD(1fdVdW)W,V(L)]HD()WV[(R(4)Lai&Yu))ds(N)du(N()uRWV()d(NFRB1101f0021fg-⋅+++⋅⋅=--))ds(N)du(N()uHD(R2201f-⋅++--(5)0uu12u2uf001f001u12u2uf001WVVW2V2V2u2u1f01f0du:du/)2/RlnuHDuRWVln(:dud:d/)2/RlnFHDWV(ln:d2:)W(EˆR:W)V(EˆR:Vσ-σσ++⎥⎦⎤⎢⎣⎡++++σ-σσ++⎥⎦⎤⎢⎣⎡+++σσρ-σ+σ=σσ---u12u2uf001f001ds:ds/)2/RlnuHDuRWVln(:dsσ-σσ++⎥⎦⎤⎢⎣⎡++++-:)(N⋅BaWHBa)]z(N)Z(N[DR)]z(N)Z(N[V)]Z(N1[FRB1f01fa---+-=--(6)V2v0fV2V0f/)2/)VRln(Dln(:z/)2/)VRln()FDln((:Zσσ+-σσ+-+GagBBG-=(7)(portfolioofloanguarantee)(Montecarloapproach)..CIR1.A.B.C.D.(lognormaldistribution)E.(riskneutralmarket)F.G.H.CIR2.WT(seniordebt)H2WsNViT(i=1,2,....N)DiTFiTi2VisiDiHViFiWG(F1)G(F2)...EViEW(t=T)iFiT(subordinateclaim)Ni(payoff)Bgi}F,])T(R,F[Min]0,D)T(V[Max{Min)T(BiTiiTiiTigiα+-=(8)Ri(T)i)T(X]0,HW[Max]}0,]0,DV[MaxF[Max,F{Min)T(RTiTiTiTiTii-⋅--α=(9)]}0,]0,DV[MaxF[Max,F{Min)T(XiTiTiTiTi--α=∑=≠--α+1jjTjTjTiTjji]}0,]0,DV[MaxF[Max,F{Min)T(R,ji,]0,DV[MaxFiiTiTiT=-≤(artificialasset)QiQiT)]T(R,F[Min]0,DV[MaxiiTiiTiTα+-=(10)]F,Q[Min)T(BiTiTgi=]0,FQ[MinFiTiTiT-+=]0,QF[MaxFiTiTiT--=)T,F,Q(PutFiTiTiT-=Bi(T)}F,]0,DV[Max{Min)T(BiTiTiTi-=(11)=iGi}F,]0,DV[Max{Min]F,Q[MinGiTiTiTiTii--=]0,QF[Max}0,]0,DV[MaxF{MaxiTiTiTiTiT----=)T,F,Q(Put)T,F,)T,D,V(Call(PutiTiiTiTiT-=(12)(iFiT)}F,]0,)0,DV(MaxF[Max{MinHWiTiiTiTiTT×α---∑=×α--+1jjTjjTjTjT}F,]0,)0,DV(MaxF[Max{Min(13)1.(MonteCarlo)A.(t=T)(GiT))G(Eˆe)eG(EˆGiTrTrTiTi--==(14)(GeometricBrownianmotion)dtdtVdVdtdtWdWViViViiiεσ+μ=εσ+μ=(15)Wμ(risk-neutralmarket)Wσ0.1Viσi0.2WεViWεεViWρB.CIRdZrdt)rb(adZrdt)r(drrrrσ+-=σ+β+α=(16)Z(standardwienerprocess)Chan,Karolyi,LongstaffandSanders(1992)2339.0,0189.0r-=β=α08544.0r=σr,Wρr,Viρ2.Hull(1993)nijj,iρnxi(1in))1(nii≤≤e∑=α=ε1kkikix(17)∑=αk2ik1∑ρ=ααkj,ijkikNCholesky(Choleskydecomposition)Cholesky(symmetricandpositive)(lowertriangularmatrix)R⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡ρρρρρρρρρρρρρρρρ=r,r2v,r1v,rw,rr,2v2v,2v1v,2vw,2vr,1v2v,1v1v,1vw,1vr,w2v,w1v,(18)'TTR=⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡ααααααα=N,N3,N2,N1,N2,21,21,100000TLMLMMMLL(19)RTNT)I,0(NXXXX,TXiidN21N21─→─⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡==⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡εεε=εMM(20)NGPW03.5Wσ0.1H2Vi02.1viσ0.2di1vj,viρvi,wρr,viρr,wρ0.3fi1iα100%T3())T,,,,f,d,H,,,,,r(PP)T,,,,f,d,H,,,,,r(GG/vwr,vr,w2v,1vw,vf/vwr,vr,w2v,1vw,vfffff-+++++++++++--+-+-++-+----ασσρρρρ=ασσρρρρ=CIR2-1-12-222-3()2-62-7-12-8()(2-7-1)2-9-132-10-12-11()12-12(15)(GeometricBrownianmotion)(drift)dtWμdtVμdtWWεσdtVVεσdt(2-1-1)(15)G1G2P)T,,,,f,d,H,,,r(GG)T,,,,f,d,H,,,r(GG)T,,,,f,d,H,,,r(PP/22vw222v,1vw,2vf22/22vw222v,1vw,2vf11/22vw222v,1vw,2vf-+-+-++-----+----------+++++++++-ασσρρ=ασσρρ=ασσρρ=CIR(GR)(GC)(GR-C)(PR)(PC)(PR-C)2-1-1(0.04)CIR(MeanreversioneffectMRE)(16)dt)rb(a-(16b)0.0840.08MRE5(GR)(PR)(2)0.08(GR)(PR)(16)dZrrσ6()dtr08544.0dt)r0808.0(2339.0dr00+-×=dtdtViWμμVWrεεεdtdtViViWWεσεσ()()dZrrσdtdtViWμμ2-1-52-1-72-1-10.04(16dZrrσ)0.080.082-1-32-1-4(16a)2-42-22-572-6(GR)(GC)(GR-C)(2)GR-C2-1-1GR-CGR-CPR-C2-7-112-1-1(2-1-1)dtViViεσdZrrσ8GR-CPR-C0.03(2-7-2)12-1-12-1-12-82-7-1GR-CPR-C2-9-1r=0.060.1()2-1-1()r=0.032-9-20.1()2-10-1GR-CPR-Cr=0.060.2(2-1-1)(15dtVVεσ)2-1-1()2-10-1GR-CPR-C2-10-2r=0.030.1()2-11GR-CPR-C(1)2-1-1GR-CPR-CGR-CGR-CPR-C2-122-13GR-CPR-CCIRCR1G-CR2G-PR-C)T,,,,f,d,H,,,,a,b(GG)T,,,,f,d,H,,,,a,b(GG)T,,,,f,d,H,,,,a,b(PP/22vw22r,2vr,wr/CR2CR2/*2*2vw2*2*r,2vr,wr/CR1CR1/22vw22r,2vr,wr/CRCR-+-+-++-+-+-+----+----+-+----++++++++++-+---ασσρρσ=ασσρρσ=ασσρρσ=b(16)abaCR2CR1GG--PR-CbaCR2CR1GG--PR-C*2-2-2-2-1-42-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-mWj(j=12.....m)Hj(j=12......m)VDFF(V-DF)jCjCjSjRj⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎨⎧⎩⎨⎧--+⎪⎩⎪⎨⎧-+--+-⎪⎩⎪⎨⎧-+--+-+-+=0HW0VFDif,0)F,VFDmin(m1HW00VFDif,HW)F,VFDmin(m1HW0VFDif,)F,VFDmin(m10VFDif,0Sjjjjjjjjj})HW,0max(,]F,)0,VFDmax(min[m1min{Sjjj--+=(21)]F,)0,VFDmax(min[m1res-+=])HW,0max(,resmin[Sjjj-=Rjj}]0,)0,HWmax(resmax[nm1,]0,res)0,HWmax(max[min{R1m1iiijjj∑-=-----=n⎩⎨⎧----==∑-=0)0,HWmax(resif,00)0,HWmax(resif,1AAnjjjjj1m1ijjjjRSC+=(22))F,VDFmin()HWmax(1iii-+-∑=0V3.2Vσ0.2DF1.5jWj01.6Hj1σWj0.1ρwj,Vρwi,wjρwj,rr,Vρ0.3T33-1-1()3-23-315dtVVεσdt2W2Wεσ3-43-53-69()3-73-8()10()dtVVεσVσ3-9-1(15)(GeometricBrownianmotion)dtWμdtVμdtWWεσdtVVεσdt(3-1-1)(15)C1C2P)T,,H,,F,D,,,r(CC)T,,H,,F,D,,,r(CC)