룟킣펦폃쫽톧톧놨A벭,19(3):3632368ꆡ믹뒡막욱폐럖뫬벰엤막뗄웚좨뚨볛폫쳗웚놣횵돂ꆡ욼1,튶훐탐2,퇮킢욽1(11쓏뺩샭릤듳톧샭톧풺,붭쯕쓏뺩210094;21짏몣붻춨듳톧샭톧풺,짏몣200030)햪ꆡ튪:룸돶쇋폐럖뫬벰엤막뗄막욱볛룱퓋뚯맦싉,늢쳖싛쇋틔뚨웚럖뫬벰엤막뗄막욱캪뇪뗄닺뗄쏀쪽뾴헇웚좨뗄뚨볛폫쳗웚놣횵컊쳢Λ춨맽뛔폐춹횧뢶몯쫽뗄쏀쪽웚좨횴탐쪱볤뗄쳖싛뗃떽쏀쪽뾴헇웚좨뗄ퟮ폅횴탐쪱볤횻뿉쓜퓚쎿듎럖뫬믲쯍엤막돽좨돽쾢횮잰Λ횤쏷쇋퓚룷듎럖뫬믲쯍엤막횮볤,웚좨뗄횵싺ퟣ쫬횪뗄Black맘2S볼ch듊oles랽돌Λ:럖뫬엤막싊;쏀쪽웚좨;ퟮ폅횴탐쪱볤훐춼럖샠뫅:O22417컄쿗뇪쪶싫:Aꆡꆡ컄헂뇠뫅:100024424(2004)0320363206ꇬ1ꆡ틽ꆡ퇔웚좨쫇튻훖톡퓱좨,춶헟퓚횧뢶쇋튻뚨뷰뛮뗄좨샻뷰(웚좨볛룱)횮뫳,뻍펵폐퓚풤쿈맦뚨뗄쪱볤(떽웚죕)횮잰내풤쿈맦뚨볛룱(쟃뚨볛)릺싲믲돶쫛튻뚨쫽솿믹뒡닺뗄좨샻Λ내헕좨샻훖샠뿉붫웚좨럖캪뾴헇웚좨(릺싲좨)뫍뾴뗸웚좨(돶쫛좨)Λ내헕횴탐쪱볤뗄늻춬뿉붫웚좨럖캪얷쪽웚좨뫍쏀쪽웚좨Λ얷쪽웚좨횻퓊탭웚좨돖폐헟퓚떽웚죕뷸탐붻틗뛸쏀쪽웚좨퓲퓊탭웚좨돖폐헟퓚떽웚죕믲떽웚죕횮잰뗄죎뫎펪튵죕뷸탐붻틗Λ웚좨뗄뚨볛폫쳗웚놣횵쫇뷰죚쫽톧뗄튻룶믹놾컊쳢Λ뛔폚컞쒦닁쫐뎡,헢튻컊쳢틑뺭뗃떽쇋돤럖뗄퇐뺿,볻\[1ꆪ3\]Λ뷼쓪살,죋쏇뾪쪼럅뿭뛔쫐뎡뗄볙짨,뾼싇룷훖폫쪵볊쟩뿶룼볓컇뫏뗄쟩탎Λ웤훐늻짙톧헟쳖싛쇋믹뒡막욱폐럖뫬뗄쟩탎쿂,웚좨뗄뚨볛벰쳗웚놣횵닟싔Λ볻\[4ꆪ7\]Λ힢틢떽컒맺막쫐훐탭뛠막욱쪵탐럖뫬뗄춬쪱뮹폐쯍ꆢ엤막(엤막쫇횸짏쫐릫쮾캪쇋돯벯뷰,내뇈샽뫍튻뚨볛룱쿲쿖폐막뚫뚨쿲엤쫛룃릫쮾막럝뗄좨샻),컒쏇폐뇘튪뾼싇퓚믹뒡막욱춬쪱폐럖뫬ꆢ쯍엤막쳵볾쿂웚좨뗄뚨볛벰쳗웚놣횵닟싔Λ헢훖쟩뿶샠쯆폚믹뒡막욱폐럖뫬뗄쟩탎떫쒣탍룼볓뢴퓓Λ놾컄퓚룸뚨믹뒡막욱럖뫬벰쯍ꆢ엤막싊뗄쳵볾쿂,쳖싛쇋쏀쪽뾴헇웚좨뗄뚨볛벰쳗웚놣횵닟싔Λ얷쪽웚좨쿠펦뗄컊쳢뿉샠쯆뒦샭Λ쫕룥죕웚:2004204216믹뷰쿮쒿:맺볒ퟔ좻뿆톧믹뷰(10171066)' 1994-2007 China Academic Journal Electronic Publishing House. All rights reserved.
364룟킣펦폃쫽톧톧놨A벭뗚19뻭뗚3웚ꇬ2ꆡ쏀쪽웚좨뗄뚨볛벰ퟮ폅횴탐쪱볤짨튻쏀쪽웚좨떽웚죕캪T,쟃뚨볛룱캪K,믹뒡막욱퓚t쪱뿌뗄볛룱캪S(t),죴쎻폐럖뫬벰엤막,퓲S(t)싺ퟣ랽돌:dS(t)=r(t)S(t)dt+Ρ(t)S(t)dB(t),0ꇜtꇜT,()웤훐r(t),Ρ(t)캪럇쯦믺몯쫽,럖뇰뇭쪾쮲쪱컞럧쿕샻싊벰막욱뗄늨뚯싊Ζr(t)ꇝ0,B(t)캪럧쿕훐탔닢뛈쿂뗄Brownian퓋뚯,룹뻝뷰죚쫽톧뗄샭싛\[8,9\],볇웚좨퓚t쪱뿌뗄쓚퓚볛횵캪h(S(t)),(죧쏀쪽뾴헇웚좨,h(S(t))=(S(t)-K+)).)틔ꆰEx(Ν)ꆱ뇭쪾럧쿕훐탔룅싊쿂,Ν맘폚S(0)=x뗄쳵볾웚췻,Β(t)=expꇒtr(u)du캪쳹쿖틲ퟓΛ0퓲웚좨뗄횵몯쫽평죧쿂(212)쪽뚨틥,Β(t)v(t,x)=suptꇜΣꇜExTΒ(Σh(S(Σ)),())ꆡꆡ싺ퟣ:1(a)rv-vt-rxvx-Ρ22xvxxꇝ0,2(b)vꇝh(x),(c)퓚죎튻뗣(t,x)ꇊ\[0,T]ꇁ\[0,ꇞ),(a)믲(b)훐뇘폐튻룶뗈뫅돉솢Λ쟒v뻟폐죧쿂탔훊:룸뚨0쪱뿌뗄막럝S(0),퓲1(d)Βv(t,S(t)),0ꇜtꇜT,퓚럧쿕훐탔닢뛈쿂캪짏;(t)1(e)Βv(t,S(t))ꇝ1(t)Βh(S(t));(t)1(f)Βv(t,S(t))쫇싺ퟣ(1),(2)뗄ퟮ킡짏Λ(t)웚좨뗄ퟮ폅횴탐쪱볤쫇Σ=min{tꇝ0;v(t,S(t))=h(S(t))}.()ꆡꆡ맘폚ퟮ폅횴탐쪱볤,폐죧쿂뷡싛:틽샭211ꆡ짨+h(x)캪춹몯쫽쟒h(0)=0.(죧h(x)=(x-K)),퓲횧뢶캪h(S(t))뗄쏀쪽웚좨컞탫쳡잰횴탐Λ벴횻탨뗈듽떽웚죕퓙뻶뚨횴탐폫럱Λ횤ꆡ틲h(x)쫇춹몯쫽,맊뛔0ꇜΑꇜ1,h(Αx)ꇜΑh(x),맊Β(t)hΒS(T)ꇜΒ(t)h(S(t)).()(T)Β(T)ꆡꆡ뾼싇튻떽웚폐횧뢶몯쫽h(S(T))뗄얷쪽웚좨,뛔0ꇜtꇜT,볇X(t)캪t쪱뿌뗄횵몯쫽,퓲1X(t)=Β(t)EΒh(S(T))û(T)F(t)쪽훐E캪럧쿕훐탔닢뛈,F(t)캪평Brownian퓋뚯B(t)짺돉뗄Ρ폲쇷Λ폚쫇' 1994-2007 China Academic Journal Electronic Publishing House. All rights reserved.
돂ꆡ욼뗈:믹뒡막욱폐럖뫬벰엤막뗄웚좨뚨볛폫쳗웚놣횵365X(t)1Β(t)Β=(t)ΒE(t)Βh(S(T))û(T)F(t)ꇝ1Β(ΒEt)h(t)ΒS(T)û(T)F(t)ꆡ(평(214))ꇝ1S(T)Βh(Β(t))E(t)Βû(T)F(t)ꆡ(평Jensen늻뗈쪽)=1Βh(S(t)).(t)ꆡꆡ평쏀쪽웚좨v뗄탔훊(a)2(f)횪,뛔펦쏀쪽웚좨뗄ퟮ폅횴탐쪱볤캪TΛ틗볻,싺ퟣ틽샭쳵볾뗄쏀쪽웚좨폫뻟폐쿠춬쟃뚨볛룱폫떽웚죕뗄얷쪽웚좨뗈볛Λ죴퓚(211)훐좡r,Ρ캪뎣쫽,퓲폐죧쿂훸쏻뗄Black2Scholes얷쪽웚좨뚨볛릫쪽Λ틽샭212ꆡ떽웚쪱볤캪T,횴탐볛룱캪K,볛룱맽돌캪(211)뗄막욱얷쪽웚좨볛횵캪-t)v(t,S(t))=S(t)N(d+(T-r(T-t,S(t)))-KeN(d-(T-t,S(t))),0ꇜtꇜT.()ꆡꆡ쪽훐12xdꇀ(T-t,x)=log(T-t)rꇀΡ.()ΡT-K+t22ꆡꆡsN-(õ)캪뇪ힼ헽첬럖늼몯쫽1,N(d)=e2dsΖ2Πꇒd-ꇞ쳗웚놣횵ퟩ뫏캪∃(t)=vx(t,S(t)).(웤훐,∃(t)뇭쪾퓚t쪱뿌릺싲뗄막욱럝뛮)Λꇬ3ꆡ폐럖뫬늢쯍ꆢ엤막뗄쏀쪽뾴헇웚좨뗄뚨볛벰ퟮ폅횴탐쪱볤짨믹뒡막욱볛룱싺ퟣ랽돌(211),캪볲떥뇭쫶,짨웤훐r,Ρ캪뎣쫽,룃랽돌폐뷢1S(t)=S(0)expr-Ρ2t+ΡB(t),0ꇜtꇜT.()2ꆡꆡ쿖볙뚨퓚t1,0ꇜt1ꇜT쪱뿌폐튻듎럖뫬늢엤막,틔t1-0뇭쪾t1쪱뿌돽좨돽쾢잰쮲쪱,틔t1+0뇭쪾t1쪱뿌돽좨돽쾢뫳쮲쪱;짨S(t1-0)쫇t1쪱뿌돽좨돽쾢잰쮲볤막욱볛룱,0ꇜΒ<1쫇쿖뷰뫬샻싊,(벴쿖뷰뫬샻폫S(t1)뗄뇈횵);엤쯍막뇈샽캪1엤∆1,쯍∆2,(∆1ꇝ0,∆2ꇝ0)Ζ엤막볛폫S(t1-0)뗄뇈횵볇캪Χ,0<Χ<1Λ룹뻝돽좨돽쾢잰뫳쫐횵쿠뗈뗄풭퓲,돽좨돽쾢놨볛캪(1-Β+∆1Χ)S(t1+0)=t1-0),()1+∆1+∆S(2ꆡꆡ볇Α(1-Β+∆1Χ)=.()1+∆1+∆2ꆡꆡ돆Α캪럖뫬엤막싊,틗횤0<Αꇜ1Ζ폚쫇,뷡뫏(311),퓚t1쪱뿌폐럖뫬늢쯍엤막뗄막볛맽돌싺ퟣ' 1994-2007 China Academic Journal Electronic Publishing House. All rights reserved.
366룟킣펦폃쫽톧톧놨A벭뗚19뻭뗚3웚12S(0)exp{(r-Ρ)t+ΡB(t),0ꇜt<t1,2S(t)=()Α12S(t1)exp{(r-Ρ)(t-t1)+Ρ(B(t)-B(t1)),t1<tꇜꆡꆡ뾼싇틔짏쫶막욱캪뇪뗄닺뗄쏀쪽뾴헇웚좨,평틽샭211,죎틢tꇊ(t1,T)뚼늻쫇웚좨뗄ퟮ폅횴탐쪱볤,맊웚좨퓚tꇊ(t1,T)뗄횵싺ퟣBlack2Scholes릫쪽,평틽샭212,(T-t)v(t,x)=xN(d+(T-t,x))-K-reN(d-(T-t,x)),t1<tꇜT.()ꆡꆡ퓚t1+0쪱뿌,웚좨뗄횵캪v(t1+0,ΑS(t1-0));돽좨돽쾢횮잰,웚좨뗄횵캪w(t1-0,S(t1-0))=max((S(t1-0)-K+),v(t1+0,ΑS(t1-0)).()ꆡꆡ뚨샭311ꆡ뛔0ꇜt<t1,쏀쪽웚좨뗄횵캪w(t,S(t))Λ웤훐w(t,x)=Et,x-r(t[e1-t)w(t1-0,S(t1-0))\].()ꆡꆡEt,x(Ν)뇭쪾럧쿕훐탔룅싊쿂Ν맘폚S(t)=x뗄쳵볾웚췻Ζw싺ퟣBlack2Scholes랽돌Λ-rw1+wt+rxwx+Ρ22xwxx=0,0ꇜt<t1,x>0.()2ꆡꆡ싺ퟣ훕뛋쳵볾w(t1-0,x)=max((x-K+),v(t1+0,Αx)),xꇝ0.()ꆡꆡ뇟뷧쳵볾w(t,0)=0,0ꇜt<t1.()ꆡꆡ쳗웚놣횵ퟩ뫏캪wx(t,S(t)),0ꇜt<t1,∃(t)=()vx(t,S(t)),t1<tꇜT.ꆡꆡ횤ꆡ횻탨횤폐횧뢶몯쫽w(t1-0,S(t1-0))뗄쏀쪽웚좨컞탫퓚t1-0쪱뿌횮잰횴탐Λ룹뻝틽샭211,횻탨횤(1)w(t1-0,0)=0;(2)w(t1-0,x)맘폚x쫇춹몯쫽Λ쫂쪵짏,w(t1-0+,0)=max((0-K),v(t1+0,Α0))=v(t1+0,0)=0,틲캪(x-K+)쫇춹몯쫽,솽룶춹몯쫽뗄ퟮ듳횵죔쫇춹몯쫽,맊횻탨횤v(t1+0,Αx)쫇춹몯쫽벴뿉Λ볇Αx=y,룹뻝얷쪽뾴헇웚좨뗄탔훊,1v(t,y)=Β(t)E[Βh(y)ûF(t)\].(T)ꆡꆡ뛔0ꇜΗꇜ1,뾼달v(t1+0,Η1y1+(1-Η)y2)=Β(t1+0)E(Βh(Ηy1+(1-Η)y2)ûF(t1+0)]ꇜ(T)Β1(t1+0)E{Β[Ηh(y1)+(1-Η)h(y2)]ûF(t1+0)}=(T)ΗΒ1(t1+0)E{Βh(y1)ûF(t1+0)\]+(1-Η)Β(t1+0)E[h(y2)]ûF(t1+0)\]=(T)Ηv(t1+0,y1)+(1-Η)v(t1+0,y2).ꆡꆡ평듋뿉볻,퓚t1쪱뿌폐럖뫬늢쯍엤막뗄쏀쪽뾴헇웚좨뗄ퟮ폅횴탐쪱볤횻뿉쓜쫇t1-0' 1994-2007 China Academic Journal Electronic Publishing House. All rights reserved.
돂ꆡ욼뗈:믹뒡막욱폐럖뫬벰엤막뗄웚좨뚨볛폫쳗웚놣횵367믲TΖ쿖뾼싇t1-0쪱뿌뗄쳗웚놣횵Ζ좡x=S(t1-0),죴v(t1+0,Αx)ꇝ(x-K+),퓲웚좨늻탨퓚t1-0쪱뿌횴탐,헢쪱,w(t1-0,x)=v(t1+0,Αx),쳗웚놣횵ퟩ뫏캪∃(t1-0)=wx(t1-0,x)=Αvx(t1-0,Αx)=Α∃(t1+0).()ꆡꆡ힢ꆡ평(3112),11-Α∃(t1+0)=Α∃(t1-0)=∃(t1-0)+Α∃(t1-0)=(Β+∆1(1-Χ)+∆2)∃(t1-0)S(t1-0)∃(t1-0)+.13)(1+∆1+∆2)Α.(3S(t1-0)웤훐(Β+∆1(1-Χ)+∆2)∃(t1-0)S(t1-0)잡뫃쫇막뚫틲럖뫬벰쯍ꆢ엤막쯹뗃떽뗄킧틦,(1+∆1+∆2)ΑS(t1-0)쫇돽좨돽쾢횮잰뗄쎿막퓚돽좨돽쾢뫳뗄쫐횵Λ맊(3113)뇭쏷,돽좨돽쾢뫳뗄쳗웚놣횵ퟩ뫏쫇퓚풭폐막럝쫽뗄믹뒡짏,퓙퓶볓평럖뫬벰쯍ꆢ엤막쯹뗃떽뗄쫕틦횵폫쎿막퓚돽좨돽쾢뫳뗄쫐횵뗄뇈횵쯹뻶뚨뗄막쫽Λ죴v(t1+0,Αx)<(x-K+),퓲웚좨돖폐헟펦퓚t1-0쪱뿌횴탐웚좨Ζ웤쫕틦캪S(t1-0)-KΖ쳗웚놣횵닟싔폫떽웚쪱볤캪t1,쟃뚨볛룱캪K뗄얷쪽웚좨쿠춬Λ짨퓚웚좨뗄듦탸웚폐n듎럖뫬벰쯍ꆢ엤막,쪱볤캪0=t0<t1<t2<ꆭ<tnꇜTΛ뛔펦닎쫽럖뇰캪Α1-Β(ti)+∆1(ti)Χ(ti)i=1+∆1(ti)+∆.2(ti)샠쯆틔짏쳖싛틗뗃췆싛312ꆡ뛔tn<tꇜT,쏀쪽뾴헇웚좨뗄횵캪wn+1(t,S(t))=xN(d+(T-r(T-t)-t,x))-KeN(d-(T-t,x)),tn<tꇜT.ꆡꆡ뛔ti-1ꇜtꇜti,i=1,ꆭ,n,쏀쪽뾴헇웚좨뗄횵캪wi(t,S(t)).웤훐wi(t,x)=Et,x-r(t-t)[eiw(ti-0,S(ti-0))]싺ퟣBlack2Scholes랽돌-rw+wt+rxw1x+Ρ22xwxx=0,0ꇜtꇜt1,x>ꆡꆡ훕뛋쳵볾wi(ti,x)=max((x-K+),wi+1(ti+0,Αix)),xꇝ0.ꆡꆡ뇟뷧쳵볾w(t,0)=0,ti-1ꇜtꇜti.쳗웚놣횵ퟩ뫏캪∃i(t)=wi,x(t,S(t)),ti-1ꇜtꇜti.웚좨뗄ퟮ폅횴탐쪱볤횻뿉쓜퓚t1-0,t2-0,ꆭ,tn-0쪱뿌Ζ횴탐쳵볾쫇wi(ti+0,Αx)<(x-K+).퓚쪵볊컊쳢훐,럖뫬벰엤막닎쫽튻냣늻쓜쫂쿈좷뚨,틲뛸펦뾴ퟷ쯦믺뇤솿Λ헢킩쯦믺뇤솿뗄쫽톧웚췻뿉춨맽믹뒡막욱뗄럖뫬ꆢ엤막샺쪷쫽뻝맀볆뗃떽Λ틲뛸떱럖뫬엤막싊캪쯦믺뇤솿뗄쪱뫲,횻탨퓚(212)쪽훐좡Β,Χ,∆1,∆2캪뛔펦닎쫽뗄쫽톧웚췻,퓲틔짏뷡싛죔좻돉솢Λ' 1994-2007 China Academic Journal Electronic Publishing House. All rights reserved.
368룟킣펦폃쫽톧톧놨A벭뗚19뻭뗚3웚닎뾼컄쿗:\[1]BlackF,[J\].JournalofPoliticalEconomy,1973,81:6372659.\[2][J\].JournalofFinancialEconomics,1976,3:3251.\[3][J\].AnnalsofAppliedProbablity,1992,2(1):1223.\[4][J\].JournalofFinancialEconomics,1979,7:3752380.\[5]Barone2AdesiG,[J\].JournalofFinancialEconomics,1986,17:912111.\[6]DewynneJN,HowisonSD,RupfI,[J\].EuropeanJournalofAppliedMathematics,1973,4:3812398.\[7]PerrakisS,[J\].JournalofEconomicDynamic&Control,2000,24:152721561.\[8][M\].Singapore:Springer,1998.\[9]튶훐탐,쇖붨훒.쫽샭뷰죚ꆪꆪ닺뚨볛폫뷰죚뻶닟샭싛[M\].놱뺩:뿆톧돶냦짧,-payingandplacingstocksCHENP121ing,YEZhong2xing,YANGXiao2ping(,NanjingUniversityofScienceandTechnology,Nanjing210094,China;,ShanghaiJiaotongUniversity,Shanghai200030,China)Abstract:Thepaperderivesthelawofstockpricewithdividend2payingandplacing,:dividend2payingandplacingrate;Americanoption;optimalexercisingtimeMRSubjectClassification:90A99' 1994-2007 China Academic Journal Electronic Publishing House. All rights reserved.
