ֻ11जֻ3௹߉ၼഽٓ࿐Vol111No132012୍9ᄅჽ࿐Б(ሱಖ॓࿐) JOURNALOFHUA IYINTEACHERSCOLLEGE(NaturalScience) 一种带有交易成本的保底型基金的定价朱海燕1,张寄洲2(1.৵ᄉۖഽٓۚ֩ህ॓࿐,ࡾ්৵ᄉۖ 222006;2.ഈݚഽٓն࿐ඔ࿐ჽ,ഈݚ 200234)ᅋ ေ:讨论了一种带有交易成本的保底型基金的定价问题.作为常见的市场利率模型Va-sicek模型的推广,本文假设市场利率服从更为一般的Hul-lWhite模型,在此基础上利用投资组合模拟基金收益和Ito^公式建立数学模型,并利用PDE方法,得到了解析表达式.ܱՍ:交易成本;随机利率;Hul-lWhite模型ᇏٳোݼ:CP31,F224 ໓ངѓ്:A ໓ᅣщݼ:1671-6876(2012)03-0235-060 ႄෛሢದૌࣁವၩ്֥҂؎ᄹ఼,ֆՂ֥ܢௐ൧ӆaᅏಊ൧ӆބၿྛԥྐྵၘ҂ିડቀದૌᄝሧҍٚ૫֥ླ.ᆌؓದૌ҂֥ڄགொݺބሧ߭Б௹ຬ,୍࣍ট,ࣁವࠏܒԛਔޓ؟ડቀ҂ླದಕ֥۲ᇕ҂ো֥ࣁವӁ,ၛડቀದૌሧҍ֥ླေ.ંؓԛ൲ٚটඪߎ൞ؓሧᆀط,ିᆞಒֹܙᆃᇕࣁವӁ֥ൌ࠽ࡎᆴ൞٤ӈᇗေ֥.ЌָࠎࣁႮႿ࠻ॖၛЌᆣሧᆀቋ֮൬ၭ(๙ӈ֮Ⴟ௹ݓᅏ֥൬ၭ),ൈႻॖିࠆ֤ۚح֥ሧ߭Б(ӑح҆ٳο၂קб২֥ٳޣ),ડቀਔ၂҆ٳሧᆀ֥ླေ,෮ၛଢభᄝ൧ӆഈޓ൳ߋ႒.໓[1]ษંਔ০ੱڛՖVasicekଆބ൧ӆޅଉ҈౦ྙ༯֥Ќָࠎࣁ֥ഡ࠹ჰބקࡎٚم.໓[2]ᆌؓགྷൌ൧ӆഈሧᆀࡼ૫ਢඔਈॖܴa҂ಸޭ൪֥ࢌၞӮЧ,۳ԛਔႵࢌၞӮЧ֥ൔ௹ಃקࡎٚم.طЧ໓ଆ֥ંࠎԤ൞Hul-lWhite০ੱ௹ཋࢲܒଆ[3](ቔູVasicekଆ֥ܼ,ఃҕඔनູൈࡗt֥ݦඔ),ᄝࢌၞݖӱᇏᆦڱ၂ק֥ࢌၞٮႨ,ᆃဢࣼᆃᇕҍӁ۷์࣍ૌ֥൧ӆൌ࠽,Ֆط۷ऎൌႨࡎᆴ.1 ႄඔ࿐ଆ ࠎЧࡌഡ1)ሧӁ֥ܒӮ:ૄٺࠎࣁ֥૫ᆴູ1ჭ;2)۴ऌࠎࣁ܄ඳ൞ڎࠆ֤ࣁವࠏܒࣉྛքЌ,ഡ࠹ೂ༯่ॻ:ೂݔႵࣁವࠏܒࣉྛքЌ,ପહa)ࠎࣁ֞௹Ќָ൬ၭູૄٺAჭ,ӑح൬ၭ҆ٳ֥ٳޣб২ູB(0[B[1);b)ࠎࣁಆ҆ሧႿᆷඔࠎࣁ(ೂܢௐ);ೂݔીႵࣁವࠏܒࣉྛքЌ,ପહ,ູಒЌ܄ᇙሧᆀ֥Ќָ൬ၭ,ؿఏದсྶೆ၂҆ٳሱႵሧࣁ,ᄝࠎࣁᇏ෮ᅝб২ູ1-X,ဢܿקഈඍ่ॻ,ᄹࡆֻ่ॻ:c)֒ࠎࣁሧؿളൈ,ؓࠎࣁ࣪ᆴഡᇂ၂۱༯ཋK(t).֒ࠎࣁ࣪ᆴԨࠣK(t)ൈ,ࠎࣁಆ҆ሧႿTൈख़֞௹֥ਬ༏ௐݓᅏ,ՖطಒЌሧᆀTൈख़֥Ќָ൬ၭ.ఃᇏK(t)=XAP(r,t).ᆃ,P(r,t)ູTൈख़֞௹֥ਬ༏ௐݓᅏᄝtൈख़֥ࡎ.۬3)൧ӆ০ੱଆ:ૌҐႨڛՖनᆴ݂֥߭Hul-lWhiteଆ drt=(H(t)-a(t)rt)dt+Rr(t)dW(1)t. ൬۠ರ௹:2012-06-06 ቔᆀࡥࢺ:ᇫݚဉ(1978-),୯,ࡾ්Ӭದ,ࢃഽ,ණൖ,࣮ٚཟູႋႨັٳٚӱ.
236߉ၼഽٓ࿐ჽ࿐Б(ሱಖ॓࿐)ֻ11जఃᇏH(t)൞Ԛ௹ཋࢲܒႵܱ֥ҕඔ,a(t),Rr(t)൞ѯੱ֥ҕඔ,ૌन൞ൈࡗt֥ݦඔ.ھଆ෮૭ඍ֥০ੱᄎྙ൞ಪູ০ੱऎႵ໗ק֥Ӊ௹ᄎ൝H(t),ᄝ҂ൈख़t,rtၛa(t)֥؇ཟ൝ौ࣍,ᆃᇕྟᇉູࠧनᆴ݂߭ྟ.4)ؓႿᆷඔࠎࣁ(ೂܢௐ):ࡌקః࣪ᆴڛՖࠫޅBrownᄎdVt =L(t)dt+RV(t)dW(2)t,VtᆃL(t),RV(t)൞ൈࡗt֥ݦඔ.ѩcov(dW(1)t,dW(2)t)=Qdt,(|Q|<1),E(dW(i)t)=0,i=1,2,var(dW(i)t)=dt,i=1,2,ᄝԚൈख़V0=)ᆦڱࢌၞٮႨ:ࢌၞٮႨॖुቔ൞ሧᆀၹઙછѓ֥ሧӁ(ܢௐ)طӁള֥ᆰࢤٮႨ,၂ϮႮ؟ᆦڱ,ѩ๙ӈၛࢌၞح֥ܥקб২Kটіൕ.ॉ੮ൈ؍[t,t+Dt],Dt൞ཬਈ,҂Ⴟ0,Վൈ০ੱrཌྷؓܥק,ॖुӮӈඔ.ູਔؓԊؿྛࠎࣁ෮ջট֥ڄག,ૄۯDtൈ؍ࣼေטᆜ၂ՑႨটؓԊ֥ѓ֥ሧӁ(ܢௐ)֥ٺحD$(D$>0ઙࣉ,D$<0છԛ),ᄵӁള֥ࢌၞӮЧູK|D$|)ࡌഡग़ӻႵ֥ૄٺЌָࠎࣁ֥ࡎᆴCt൞ܱႿൈࡗa০ੱބᆷඔࠎࣁࡎᆴ֥ݦඔ,ࠧCt=C(Vt,rt,t),ѩ࠺ᄝTൈख़֞௹ರ֥ਬ༏ௐᅏಊࡎູ۬Pt=P(rt,t). ࡹ৫ٚӱᄝ[t,t+Dt]ൈ؍ଽܒᄯሧቆކ0,Ⴎ1ٺࠎࣁ؟Cބ$1ٺᆷඔࠎࣁVࠣ$2ٺਬ༏ௐॢ5CPቆӮ,ࠧ0t=CCt-$1Vt-$2Pt,ఃᇏ$51=,$=5P5rႮࡌഡ่ࡱ,ࢲކIto^܄ൔ֥ྙൔ,Dtൈ؍ުሧቆކ֥ࡎᆴູ: D0t=DCt-$1DVt-$2DPt-KVt|D$1|=5C22215C Dt+R2r(t)+5CV25Ct+2V5t252t5V2R2V(t)5r5VtQRr(t)RV(t)Dt-5C KVt|D$1|-5r5PD152Pt+5PR25r(t)Dt,t252r5rၹЌᆴטᆜҦطӁള֥ࢌၞٺحD$1ູ D$5C25C5C1=5V(Vt+DVt,rt,t+Dt)-5V(Vt,rt,t)URV(t)Vt5Dt2,5Vఃᇏ5~N(0,1).ၹՎࢌၞٮႨ֥ඔ࿐௹ຬູ $E(KVt|D$1|)UK2VtRV(t)D52Ct5V2.ࣉ၂֤҄֞D0֥ඔ࿐௹ຬູ5C E(D0)=D152C52CCt+r(t)+V5t252R2t5V2R2V(t)V252t+25r5VtQRr(t)RV(t)Dt-52C KV2tR1V(t)D2Dtt5V5C5r5PD152P -t+5PR25r(t)Dtt252r5r۴ऌส০ჰ,ՎൈE(D0)=r(t)0Dt=r(t)(Ct-$1Vt-$2Pt)Dt.ᆜ֤5P152P+5t252R2r(t)-r(t)Pr 5P=5r
ֻ3௹ᇫݚဉ֩:၂ᇕջႵࢌၞӮЧ֥Ќָࠎࣁ֥קࡎ23725C5C15C52C52C+r(t)V(t)+V2t+2V5tQR5V2R2V(t)r(t)RV(t)5t+t5V252R2rtr5V -5C5r2K15CV2tRV(t)Dt5V2-r(t)C 5C5tႄೆڄག֥൧ӆࡎ۬K(t),ᄝHul-lWhiteଆ༯Ⴕ5P25P15P 5+(H(t)-a(t)r(t))5+2R2r(t)-r(t)P=0,tr25rC5C5C LC5=+(H(t)-a(t)r(t))+r(t)V5Vt+5t5r152C52C52C 252R2r(t)++2Vt5V2R2V(t)V2t5r5VtQRr(t)RV(t)-152C K2VtRV(t)Dt5V2-r(t)C=ఃᇏHC(t)=H5(t)-K(t)Rr(t).5V2ӫູCЌᆴၹሰ.ؓႿࠎࣁ؟ط,Cᇔູᆞ,طؓॢٚ,Cᆴ֥նཬა؟ٚཌྷ,ٚཟཌྷّ,ࠧCᇔູڵ.ܣھࠎࣁ؟֥קࡎٚӱູ LC=5C+(H(t)-a5C5C(t)r(t))+r(t)V5t5t+r5V22215C5C2 252R2r(t)+V(t)V2t+25CV5V2Rt5r5VtQRr(t)RV(t)-r(t)C=0.ఃᇏR21V(t)=R2V(t)-2KRV(t) קࢳ໙ีೂݔႵၿྛิ܂քЌ,҂թᄝິჿ,Վൈࣁವࠏܒླิ܂ሱႵሧࣁ,ᄝ 2:-]<r<+],0[V[+],0[t[Tଽקࢳ໙ีູLC=0 (1)C(V,r,T)=A+B(V-A+)ೂݔીႵၿྛิ܂քЌ,թᄝິჿॖିྟ,Վൈࣁವࠏܒсྶิ܂ሱႵሧࣁ,۴ऌࠎࣁ่ॻ(3),ᄝ2:-]<r<+],0[V[+],0[t[T,ଽקࢳ໙ีູLC=0 C(V,r,T)=A+B(V-A+)(2)C(V,r,t)|V=XAP(r,t,T)=AP(r,t,T)ఃᇏ֞௹ರູT֥ਬ༏ௐࡎ۬Pડቀ༯ਙшᆴ໙ี(໓ང[4])5P5P152P+(H(t)-a(t)r(t))+5t5r252R2r(t)-r(t)P=0 r(3)P(r,T,T)=1קࢳ໙ีA(t)-B(t)r((3)Ⴕື၂ཁൔࢳPt)(rt,t,T)=e,ఃᇏTa(s)dstBeQT A(t)=QH1(s)B(s)-B2(s)R2r(s)+r(s)ds,(t)=.t2r(t)
238߉ၼഽٓ࿐ჽ࿐Б(ሱಖ॓࿐)ֻ11ज2 ႄקࢳ໙ีࢳၛഈקࢳ໙ี൞ؽົ໙ี,ҐႨሇߐ࠹ࡎֆ໊֥ٚم,ၛ֞௹ರT֥ਬ༏ௐPtቔູ࠹ࡎֆ໊,ॉ੮ཌྷؓႿPt֥ཌྷؓࡎ۬CtPބVttP,Ϝഈඍקࢳ໙ีሇ߄ູ၂ົ໙ีটԩ.t z=VP,WC(V,rt,t)(z,t)=(rt,t,T)P,(rt,t,T)ᄝՎэߐ༯,ࣜ࠹ෘ߄ࡥॖࡼLCэູ5W1 +2zR252W5W(t)+L(5t252t)z-r(t)W=0,z5zఃᇏ R215P215P(t)>P2R25r(t)+R2V(t)-2QRr(t)RV(t)rP,5r15P5r15V15P5r L(t)>-P+,r(t)=Ⴟ൞1)ؓႿႵၿྛࠇఃࣁವࠏܒքЌ౦ঃ,ᄝഈඍэߐ༯,ჰקࢳ໙ี(1)эູ5W1+2252WzR(t)252+L5W(t)z-r(t)W=0,5t r5zW(z,T)=A+B(z-A+)ՎൈໃקಃၭW൞གྷࣁੀAაBٺీקࡎູ۬A֥ुᅨ௹ಃ֥ބ,Ⴟ൞Ⴕ(໓[5]) Wr(S)-L(S))dS(z,t)=A+B-zeQT(tN(d^S1)-A-eQTr(S)dtN(d^2),ఃᇏzR2(S)lnA+QTL(S)+dSt2 d^1=,d^2=d^1-QTQTR2(S)dS,N(x)=-X2et-](S)dStܣ C(Vt,rt,t)=AP(rt,t)+BV-(S)-L(SeQT(r))dStN(d^-r(S)dS1)-AeQTtP(rt,t)N(d^2).2)ؓႿીႵၿྛࠇఃࣁವࠏܒքЌ౦ঃ,ᄝഈඍэߐ༯,קࢳ໙ี(2)эູ5W1252W5+zR25W2+L(t)z5-r(t)W=0t2(t)5zz W+(4)(z,T)=A+B(z-A)W(z,t)|z=XA=Azx=lnXA,W(x,t)=W(z,t),ᄵקࢳ໙ี(4)эູ5W152W1W+R2(t)+L(t)-R25(t)-r(t)W=05t252x25x W+x+(x,T)=A(5)+Bx(XAe-A)=A+ABX(e-k)W(x,t)|x=0=Aఃᇏ1 k=X.ࣉ၂҄ླؓшࢸ่ࡱࣉྛԩ,ູՎU(x,t)=W(x,t)-AAxe,ఃᇏҕඔAડቀٚӱ11 R2(t)A2+L(t)-R2(t)A-r(t)=
ֻ3௹ᇫݚဉ֩:၂ᇕջႵࢌၞӮЧ֥Ќָࠎࣁ֥קࡎ239Ⴟ൞קࢳ໙ี(5)эູ5U152U+R2(t)L125U(t)-R(t)-r(t)U=05t252+x25x U(x,T)=A(6)BXx+Ax(e-k)+A(1-e)U(x,t)|x=0=0ᄜt)+bx U(x,t)=a(T-eU(x,N),ఃᇏ a=-1R2L(t)(t)+1L(t)-1L2(t)-r(t),b=1-,N=822R2(t)2R2(t)QTR2(s)ᄵקࢳ໙ี(6)эູ5U152U5N-252=0x U(7)(0,N)=0Ubxx+bxAx(x,T)=ABX-e(e-k)+A-e(1-e)ູࢳקࢳ໙ี(7),০Ⴈࣤཞم,קၬABX-bxx+Axe(e-k)+A-bxe(1-e),x>0, U(x)=-ABXbx-xe(e-+k)-Abx-Axe(1-e),x<0.ᄵႮPoisson܄ൔॖ֤2U1(x,N)=2NU(s)ds=2PNQ+]-(x-s)e-]ABX+]22-(x-s)x+s)e2N-(-e2N-bssA+]22-(x-s)-(x+s)Ase(e-k)ds+e2N-e2N-bse(1-e)-lnX2PNQ0ܣN(b-12)Nb2N(b-12)Nb2a(T-t)x+ee2 U=ABXNx(2b-1)+(d1)-ke2N(d2)-e2N(d+3)-2xbke2N(d4)+Nb2N(b-A2)Nb2N(b-A2)-t)e2 Aa(TeN(d5)-x+e2N(d6)-2xb+e2N(d7)-x(2b-A)+ke2N(d8)ఃᇏx+lnX-N(b-1)x-lnX+N(b-1) d1=,dN2=d1-N,d3=,dN4=d3-N, d5=x-bN,d6=d5+AN,d7=-x+bN,d8=d7+AN,NNႮՎ֤֞C(Vt,rt,t)=P(rt,t,T)(U(x,t)+AAxe).3 ႄඔᆴٳ༅ई২ᄝHul-lWhiteଆᇏ,ૌᄝt=0ൈख़౼Lr=0106,H=1,Rr=01001,ᄝܢᆷଆᇏ౼RV=014,Q=-0125,ࣜ࠹ෘ,ॖၛٳљ֤֞ᄝ൧ӆႵଉ҈ࠧᆦڱࢌၞٮ౦ྙ༯,ႵքЌބքЌൈЌָٳޣࠎࣁࡎᆴCაٳޣб২BࠣЌָحAᆀᆭࡗ֥ܱ༢.1a2ٳљ൞ഈඍਆᇕ౦ঃ֥ܱ༢.ਸ਼ຓ,οࢌၞح֥115%টᆦڱࢌၞٮႨ,ૌ္ॖၛٳљ֤֞ᄝٳޣб২ބЌָحཌྷ่֥ࡱ༯,Ⴕࢌၞٮބࢌၞٮ౦ঃ༯Ќָࠎࣁ֥ࡎᆴ.২ೂ,ЌָحA=019,ٳޣб২B=015,ᄵཌྷႋֹࢌၞٮႵքЌ౦ྙ֥ࠎࣁࡎᆴC=017315,ႵࢌၞٮႵքЌ౦ྙ֥ࠎࣁࡎᆴC=017318.ࣜݖა൧ӆඔऌ֥бࢠॖၛؿགྷ,ھඔऌა൧ӆൌ࠽ࢠ໖ކ.
240߉ၼഽٓ࿐ჽ࿐Б(ሱಖ॓࿐)ֻ11ज4 ႄࢲંᄝनᆴ݂߭Hul-lWhite০ੱଆࠣܢᆷڛՖࠫޅBrownᄎ,ѩᆦڱ၂קࢌၞٮႨ֥౦ྙ༯,ૌ֤֞ਔაܢᆷܫ܌֥Ќָٳޣࠎࣁ(ႵքЌބքЌਆᇕ౦ঃ)֥ཁൔקࡎ܄ൔ.๙ݖඔᆴٳ༅,ૌࣉ၂҄ؿགྷ,ᄝࡌഡ่ࡱ۷ऎ၂Ϯྟ֥ࡌഡ༯,ࢲݔაൌ࠽ࡼ۷ࡆ໖ކ.1 ࢌၞٮႵքЌ౦ྙ 2 ႵࢌၞٮႵքЌ౦ྙҕॉ໓ང:[1] ࿐ૹ,ഒ.Ќָࠎࣁ֥ഡ࠹აקࡎ[J].༢۽ӱંაൌ,2005,9:2228.[2] HullJC,OptionsFuturesandOtherDerivatives[M].China:TsinghuaUniversityPress,2001.[3] KwokYK,MathematicalModelsofFinancialDerivatives[M].Springer:Singapaore,1998.[4] HullJ,[M].ReviewofFinancialStudies,1990,3:573592.[5] ࡻড়ഉ.௹ಃקࡎ֥ඔ࿐ଆބٚم[M].Кࣘ:ۚ֩࢝ტԛϱഠ,-iyan1,ZHNAGJ-izhou2(,LianyungangTeachersCollege,LianyungangJiangsu222006,China)(,ShanghaiNormalUniversity,Shanghai200234,China)Abstract: ,thepricingmod-elisestablishedbyreplicatingthepayoffundwithportfoliocombinationandusingIto^,: transactioncosts;stochasticinterestrate;hul-lwhitemodel[ᄳщࠠ:Խޣ]
