ֻ30जֻ3௹ޙဝഽٓ࿐ჽ࿐Б୍6ᄅ随机市场模型下美式看跌期权的定价易艳春,吴雄韬(衡阳师范学院数学与计算科学系,湖南衡阳 421008)ᅋ ေ:文章讨论银行利率a期望收益率a分红率以及波动率都是随机变量时美式看跌期权的定价问题,利用Fourier变换得出美式看跌期权的价格表达式,并给出有交易成本的美式看跌期权的定价公式bܱՍ:美式看跌期权;交易成本;期权定价ᇏٳোݼ:O211 4໓ངѓ്:A໓ᅣщݼ:1673 0313(2009)03 0007 041 ႄ ௹ಃ൞20ൗࡀ70୍սᇏ௹൮༵ᄝૅݓԛགྷ֥၂ᇕࣁವစള۽ऎ,20؟୍ট,ቔູ၂ᇕٝٓڄགࠇࠏ֥Ⴕི൭؍֤֞ਔ֥ؿᅚ,طೂޅؓ௹ಃקࡎࣼӮਔ௹ಃࢌၞ֥ނྏ໙ีbෛሢ௹ಃקࡎં֥҂؎ؿᅚ,۲ᇕྙൔ֥௹ಃקࡎ໙ีၘФֹܼ࣮ٗbؓႿൔ௹ಃ,ૌၘࣜႵਔ఼Ⴕ৯֥۽ऎ Black Scholes܄ൔ[1],ၘӮູ௹ಃࢌၞ֥ѓሙ۽ऎbطૅൔ௹ಃ,ᄵႮႿఃॖิభᆳྛ֥หᆘ,مᅳ֞ࢳ༅ࢳ,ఃંࢲݔ൳֞ཋᇅ,௧ದૌ҂؎࿙ᅳૅൔ௹ಃ֥ඔᆴࢳم,ѩࠆ֤ਔཌྷ֥֒ࢲݔ[2 9]bᄝၘႵ֥࣮ᇏ,ն؟ඔ൞ࡌഡѯੱa௹ຬ൬ၭੱaٳޣੱູ֩ӈඔ,ೂ໓ང[3 6]ޓݺֹॉ੮ਔѯੱaၿྛ০ੱࠣٳޣੱູӈඔൈؓૅൔ௹ಃקࡎ֥႕ཙbᄝൌ࠽֥ࣁವ൧ӆᇏ,႕ཙ௹ಃקࡎ֥ၿྛ০ੱa௹ຬ൬ၭੱaٳޣੱၛࠣѯੱ൞၂۱ෛࠏэਈ,ၹط࣮ᆃᇕෛࠏࣁವ൧ӆଆ༯֥௹ಃקࡎ۷Ⴕൌ࠽ၩၬb໓ང[7 9]࣮ਔෛࠏѯੱ༯֥ૅൔुᅨ௹ಃ֥קࡎbЧ໓ᄝՎࠎԤഈ,ษંၿྛ০ੱa௹ຬ൬ၭੱaٳޣੱၛࠣѯੱ൞ෛࠏэਈൈ֥ૅൔुי௹ಃ֥קࡎ,০ႨFourierэߐ֤ԛૅൔुי௹ಃ֥ࡎᆴіղൔ,ѩ۳ԛႵࢌၞӮЧ֥ૅൔुי௹ಃ֥קࡎ܄ൔb2 ࣁವ൧ӆଆॉ੮၂۱ࣁವ൧ӆ,ᆺႵਆᇕሧӁ,၂ᇕ൞ڄགሧӁ,ӫູᅏಊ,֥ࡎ۬ݖӱູB=(Bt)0 t T,ડቀ:dBtB=rtdt,BT=1,t![0,T],tਸ਼၂ᇕູڄགሧӁ,ӫູܢௐ,֥ࡎ۬ݖӱູS=(St)0 t T,ડቀ:dSt= tdt+ tdMt,t![0,T],St൬۠ರ௹:2009 03 14ࠎࣁཛଢ:ଲസ࢝ტ๏ሧᇹཛଢ(08C175)ቔᆀࡥࢺ:ၞဇԽ(1975 ),୯,ଲӈୡದ,ޙဝഽٓ࿐ჽඔ࿐ა࠹ෘ॓࿐༢,ࢃഽ,ණൖ,࣮ٚཟ:ෛࠏٳ༅aࣜ࠶ٳ༅აࣁವ.
8ޙဝഽٓ࿐ჽ࿐Б2009୍ֻ30जఃᇏ{Mt,t![0,T]}ູѓሙ֥҃ᄎbrtູڄག০,ੱ tູѯ,ੱ tູ௹ຬ൬ၭੱ,ૌ൞ෛࠏэਈ,ડቀ:ttt∀rtdt<#,tdt<#,∀utdt<#,0∀ 200ູ֤൧ӆ҂թᄝส০ࠏ߶,ܒᄯೂ༯֥ሧቆކ:Vt=mtSt+ntBt,ఃᇏmt,ntٳљіtൈख़ႵڄགሧӁބڄགሧӁ֥ٺඔ,ᄵ֥ັٳٚӱູ:dVt=mtdSt+ntdBt+ tmtStdt,ఃᇏ tູܢௐሧӁ֥ٳޣੱ,ડቀ∀t0rtdt<#bႨC(St,t)іൕ௹ಃ֥ࡎᆴ,ᄵૅൔुי௹ಃႮႿॖၛᄝ௹ಃ֞௹ರTᆭభ֥ޅൈख़ᆳྛ,ᄵсႵC(T,S)=max(K-S,0),0 S<#,0<t<TC(t,0)=K,C(t,#)=0,ఃᇏKູTൈख़௹ಃ֥ᆳྛࡎ۬bᄵૅൔुי௹ಃࡎ۬ડቀ༯ਙொັٳٚӱ: C C1-(rt- 2tSt)+ S2 2C t S2 S2-rtC=0;ડቀшࢸ่ࡱC(T,S)=max(K-S,0),0 S<#,0<t<TC(t,0)=K,C(t,#)=0ഡ!t,t![0,T]սіᄪ௹ᆳྛ౷ཌ,ᄵC(!t,t)=K-!t,CS(!t,t)=-1bགྷࡌഡdV(St∀=T-t,V(St,∀)=C(St,t),౼m,t)t=1,nt=,ᄵऌItoႄॖ֤ૅൔुי௹ಃࡎ۬ડቀ༯ਙdSொັٳٚӱ:2 V V-1 22 VtS S22+rtV=0;(1) ∀-(rt- tSt) Sડቀקࢳ่ࡱV(S,0)=G(S)=max(K-S,0),0 S #,0<t<TV(0,∀)=G(0)=K,V(!∀,∀)=K-!∀,VS(!∀,∀)=-1b3 ૅൔुי௹ಃ֥קࡎႄࣉэਈэߐ:St=!xte,V(St,t)=u(x,t),Ֆط:֤ut=Vt+VS(St),ux=VS(St),uxx=VS(St)+VSSS2t,Ֆطॖ֤!∃tVt=u1t-!ux,VS=ux,VSS=12(uxx-ux),tStStႿ൞ٚӱ(1)эູೂ༯֥ෛࠏٚӱ: u!∃t1 u12-rt- t+ t!- 2t- 2 ut2+rtu=0b(2)t2 x2 xഡxf(x,t)=u(x,t)-g(x,t),ఃᇏg(x,t)=K-!te,Ⴎ(2)֤∃2 f- rt- !t12 ft+-1 2 ftrtf=h(x,t),(3) 2+t!- tt2x2 xఃᇏ
2009୍ֻ3௹ၞဇԽ,ྦᡐ:ෛࠏ൧ӆଆ༯ૅൔुי௹ಃ֥קࡎ 92∃ g1t12h(x,t)=- + 2 gt2+rt- !t+t2 x!- tgx-rtg,t2ഡF(#,t)ູf(x,t)֥Fourierэߐݦඔ,ᄵ(3)ॖэູ F1 +t2 2t#2!∃t+irt- 1t+!- 2t#+rtF(#,t)=H[h(x,t)](4)t2ࢳഈඍ၂ࢨ٤ཌྟັٳٚӱॖၛ֤֞:tF 2t 2t-∀#2(#,t)=exp+irt- t+!∃t02!-#+rtdtF(#,0)t2tt2t+∀H(#,T)exp-∀ #2!∃t+ir∀- 2∀∀+!-#+r∀d∀d∀0∀2t2Ֆط০ႨFourierэߐॖၛ֤t∃t# !t 22tt+!-dt+y-xt2xp∀rt-∀tf(x,t)=1e-rtdth(y,∀2∀)dyd∀t2∃∀0∀10t∀∀ 22tdt∀∀ tdt∀ՖطႮf(x,t)=V(St,t)-max(K-S,0)ॖၛ֤֞ૅൔुי௹ಃ֥ࡎ۬іղൔູttV(St,t)=max(K-S,0)+∀rtKexp∀rsds(%(d1)-%(d2))0∀t 2tKdt1- tStexp-∀ sds(%(d1)-%(d2))+exp-∀t222∃∀∀rtdtdt,∀ 2tdt∀ఃᇏ%(x)іѓሙᆞٳ҃ݦඔ,t!∃t 2tt- Sttd1=∀rt+∀!-dt+lnt2K/∀ 2tdt,∀tttd2=∀rt- t+!∃t- 2Stdt+ln∀!t2!2∀/∀ 2tdt∀4 ႵࢌၞӮЧ֥ૅൔुי௹ಃ֥קࡎഡࣁವ൧ӆ֥ࢌၞӮЧၛࢌၞح֥ܥקб২lটіൕ,ഡܢௐժؿളjٺح֥э߄,ᄵܓઙૅൔुי௹ಃ֥ࡎᆴэ߄ູVljSt,ᄝ&tൈࡗ؍ଽሧቆކ ∋=V-S֥э߄ਈູ S&V∋= tS t+1 2tS2t+ V+ tSt- tSt&t-ljSt, S2 tູЌӻส০ࠏ߶,ҐႨЌᆴҦ,ॖၛԛࣜݖ&tൈࡗ؍ުཌྷႋ֥ࢌၞٺحj= V(S+&tS,t+&t)- V(S,t) S SҐႨTaylorᅚषൔѩޭۚࢨཛॖ֤ 2Vj%St S2 t((&t),ႮՎॖ֤ૅൔुי௹ಃ֥ࡎᆴэ߄֥௹ຬູE(lStj)=∀lS1texp-1(2(t)S 2Vt2∃2 S2 t((&t)dtՖطE 2V(&∋)= V+1 2tS22-2 2V2+ tSt&t t2 S∃&l tS2tt S০Ⴈส০ჰॖၛ֤֞ႵٳޣބࢌၞӮЧ֥ૅൔुי௹ಃડቀ༯ਙٚӱ:2 V-2 V(rt- tS Vt)-! tS2 S2+rtV=0, t S
1 0ޙဝഽٓ࿐ჽ࿐Б2009୍ֻ30जఃᇏ! 122= -2∃&l t,భ૫֥ษંॖၛ֤֞tTTC(St,t)=Stexp-∀ sds%(d3)-Ktexp-∀rsds%(d4),ttఃᇏKtіtൈख़֥ᆳྛࡎ,۬STt! 2TT-td3=lnK+rs- s) T-t,d4=d3-∀ 2sdst∀(ds+!t2/tҕॉ໓ང:[1]BlackF,[J].,1973,81:637 654 [2]DuanJC,[J].JournalofEco nomicDynamicsandControl,2001,25:1689 1718.[3] formsolutionforwithstochasticvolatilitywithapplicationtobondandcurrencyoption[J].ReviewofFinancialStudies,1993(6):327 343.[4]UnderwoodR,WangJun [J].JournalofNonlinearAnalysis:RealWorldApplications,2002,3(2):259 274.[5]ᅦํ.ૅൔ௹ಃקࡎ໙ี֥ඔᆴٚم[J].ႋႨඔ࿐࿐Б,2002,25(1):113 121.[6]ზ৫,ࣁӔᗘ.ૅൔुי௹ಃ֥ҵٳ۬ൔ[J].ᇗ౩ࡹᇽն࿐࿐Б,2004,26(4):100 114.[7]ӧ໓,ૼ.ෛࠏ൧ӆ༯ૅൔुᅨ௹ಃ֥קࡎ[J].ᆢᇜն࿐࿐Б(࿐ϱ),2006,38(3):115 119.[8]נਪ,ဗࡀ.ջෛࠏѯੱ֥Levyଆ༯ૅൔुᅨ௹ಃ֥קࡎ[J].ଲࣘഽն࿐Б(ሱಖ॓࿐ϱ),2008,31(3):84 90.[9]઼ᆞဝ,ਾՑ.၂োෛࠏѯੱ֥ૅൔ௹ಃקࡎ[J].༆ն࿐࿐Б(ሱಖ॓࿐ϱ),2008,31(3):443 chun,WUXiong tao(DepartmentofMathematicsandComputionalScience,HengyangNormalUniversity,HengyangHunan421008,China)Abstract:ThispaperfocusesonthepricingofAmericanputoptioncontributingtobankinterestrate,returnrate,,:Americanputoption;transactioncosts;optionpricing
