有交易费和连续红利时的期权定价公式

222Vol.22No.220096JOURNALOFNINGBOUNIVERSITY(NSEE)June2009:1001-5132200902-0230-05,*,315211:,,.2B-S,,,B-S.:Black-Scholes;;;:O175.23:A,ScholeMMertonR1997.Black-Schole(B-S).,.1900“”.,(Kassouf)(Sprekle)(Boness),1964(PaulSamuelson)BachelierL,,.BlackFischerScholesMyronB-S,,.1B-S1[1],,,.2:,,.3:,,.,,.,,,,ΔΔ-.2008-01-11.::.E-mail:[e-mail]*1965,,,/,:.E-mail:[e-mail]2,:2314[2]V,ΔS,Π:,VSΔΠ=−(1),Δ-(Δ-hedging).,Πr,ρ,0TρΠ=Π,1rTρ=+().:V,V=(,)VSt;S();K;T;t;μ;σ;r;dtWBrown.:(1)Brown:d/dd,tttSStWμσ=+(2)d/ttSS;(2)r;(3);(4);(5),,.B-S[3]:222210.2VVVSrSrVtSSσ∂∂∂++−=∂∂∂(3)[5]:()12()()().rTtcStSNdKeNd−−,=−(4):()21()()(),rTtpStKeNdSNd−−,=−−−(5):22()1/2d,wxNxeω−−∞=π∫21(ln(/)(/2)())/()dskrTtTtσσ=++−−,21.ddTtσ=−−2B-S,,2:.,,.,,.B-S,dt,dt→0.,,,.,.:(1)[4]:()(()())()()(),SttqtStttSttδμδσΦδ=−+(6),Φ,212()1/2feΦΦ−=π,()tδ,0.(2)().rrt=(3)(),();qt(4)[5],,M.ωS,Mω||S(0ω,0ω).()VVSt=,,Ito:22221(())2VVVVqSStSSδμσ∂∂∂=+−+⋅∂∂∂,VtStSδσΦδ∂+∂(7)Δ-:,VSΔΠ=−Δ,Π[]tttδ,+.tΠ,[]tttδ,+,Δ,Π,[6]:(1)tttrtδδ+Π=+Π,,ttttrtδδ+Π−Π=Π2322009.rtδδΠ=Π,,.tδqStδ,()VSqStδδΔδδΠ=−+=.VSqStδΔδΔδ−−(8)(6)(7)(8),:22221(()(2VVVqSStSSδμσΔμ∂∂∂Π=+−+−−∂∂∂))().VqSqStSStSΔδσΔσΦδ∂−+−∂rtδδΠ=Π,22221(()2VVVqSStSSμσ∂∂∂+−+−∂∂∂())(VqSqStSSΔμΔδσ∂−−+−∂),StrtΔσΦδδ=Π,Φ0,/VSΔ=∂∂,δΠ:22221().2VVSqSttSδσΔδ∂∂Π=+−∂∂MSω||,:22221().2VVSqStMStSδσΔδω∂∂Π=+−+||∂∂(9)Π[]tttδ,+,.ω:()().VVSSttStSSωδδ∂∂=+,+−,∂∂(10)Taylor,(10)1:()()VVSSttStSSδδ∂∂+,+=,+∂∂222()().VVSSttStSStδδ∂∂,+,+∂∂∂(11)(6),(11)tδ,:2222()().VVSStStStSSωδσΦδ∂∂≈,≈,∂∂()2/,EΦ=πMSω||:2222().VEMSMStSωσδ∂||=||π∂δΠ:222221()(2VVESMStSδσσ∂∂Π=++⋅∂∂222)VVSqttSSδδ∂∂||−=π∂∂().rtrVStδΔδΠ=−tδ,B-S:22222122VVSMStStσσδ∂∂++⋅∂∂π22()0.VVrqSrVSS∂∂||+−−=∂∂(12)3B-S:2222222122()0().tTTVVVSMStStSVrqSrVSVSKσσδ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪+⎪=⎩∂∂∂++||+∂∂π∂∂−−=,∂|=−(13)22/VS∂∂=γγ[7],(),γ,(),γγ,.,22/0VS∂∂,.()tuVeβ=,()tySeα=,()[()()]tVuuyttuettyβαβ−∂∂∂′′=+−,∂∂∂22()()2()()22.ttttVuVueeeeSySyαβαβ−−∂∂∂∂=,=∂∂∂∂(12)()teβ−,:2,:233222212[]2uuMyttyσσδ∂∂+++∂π∂[()](())0.urqtyrtuyαβ∂′′−+−+=∂(14)()()ttαβ,:()()()0()()0()()0,rtqttrttTTαβαβ′−+=,⎧⎪′+=,⎨⎪==⎩:()[()()]d()()d.TTtttrqtrατττβττ=−=∫∫(15)()()ttαβ,(15),(14)20,2222/()LMtσσσδ=+π,(14):222210.2Luuytyσ∂∂+=∂∂(16)2()d,TLtτσωω=∫ln,xy=:22222211[]Luuuuuuuyxyyyxxtστ∂∂∂∂∂∂∂=,=−,=−∂∂∂∂∂∂∂,(16),2Lσ,:221()0.2uuuxxτ∂∂∂−+−=∂∂∂(17),abxumeτ+=,:[][]abxabxumumeamebmxxττττ++∂∂∂∂=+,=+,∂∂∂∂22222[2],abxummebbmxxxτ+∂∂∂=++∂∂∂(17),:2211()22mmmbxxτ∂∂∂−+−+∂∂∂211()0.22abbm+−=1/81/2,ab=−,=(13)Cauchy:220012()(),bxbxbxxmmxmueeyKeeKτττ⎧⎪⎪⎪⎪⎨−−+⎪==⎪⎪−+⎪⎩∂∂=,∂∂|=|=−=−(18)(18)Poisson:2()11222ln1()[]d,2xKmxeeKeξτξξτξτ−+∞−−,=−π∫(19):118212()()xuxmxeIIτττ−+,=,=+.1I,:21122[()]1ln1d.2xKIeτξτξξτ+∞−−−+=π∫(1/2)/,wxξττ=−+1I,1ln222121ln1d(),2xKwxxxKIeeweNττττ−+−−∞−+==π∫(20),()Nx.2I.(/2)/,wxξττ′=−−:1ln2222ln/2d().2xKwKxKIewKNττττ−−′−−∞−−−′==−π∫(21)(20)(21):12ln/2()()xxKuxIIeNτττ−+,=+=−ln/2().xKKNττ−−121ln/2,xKdddτττ−+=,=−12()()()xuxeNdKNdτ,=−.:[()()]d(),ln[()()]d,TtrqxtTteySeSexSrqτττατττ−∫====+−∫:()d()1()()TtqtVStueSeNdττβ−−∫,==−()d2().TtrKNdeττ−∫(22)[8]:()d()d()(),TTttrqcStKepStSeττττ−−∫∫,+=,+:()d()d21()()(),TTttrqpStKeNdSeNdττττ−−∫∫,=−−−12dd,:21((ln(/)[()()()/2]d)/TLtdSKrqττσττ=+−+∫23420092221()d)()d.TTLLttddσττσττ=−∫∫,22Lσσ=+22/()Mtσδπ2222/()LMtσσσδ=−π.4,Black-Scholes,.B-S,,B-S.B-S2,B-S.,.:[1],.Black-Scholes[J].:,2006,23(4):351-353.[2]HullJC.Option,futuresandotherderivatives[M].4thed.NewJersey:Prentice-Hall,2000.[3].[M].:,2003.[4].Black-Scholes[J].,2007,24(2):43-45.[5],.[J].:,2002,4(2):22-25.[6]Miklavž,Mastinšek.Discrete-timedeltahedgingandtheBlack-Scholesmodelwithtransactioncosts[J].MathematicalMethodsofOperationsResearch,2006,64(2):227-236.[7],.[J].:,2000,2(2):110-112.[8]SheldonMRoss.Anelementaryintroductiontomathe-maticalfinance:Optionsandothertopics[M].2nded.England:CambridgeUniversityPress,2003.OptionPricingFormulawithTransactionCostsandContinuousDividendsWUYi-ling,TAOXiang-xing*(FacultyofScience,NingboUniversity,Ningbo315211,China)Abstract:Black-Scholesoptionpricingmodelhasbroughtinnovationtothestudyofoptionsandotherfinancialderivatives’pricing.Butsomebasicassumptionsofthemodelareinconsistentwiththereality,makingoptionpricecalculatedfromthemodeldigresswiththeactualpriceinfinancialmarket.ThispaperimprovesB-Smodelbychangingtwoconditions,thatis,notransactioncostsandnodividendpayments,tomakeitmorecorrelated.Usingthebasicsolutionmethodofpartialdifferentialequations,thepricingformulaofEuropeancall-putoptionsisderivedfromtherevisedB-Smodel.Keywords:Black-Scholesmodel;optionpricing;transactioncosts;dividendCLCnumber:O175.23Documentcode:A