Vasiek利率模型下的亚式期权的定价问题和数值分析

263Vol.26No.3 20037ACTA MATHEMATICAE APPLICATAE SINICA July,2003 Vasiˇc ek∗(200092)(230026)Vasiˇc ekCauchyCauchy1(Call/Put Option)(Exotic Option).Black-ScholesVasiˇc ek T,[0,T]2001107∗(10201029)46826Monte Carlo[1,2],[3–5].Turnbull &Wakeman (1991)Levy (1992).LaplaceTaylor([6–9]),[3,10,11].Cauchy[12].1CauchyCauchy2T ,[0,T ]T 0(Zero-Coupon).(Ω,F,P )rSd r t =(β−αr t )d t +γd Z t ,d S t =S t (r t dt +σB t ).(2.1)(Z t ,B t )(Ω,F,P )2(F t )t ≥0σ-α,β,λ=0σ=0T ,1TTS (τ)d τTξ=S T −1TTS (τ)d τ+.(2.2)C (t )C (t )=E p ξexp−Ttr s d s F t .(2.3)I t =tS (τ)d τ,(t,r t ,S t ,I t )MarkovianC (t )(t,r,S,I )C (t,r,S,I ).Feymann-kac3Vasiˇc ek469 PDE Cauchy⎧⎪⎨⎪⎩∂V∂t+12σ2S2∂2V∂S2+12γ2∂2V∂r2+rS∂V∂S+(β−αr)∂V∂r+S∂V∂I−rV=0,V(T,S,r,I)=S−IT+,(2.4)0≤t<T,−∞<r<+∞,0≤S,I<+∞.(2.4),x=IT S ,V(t,S,r,I)=Sf(t,x,r),⎧⎨⎩∂f∂t+1T−rx∂f∂x+12σ2x2∂2f∂x2+(β−αr)∂f∂r+12γ2∂2f∂r2=0,f(T,x,r)=(1−x)+.(2.5)t→T∂2f(x,t,r)∂x2→δ(1−x),δ(ξ)0DiracT(2.5)f=f1+f2,f1f2PDE:⎧⎨⎩∂f1∂t+rx∂f1∂x+12σ2x2∂2f1∂x2+(β−αr)∂f1∂r+12γ2∂2f1∂r2−rf1=0,f1(T,x,r)=(1−x)+(2.7)⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩∂f2∂t+1T−rx∂f2∂x+12σ2x2∂2f2∂x2+(β−αr)∂f2∂r+12γ2∂2f2∂r21 =−1T−2rx∂f1∂x−rf,f2(T,x,r)=0,(2.7)0≤t<T,−∞<r<+∞,0≤x<+∞.f1Vasiˇc k1Call-PutC(t,r t,S t,K)−P(t,r t,S t,K)=S t−KP(t,T),(2.8) P(t,T)T0t Vasiˇc k([13]):C(t,r t,S t,K)=S t N(d1)−Ke−C1(t,T,r t)+σ2X2N(d2).(2.9)d1=log S tK+12σ2Y+C(t,T,r t)σ2X+σ2Y,d2=log S tK−σ2X+12σ2Y+C(t,T,r t)σ2X+σ2Y,σ2X=1αTtγ2e2α(T−u)d u,σ2Y=σ2(T−t),C1(t,T,r t)=γα(eα(T−t)−1)−γα2(T−t+1)+γα2eα(T−t).47026(2.6)f1(t,x,r)=xN(d1)−e−C1(t,T,r)+σ2X2N(d2)+e A(t,T)−B(t,T)r−x,(2.10)B(t,T)=1α[1−e−α(T−t)],A(t,T)= Tt12γ2B(s,T)−βB(s,T)d s.∂f1∂x=N(d1)−1,τ=T−t,(2.7)⎧⎪⎪⎪⎨⎪⎪⎪⎩∂f2∂τ−1T−rx∂f2∂x−12σ2x2∂2f2∂x2+(β−αr)∂f2∂r−12γ2∂2f2∂r2 =1T−2rxN(d1)−1+rf1=F(τ,x,r),f2(0,x,r)=0.(2.11)Cauchyx∈[0,X],r∈[−R,R],τ∈[0,T],∆x=XN,∆r=2RM,∆τ=TK.∂f∂τki,j=f k+1i,j−f k i,j∆τ+O(∆τ),∂f∂xki,j=f k i+1,j−f k i−1,j2∆x+O(∆x2),∂f∂rki,j=f k i,j+1−f k i,j−12∆r+O(∆r2),∂2f∂xki,j=f k i+1,j+f k i−1,j−2f k i,j∆x+O(∆x2),∂2f∂r2ki,j=f k i,j+1+f k i,j−1−2f k i,j∆r2+O(∆r2),(2.11),f k+1 i,j =1−σ2x2∆τ∆x2−γ2∆τ∆r2f k i,j+∆τ2∆xσ2x2i∆x+1−r j x if k i+1,j+∆τ2∆rγ2∆r+β−αr jf k i,j+1+∆τ2∆xσ2x2i∆x−1+r j x if k i−1,j+∆τ2∆rγ2∆r+αr j−βf k i,j−1+∆τF k i,j,(2.12)1≤i≤N−1,−M+1≤j≤M−1,1≤k≤K−1.∆r=∆x,∆x≤min{σ2x2i|1−r j x i|,γ2β−αr j},∆τ∆x2≤1γ2+σ2x2i3Vasiˇc ek471(2.12)O (∆τ+∆x 2),f k N,j ,f k 0,j ,f k i,M ,f ki,−M ,0≤i ≤N ,−M ≤j ≤M ,1≤k ≤K∀0<p <1,f k N,j =pf k −1N,j +(1−p )f kN −1,j ;(2.13)f k 0,j =pf k −10,j +(1−p )f k 1,j ;(2.14)f k i,M =pf k −1i,M +(1−p )f k i,M −1;(2.15)f k i,−M =pf k −1i,−M +(1−p )f k i,−M +1;(2.16)3f1S =100,I =1t t0S (τ)d τ=1003,T =3t =0.03,σ=30%,α=1,β=0.05γ=0.1γ=0.05r =Vasick Constant Vasick 0.115.847818.673315.63400.318.291134.416118.17360.520.814347.859820.74790.723.344257.774423.3150S =50,I =1t t0S (τ)d τ=1003,T =3t =0.03,σ=30%,α=1,β=0.05γ=0.1γ=0.05r =Vasick Constant Vasick 0.18.00239.39147.89230.39.249317.33959.18640.510.541324.203010.50410.711.840029.258711.82202β=0.1S =100,I =1t t0S (τ)d τ=1003,T =3t =0.03,σ=30%,α=1,β=0.1γ=0.1γ=0.05r =Vasick Constant Vasick 0.118.794018.673318.69760.321.396234.416121.34730.521.396247.859823.9827S =50,I =1t t0S (τ)d τ=1003,T =3t =0.03,σ=30%,α=1,β=0.1γ=0.1γ=0.05r =Vasick Constant Vasick 0.19.45869.39149.40540.310.793417.339510.76440.512.132524.203012.1204γ47226γβ<r,β=rβ1,23T=3,t=0.03,σ=30%,r=10%,α=1,β=0.1x1T=3,t=0.03,σ=30%,x=1,α=1,β=0.05r23Vasiˇc ek473 T=3,t=0.03,σ=30%,r=10%,α=1,β=0.05x34VasicekCauchy1Kemna A G Z,Vorst A C F.A Pricing Method for Options Based on Average Asset Values.Journal of Banking and Finance,1990,14:113–1292Carverhill A,Clewlow L.Flexible Convolution.RISK,1990,5:25–293Rogers L,Shi Z.The Value of an Asian Option.Journal of Applied Probability,1995,32:1077–1088 4Alziary B,Decamps J,Koehl P,A P.D.E.Approach to Asian Option:Analytical and Numerical Evidence.Journal of Banking and 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Av-erage Rate Options.Journal of Computational Finance,2001,5(1):1–2013Wang L J,Zhang S.Princing the Reset Option under Stochastic Interest Rate.Applied Mathematics,A Journal of Chinese Universities,2002,17:471–478PRICING THE ASIAN OPTION UNDERV ASIˇCEK INTEREST RATEWANG Lijun(Department of Applied Mathematics,Tong-Ji University,Shanghai200093)ZHANG Shuguang(Department of Statistics and Finance of University of Science&Technology of China,Hefei230026) Abstract This paper presents a theory of continuous sampled Asian option pricing when the interest rate is modeled by Vasiˇc ek model.For arithmetic Asian option,we subtract an explicit formula from the solution of the price and get a PDE satisfied by the residue with smooth coefficients and0initial condition.We adopt infinite difference scheme to calculate the solution numerically.Key words Asian option,floating strike price,infinite difference scheme。