若干期权定价模型研究
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摘要
期权定价问题是金融数学中一个非常重要的问题.期权的价格,即期权金,指的是期权买卖双方在达成期权交易时,由买方向卖方支付的购买该项期权的金额.期权的定价决定于标的资产价格的变化.由于标的资产是一种风险资产,因而它的价格变化是随机的.进而由此产生的期权价格也是随机的.在数学上,期权的价格表现为一个倒向抛物形偏微分方程解.
1973年,Fischer Black和Myron Scholes在假设标的资产的价格服从几何布朗运动时,建立了经典的Black-Scholes模型,得到了欧式期权所满足的偏微分方程,即Black-Scholes方程(简称B-S方程).并给出了欧式看涨期权定价公式,即著名的Black-Scholes公式.随后,Cox等又提出了期权定价的离散时间模型,即期权定价的二叉树方法.这两种模型仍旧是现在世界上常用的金融衍生产品定价模型.
然而,近数十年来,大量实证研究表明,标的资产的价格并不是一定遵循几何布朗运动.因此,研究者们把研究期权定价的重点放在改进Black-Scholes模型上.例如：分数Black-Scholes模型,次扩散(时变)Black-Scholes模型.
本文致力于研究在不同时变模型下欧式期权的定价问题.具体地说,我们主要研究了如下四种模型：
在第三章里,我们提出并研究了时变混合布朗-分数布朗模型.该模型假设标的股票的价格St满足这里,S0,μ,σ都是常数,Tα(t)为逆α-稳定从属过程,其中α∈(2/3,1)；MαH(t)aB(Tα(t))+BBH(Tα(t))为时变过程,H∈(1/2,1).并且假定B(τ),BH(τ)和Tα(t)都是相互独立的.
我们首先得到了在离散时间下欧式看涨期权所满足的偏微分方程,并给出了欧式看涨期权的定价公式及欧式看涨-看跌期权平价公式.具体定理如下：
定理0.0.1.当股票价格St满足方程(3.2)时,欧式看涨期权的价值V=V(t,St)满足边界条件V(T,ST)=max{ST-K,0｝
期权价格这里其中
定理0.0.2.欧式看涨期权-看跌期权平价公式
设V(t,S)和P(t,S)分别为到期日为T,敲定价格为K,无风险利率为r的具有相同标的股票欧式看涨和看跌期权,那么当标的股票的价格S满足(3.4)式时,看涨看跌期权的平价式为
其次,我们还给出了相应的数值计算结果.最后,我们讨论了隐含波动率问题,并给出了隐含波动率的偏斜曲线.
在第四章里,我们研究了时变Merton期权定价模型,该模型假设标的股票的价格St满足S0,μ和σ都是常数,B(Tα(t))为一时变过程,B(τ)是标准的布朗运动.α∈(2/3,1),Tα(t)是逆α-稳定从属过程.Nt是跳跃强度为λ>0的泊松过程,J是取正值的随机变量.这里我们假设Tα(t),B(T),Nt和J都是相互独立的.
我们得到了在该模型下欧式看涨期权所应满足的偏微分方程,并给出了期权定价公式.具体定理如下
定理0.0.3.期权价格V=V(t,St)满足下面的偏微分方程边界条件为进一步,期权的价格为here
在第五章里,我们提出了时变CEV期权定价模型.该模型假设标的股票的价格Zt满足这里,μ,σ,β,Z0都是常数,dbH(T)=w(T)(dT)H为修正的分数布朗运动.H∈[1/2,1),w(T)是均值为0方差为1的高斯白噪声.特别地,这里假设Sα(t)和bH(t)是相互独立的.
在一定假设条件下,我们首先得到了欧式看涨期权所满足的偏微分方程并给出了定价公式.其次,我们又给出了期权价格的一种渐进展开式.具体结论如下：
定理0.0.4.假设欧式看涨期权的价值C(t,Zt)属于空间C1,2([0,T)×[0,+∞)),则C(t,Zt)满足下面的偏微分方程边界条件为这里r为无风险利率.
定理0.0.5.假设αH>1/2,则问题(2.12)-(2.2)的解,即欧式看涨期权的价值C(t,Z(t))可由下式给出.这里Γ(ξ)为伽玛函数.
定理0.0.6.假设欧式看涨期权的价格C(t,z)有如下的渐进表达式则C0(t,Z)伴随边界条件C0(T,Z)=(Z-K)+可由下式给出这里且每一个Cn(t,Z),n=1,2,...,伴随终止条件Cn(T,Z)=0可由下式迭代给出
在第六章里,我们研究了基于时变过程的随机Merton利率模型下欧式期权的定价问题.首先我们得到了零息票债券的定价公式,然后再此基础上得到了欧式期权的定价公式以及看涨-看跌期权平价公式.具体结论如下
定理0.0.7.在时变Merton随机利率模型下,欧式看涨期权V(S,r,t)满足下面的偏微分方程并且满足边值条件其中进一步,我们可得欧式看涨期权的价值V(S,r,t)公式为这里
欧式看涨-看跌期权的平价公式为
定理0.0.8.设c(S,r,l),p(S,r,l)分别为具有相同到期日T,相同敲定价格K的欧式看涨期权和看跌期权的价格,P(S,r,l)为零息票债券的定价,则我们有欧式看涨与看跌期权的平价公式为
Option pricing is a very important problem in financial mathematics. The price ofoption, i.e. premium, is the amount of money the buyer paid to buy the option in aoption trading. The price of option depends on the changes of underlying asset price.Due to the underlying asset is a risk asset, therefore, its price changes are random. Thus,the price of the option is random. In mathematics, the price of option is the solution ofa backward parabolic partial diferential equation.
1973, Fischer Black and Myron Scholes established the classical Black-Scholes modelbased on the assumption that the price of the underlying asset follows the geometricBrownian motion. They obtained the Black-Scholes (B-S in short) equation, and gavethe pricing formula for the European call option, i.e. the classical Black-Scholes formula.Then, Cox et al. proposed a discrete time option pricing model, i.e. binomial treemethods. The two models are still popular in the world now.
However, in the recent decades, a large number of empirical studies show that theprice of the underlying asset is not necessarily follow the geometric Brownian motion. So, researchers focus their research on improving the Black-Scholes model. For example, fractional Black-Scholes model, subdiffisive (time changed) Black-Scholes model.
This thesis is devoted to studies the option pricing problem under different time-changed Black-Scholes model. To be specific, we consider the following four models.
In Chapter3, we proposed a time changed Brownian-fractional Brownian Black-Scholes model. In this model, we assume that the price of the underlying stock satisfies here S0,μ,σ are constants, Tα(t) is α-stable subordinator, α∈(2/3,1); Mα,H(t) αB(Tα(t))+bBH(Tα(t)), H∈(1/2,1). We assume that B(τ),BH(τ)和Tα(t) are all inde-pendent.
We first obtained the partial differential equation which the European call option satisfied in a discrete time setting. Then we obtained the pricing formula for European option and call-put parity equation. The theorems are:
Theorem0.4.1. When the underlying stock price satisfies(3.4), then the value of European call option V=V(t, St) satisfies where here
Theorem0.4.2. call-put parity
Let V(t, S) and P(t, S) are European call and put option respectively, and have the same expiry date T, strike price K, risk-free rate r. Then when the price of the underlying stock satisfies (3.4), we have
We also gave the corresponding numerical results. In the last, we discussed the implied volatility, and obtained the volatility skrew.
In Chapter4, we discussed the time changed Merton model with transaction costs. The model assumed that the price of the underlying stock St satisfies S0,μ and a are constants. B(Tα(t)) is a subdiffusive process, B(τ) is the standard Brownian motion. α∈(2/3,1), Tα(t) is the inverse α-table subordinator. Nt is a Poisson process with intensity λ>0, and J is a positive random variable. Assume that Ta(t), B(τ), Nt and J are independent.
We obtained the partial differential equation which the European call option satisfied, and gave the option pricing formula. The theorem is:
Theorem0.4.3. The value of the European call option V=V(t, St) satisfies with boundary condition Furthermore, Vt, St is given by here
In Chapter5, we proposed a time-changed CEV model. This model assumes that the underlying stock priceZt satisfies where μ,σ,β, Zo are all constants. dbH(τ)=ω(τ)(dτ)H is a modified fractional Brownian motion, H∈[1/2,1),ω(τ) is the companion Gaussian white noise with zero mean and unit variance. In particular, it is assumed that the Sa(t) is independent of bH(t).
Under some assumptions, we obtained the partial differential equation which the European call option satisfied, and gave the pricing formula. Then, we get an asymptotic representation of the European call option price. The theorems are:
Theorem0.4.4. Assume that the price of option C(t,Zt) belongs to C1,2([0,T)× [0,+∞)), then C(t, Zt) satisfies the following partial differential equation with boundary condition
Theorem0.4.5. Suppose that αH>1/2, then the solution of problem (2.12)-(2.2),i.e. the value of a European call option C(t, Z(t)) is given by where Here Γ(ε) is the Gamma function.
Theorem0.4.6. Suppose the European call option price C(t, Z) which is the solu-tion of (2.1)-(2.2) has an asymptotic expansion price such as
Then Co(t, Z) with the final condition C0(T, Z)=[Z-K)+is given by where and each Cn(t, Z), n=1,2,..., with the final condition Cn(T, Z)=0is recursively given by In Chapter6, we discussed the European option pricing under the Merton model of the short rate which based on the time-changed process. We first obtained the pricing formula for a zero-coupon bond, then based on it the pricing formula for European options and the call-put parity were obtained. The results are: Theorem0.4.7. Under the time-changed Merton model of the short rate, the price of the European call option V(S,r,t) satisfies the following partial differential equation and the boundary condition is here Furthermore, the pricing formula for V(S,r,t) is where
The call-put parity is Theorem0.4.8. Let c(S,r,t), p(S,r,t) are the price of the European call and put option with the same strike price K and expiry date T. P(S, r, t) is the price of a zero-coupon bond. The the call-put parity is
1973年,Fischer Black和Myron Scholes在假设标的资产的价格服从几何布朗运动时,建立了经典的Black-Scholes模型,得到了欧式期权所满足的偏微分方程,即Black-Scholes方程(简称B-S方程).并给出了欧式看涨期权定价公式,即著名的Black-Scholes公式.随后,Cox等又提出了期权定价的离散时间模型,即期权定价的二叉树方法.这两种模型仍旧是现在世界上常用的金融衍生产品定价模型.
然而,近数十年来,大量实证研究表明,标的资产的价格并不是一定遵循几何布朗运动.因此,研究者们把研究期权定价的重点放在改进Black-Scholes模型上.例如：分数Black-Scholes模型,次扩散(时变)Black-Scholes模型.
本文致力于研究在不同时变模型下欧式期权的定价问题.具体地说,我们主要研究了如下四种模型：
在第三章里,我们提出并研究了时变混合布朗-分数布朗模型.该模型假设标的股票的价格St满足这里,S0,μ,σ都是常数,Tα(t)为逆α-稳定从属过程,其中α∈(2/3,1)；MαH(t)aB(Tα(t))+BBH(Tα(t))为时变过程,H∈(1/2,1).并且假定B(τ),BH(τ)和Tα(t)都是相互独立的.
我们首先得到了在离散时间下欧式看涨期权所满足的偏微分方程,并给出了欧式看涨期权的定价公式及欧式看涨-看跌期权平价公式.具体定理如下：
定理0.0.1.当股票价格St满足方程(3.2)时,欧式看涨期权的价值V=V(t,St)满足边界条件V(T,ST)=max{ST-K,0｝
期权价格这里其中
定理0.0.2.欧式看涨期权-看跌期权平价公式
设V(t,S)和P(t,S)分别为到期日为T,敲定价格为K,无风险利率为r的具有相同标的股票欧式看涨和看跌期权,那么当标的股票的价格S满足(3.4)式时,看涨看跌期权的平价式为
其次,我们还给出了相应的数值计算结果.最后,我们讨论了隐含波动率问题,并给出了隐含波动率的偏斜曲线.
在第四章里,我们研究了时变Merton期权定价模型,该模型假设标的股票的价格St满足S0,μ和σ都是常数,B(Tα(t))为一时变过程,B(τ)是标准的布朗运动.α∈(2/3,1),Tα(t)是逆α-稳定从属过程.Nt是跳跃强度为λ>0的泊松过程,J是取正值的随机变量.这里我们假设Tα(t),B(T),Nt和J都是相互独立的.
我们得到了在该模型下欧式看涨期权所应满足的偏微分方程,并给出了期权定价公式.具体定理如下
定理0.0.3.期权价格V=V(t,St)满足下面的偏微分方程边界条件为进一步,期权的价格为here
在第五章里,我们提出了时变CEV期权定价模型.该模型假设标的股票的价格Zt满足这里,μ,σ,β,Z0都是常数,dbH(T)=w(T)(dT)H为修正的分数布朗运动.H∈[1/2,1),w(T)是均值为0方差为1的高斯白噪声.特别地,这里假设Sα(t)和bH(t)是相互独立的.
在一定假设条件下,我们首先得到了欧式看涨期权所满足的偏微分方程并给出了定价公式.其次,我们又给出了期权价格的一种渐进展开式.具体结论如下：
定理0.0.4.假设欧式看涨期权的价值C(t,Zt)属于空间C1,2([0,T)×[0,+∞)),则C(t,Zt)满足下面的偏微分方程边界条件为这里r为无风险利率.
定理0.0.5.假设αH>1/2,则问题(2.12)-(2.2)的解,即欧式看涨期权的价值C(t,Z(t))可由下式给出.这里Γ(ξ)为伽玛函数.
定理0.0.6.假设欧式看涨期权的价格C(t,z)有如下的渐进表达式则C0(t,Z)伴随边界条件C0(T,Z)=(Z-K)+可由下式给出这里且每一个Cn(t,Z),n=1,2,...,伴随终止条件Cn(T,Z)=0可由下式迭代给出
在第六章里,我们研究了基于时变过程的随机Merton利率模型下欧式期权的定价问题.首先我们得到了零息票债券的定价公式,然后再此基础上得到了欧式期权的定价公式以及看涨-看跌期权平价公式.具体结论如下
定理0.0.7.在时变Merton随机利率模型下,欧式看涨期权V(S,r,t)满足下面的偏微分方程并且满足边值条件其中进一步,我们可得欧式看涨期权的价值V(S,r,t)公式为这里
欧式看涨-看跌期权的平价公式为
定理0.0.8.设c(S,r,l),p(S,r,l)分别为具有相同到期日T,相同敲定价格K的欧式看涨期权和看跌期权的价格,P(S,r,l)为零息票债券的定价,则我们有欧式看涨与看跌期权的平价公式为
Option pricing is a very important problem in financial mathematics. The price ofoption, i.e. premium, is the amount of money the buyer paid to buy the option in aoption trading. The price of option depends on the changes of underlying asset price.Due to the underlying asset is a risk asset, therefore, its price changes are random. Thus,the price of the option is random. In mathematics, the price of option is the solution ofa backward parabolic partial diferential equation.
1973, Fischer Black and Myron Scholes established the classical Black-Scholes modelbased on the assumption that the price of the underlying asset follows the geometricBrownian motion. They obtained the Black-Scholes (B-S in short) equation, and gavethe pricing formula for the European call option, i.e. the classical Black-Scholes formula.Then, Cox et al. proposed a discrete time option pricing model, i.e. binomial treemethods. The two models are still popular in the world now.
However, in the recent decades, a large number of empirical studies show that theprice of the underlying asset is not necessarily follow the geometric Brownian motion. So, researchers focus their research on improving the Black-Scholes model. For example, fractional Black-Scholes model, subdiffisive (time changed) Black-Scholes model.
This thesis is devoted to studies the option pricing problem under different time-changed Black-Scholes model. To be specific, we consider the following four models.
In Chapter3, we proposed a time changed Brownian-fractional Brownian Black-Scholes model. In this model, we assume that the price of the underlying stock satisfies here S0,μ,σ are constants, Tα(t) is α-stable subordinator, α∈(2/3,1); Mα,H(t) αB(Tα(t))+bBH(Tα(t)), H∈(1/2,1). We assume that B(τ),BH(τ)和Tα(t) are all inde-pendent.
We first obtained the partial differential equation which the European call option satisfied in a discrete time setting. Then we obtained the pricing formula for European option and call-put parity equation. The theorems are:
Theorem0.4.1. When the underlying stock price satisfies(3.4), then the value of European call option V=V(t, St) satisfies where here
Theorem0.4.2. call-put parity
Let V(t, S) and P(t, S) are European call and put option respectively, and have the same expiry date T, strike price K, risk-free rate r. Then when the price of the underlying stock satisfies (3.4), we have
We also gave the corresponding numerical results. In the last, we discussed the implied volatility, and obtained the volatility skrew.
In Chapter4, we discussed the time changed Merton model with transaction costs. The model assumed that the price of the underlying stock St satisfies S0,μ and a are constants. B(Tα(t)) is a subdiffusive process, B(τ) is the standard Brownian motion. α∈(2/3,1), Tα(t) is the inverse α-table subordinator. Nt is a Poisson process with intensity λ>0, and J is a positive random variable. Assume that Ta(t), B(τ), Nt and J are independent.
We obtained the partial differential equation which the European call option satisfied, and gave the option pricing formula. The theorem is:
Theorem0.4.3. The value of the European call option V=V(t, St) satisfies with boundary condition Furthermore, Vt, St is given by here
In Chapter5, we proposed a time-changed CEV model. This model assumes that the underlying stock priceZt satisfies where μ,σ,β, Zo are all constants. dbH(τ)=ω(τ)(dτ)H is a modified fractional Brownian motion, H∈[1/2,1),ω(τ) is the companion Gaussian white noise with zero mean and unit variance. In particular, it is assumed that the Sa(t) is independent of bH(t).
Under some assumptions, we obtained the partial differential equation which the European call option satisfied, and gave the pricing formula. Then, we get an asymptotic representation of the European call option price. The theorems are:
Theorem0.4.4. Assume that the price of option C(t,Zt) belongs to C1,2([0,T)× [0,+∞)), then C(t, Zt) satisfies the following partial differential equation with boundary condition
Theorem0.4.5. Suppose that αH>1/2, then the solution of problem (2.12)-(2.2),i.e. the value of a European call option C(t, Z(t)) is given by where Here Γ(ε) is the Gamma function.
Theorem0.4.6. Suppose the European call option price C(t, Z) which is the solu-tion of (2.1)-(2.2) has an asymptotic expansion price such as
Then Co(t, Z) with the final condition C0(T, Z)=[Z-K)+is given by where and each Cn(t, Z), n=1,2,..., with the final condition Cn(T, Z)=0is recursively given by In Chapter6, we discussed the European option pricing under the Merton model of the short rate which based on the time-changed process. We first obtained the pricing formula for a zero-coupon bond, then based on it the pricing formula for European options and the call-put parity were obtained. The results are: Theorem0.4.7. Under the time-changed Merton model of the short rate, the price of the European call option V(S,r,t) satisfies the following partial differential equation and the boundary condition is here Furthermore, the pricing formula for V(S,r,t) is where
The call-put parity is Theorem0.4.8. Let c(S,r,t), p(S,r,t) are the price of the European call and put option with the same strike price K and expiry date T. P(S, r, t) is the price of a zero-coupon bond. The the call-put parity is
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[21] Lo A.W. Long term memory in stock market Prices [J]. Econometria,1991,59:1279-1313.
[22] Daiand Q., Singleton K.J. Specification analysis of afne term strcture model [J].Journal of Finance,2000,55:1943-1978.
[23] Lin S.J. Stochastic analysis of fractional Brownian motion, fractional noises andapplication [J]. SIAM Review,1995,10:422-437.
[24] Rogers L.C.G. Arbitrage with fractional Brownian motion [J]. Mathematical Fi-nance,1997,7:95-105.
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[29] Hu Y., ksenda. Fractional white noise caleclus and application to finance [J]. JInf Dim Anal. Quantum Probab Rel Top,2003,6:1-32.
[30] Necula C. Option pricing in a fractional brownian motion environment [J]. Advancesin Economic and Financial Research-DOFIN Working Paper Series2002,2.
[31] Bj¨ork T., Hult H. A note on Wick Products and the fractional Black-Scholes model[J]. Finance Stochastic,2005,9:197-209.
[32] Biagini F., ksendal B. Minimal variance hedging for fractional Brownian motion[J]. Methods and Applications Analysis,2003,10:347-362.
[33] Biagini F., ksendal B. Forward integrals and an Ito formula for fractional Brownianmotion [J]. Dept. of Math, University of Oslo,2004,22.
[34] Elliott R.J., van der Hoek J. A general fractional white noise theory and applicationsto finance [J]. Mathematical Finance,2003,13:301-330.
[35] Hu Y., ksenda, Sulem A. Optional consumption and portfolio in Black-Scholesmarket driven by fractional Brownian motion [J]. University of oslo, PrePrint,2000,23.
[36] Hu Y., ksenda, Sulem A. Optimal portfolio in a fractional Black-Scholes market[J]. In Mathematical physics and Stochastic Analysis: Essays in Honor of Ludwigstreit, World Scientific,2000,267-279.
[37] Janczura J., Wyloman′ska A. Subdynamics of financial data from fractional Fokker-Planck equation [J]. Acta Physica Polonica B,2009,40(5):13411351.
[38] Janczura J., Orzel S., Wyloman′ska A. Subordinated Ornstein-Uhlenbeck processas a tool of financial data description [J]. Preprint2011.
[39] Orzel S., Wyloman′ska A. Calibration of the subdifusive arithmetic Brownian mo-tion with tempered stable waiting-times [J]. Journal of Statistical Physics,2011,143(3):447454.
[40] Magdziarz M. Black-Scholes formula in subdifusive regime [J]. Journal of StatisticalPhysics,2009,136:553-564.
[41] Magdziarz M., Orzel S., Weron A. Option pricing in subdifusive Bachelier model[J]. Journal of Statistical Physics,2011,145:187-203.
[42] Gu H., Liang J.R., Zhang Y.X. Time-changed geometric fractional Brownian motionand option pricing with transaction costs [J]. Physica A,2012,391:3971-3977.
[43] Rostek S. Option pricing in fractional Brownian markets [M]. Springer-Verlag,2009.
[44] Mandelbrot B.B. Fractals and Scaling in Finance:Discontinuity, Concentration,Risk [M]. Springer-Verlag, New York,1997.
[45] Fraktale W.B. Long-memory und Aktienkurse-eine statistischen Analyse fur dendeutschen Aktienmarkt [M]. Eul-Verlag. Bergisch-Gladbach, Koln,1996.
[46] Sato K.I. Levy Processes And Infinitely Divisible Distributions [M]. English Edition,Cambridge University Press, Cambridge,1999.
[47] Bertoin J. Le′vy Processes [M]. Cambridge University Press, Cambridge,1996.
[48] Magdziarz M. Path properties of subdifusion-a martingale approach [J]. StochasticModels,2010,26:256-271.
[49] Lageras A.N. A renewal-process-type expression for the moments of inverse subor-dinators [J]. Journal of Applied Probability,2005,42:11341144.
[50] Zili M. On the mixed fractional Brownian motion [J]. J. Appl. Math. Stoch. Anal.,2006,1-9.
[51] Wang X.T., Zhu E.H., Tang M.M., et al. Scaling and long-range dependence inoption pricing II: pricing European option with transaction costs under the mixedfractional Brownian model [J]. Physica A2010,389:445-451.
[52] Adams R.A. Sobolev space [M]. Academic Press, New York,1975.
[53] Liang J.R., Wang J., Lv L.J., et al. Fractional Fokker-Planck Equation and Black-Scholes Formula in Composite-Difusive Regime [J]. Journal of Statistical Physics,2012,146:205216.
[54] Park S.H., J.H. Kim. Asymptotic option pricing under the CEV difusion [J]. Jour-nal of Mathrmatical Analysis and Applications,2011,375:490-501.
[55] Jarrow R.J., Rosenfeld E. Jump risks and intemporal capital asset pricing model[J]. Journal of Business,1988,57:337-351.
[56] Wang X.T., Yan H.G., Tang M.M., et al. Scaling and long-range dependence inoption pricing III: A fractional version of the Merton model with transaction costs[J]. Physica A2010,389:452-458.
[57] Cox J. Notes on option pricing I: Constant elasticity of variance difusions [J].Working paper, Stanford University,1975.
[58] Cox J., Ross S. The valuation of options for alternative stochastic process [J].Journal of Financial Economics,1976,4:145-166.
[59] Hauser S., Lauterbach B. Tests of warrants pricing model: The trading profitsperspective, Journal of Derivatives, Winter,1996,71-79.
[60] Lautervach B., Schultz P. Pricing warrants: An empirical study of the Black-Scholesmodel and its alternatives, Journal of Finance,1990,45:1181-1209.
[61] Beckers S. The constant elasticity of variance model and its implications for optionpricing, Journal of Finance,1980,35:661-673.
[62] Hinch E.J. Perturbation Methods [M]. Cambridge University Press, Cambridge,1991.
[63] Jumarie G. On the solution of the stochastic diferential equation of exponentialgrowth driven by fractional Brownian motion [J]. Applied Mathmatics Letter,2005,18:817-826.
[64] Chan N.H., Ng C.T. Fractional constant elasticity of variance model [J]. IMS lecturenotes-monograph series time series and related topics,2006,52:149-164.
[65] Kung J.J., Lee L.S. Option pricing under the Merton model of the short rate [J].Mathematics and Computers in simulation,2009,80:378-386.
[66] Merton R. C., A dynamic general equilibrium model of the asset market and itsapplication to the pricing of the capital structure of the firm [J]. Working PaperNo.497, Sloan School of management, MIT, Cambridge,1970.
[67] Wang J., Liang J.R., Lv L.J. et al., Continuous time Black-Scholes equation withtransaction costs in subdifusive fractional Brownian motion regime [J]. Physica A:2012,391:750-759.
[68] Cui Z.Y., Mcleish D., Comment on”Option pricing under the Merton model ofthe short rate” by Kung and Lee [Math. comput. simul.][J]. Mathematics andComputers in simulation,2010,81:1-4.
[69] Jiang L.S., Mathematical Modeling and Methods of option pricing [M]. Singapore:Word Sxientific Publishing,(2003).