AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 194286 10.1155/2013/194286 194286 Research Article Solution of the Fractional Black-Scholes Option Pricing Model by Finite Difference Method 0000-0001-7196-1536 Song Lina Wang Weiguo Chun Changbum Center for Econometric Analysis and Forecasting School of Mathematics and Quantitative Economics Dongbei University of Finance and Economics Dalian 116025 China dufe.edu.cn 2013 26 6 2013 2013 17 03 2013 02 06 2013 16 06 2013 2013 Copyright © 2013 Lina Song and Weiguo Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This work deals with the put option pricing problems based on the time-fractional Black-Scholes equation, where the fractional derivative is a so-called modified Riemann-Liouville fractional derivative. With the aid of symbolic calculation software, European and American put option pricing models that combine the time-fractional Black-Scholes equation with the conditions satisfied by the standard put options are numerically solved using the implicit scheme of the finite difference method.

http://dx.doi.org/10.13039/501100001809 National Natural Science Foundation of China 71171035 71273044 71271045
1. Introduction

The theory and methodology of partial differential equation started to become popular to study option pricing problems, after the classical Black-Scholes equation was proposed. The available research results include mainly two aspects: one is to give values of options using more powerful numerical and analytic methods; the other is to derive new pricing models that reflect the actual financial market more closely.

The Black-Scholes equation has been increasingly attracting interest over the last two decades since it provides effectively the values of options. But the classical Black-Scholes equation was established under some strict assumptions. Therefore, some improved models have been proposed to weaken these assumptions, such as stochastic interest model , Jump-diffusion model , stochastic volatility model , and models with transactions costs [4, 5]. With the discovery of the fractal structure for financial market, the fractional Black-Scholes models  are derived by replacing the standard Brownian motion involved in the classical model with fractional Brownian motion. These fractional Black-Scholes models are still partial differential equations with integer order derivative, which are reduced to the classical Black-Scholes equation if we let the Hurst exponent H=1/2. The differential equation involving derivatives of fractional order is a powerful tool for studying fractal geometry and fractal dynamics. As a generalization of the integer-order differential equation, fractional differential equation is used to model important phenomena in various fields such as fluid flow, electromagnetic, acoustics, electrochemistry, cosmology, and material science. Recently, fractional partial differential equation was introduced more and more into financial theory. Wyss  gave the fractional Black-Scholes equation with a time-fractional derivative to price European call option. Cartea and del-Castillo-Negrete  gave several fractional diffusion models of option prices in markets with jumps and priced barrier option using fractional partial differential equation. Jumarie [12, 13] derived the time- and space-fractional Black-Scholes equations and gave optimal fractional Merton's portfolio. The aim of this paper is to try to combine Jumarie's time-fractional Black-Scholes equation with the terminal and boundary conditions satisfied by the standard put options to study the pricing problems for European and American put options.

In the present work, the option price V=V(S,t) is suggested to be subject to the time-fractional Black-Scholes equation [12, 13] with the following form: (1)αVtα=(rV-rSVS)t1-αΓ(2-α)-Γ(1+α)2σ2S22VS2,t>0,0<α1, where r,σ denote risk-free interest rate and volatility, respectively. αV/tα involved in (1) is a modified Riemann-Liouville fractional derivative , which is defined using the equality (2)Dtαf(t)=limh0Δαf(t)hα=limh0k=0(-1)k(αk)f(t+(α-k)h),0<α1 and can be represented as (3)Dtαf(t)={1Γ(-α)0t(t-ξ)-α-1(f(ξ)-f(0))dξ,ifα<0,1Γ(1-α)ddt0t(t-ξ)-α(f(ξ)-f(0))dξ,if0<α<1,(f(α-n)(t))(n),ifnα<n+1,n1.

The value of European put option is taken as a solution of (1) with the following terminal and boundary conditions: (4)V(S,T)=max(K-S,0),V(0,t)=Kexp(-r(T-t)),V(S,t)0,S+.

Pricing European put option based on (1) and (4) can be implemented by finite differences approximation. For American put option, this idea is operable. The main problem for pricing an American put option is to consider the possibility of early exercise. To avoid arbitrage, the option value at each point in the (S,t) space cannot be less than the intrinsic value . For that, the principle of dynamic programming (at a given time, the optimal strategy corresponds to the maximum of either the exercise value or the value associated with selecting an optimal strategy an instant later)  expressed by the following equation is applied, which can be implemented using finite difference method and successive overrelaxation (SOR) method (5)V(S,t)=max(K-S,V(S+dS,t+dt)).

Fractional Black-Scholes equation (1) is actually a special kind of fractional advection-dispersion equation, which has time-varying coefficient and time-fractional derivative. A general fractional advection-dispersion equation has been studied in many works in the literature. Sousa  employed a second-order explicit finite difference method derived by a Lax-Wendroff-type time discretization procedure to solve space-fractional advection-diffusion equation. Mohebbi and Abbaszadeh  applied compact finite difference scheme to deal with time-fractional advection-dispersion equation with constant coefficient. Meerschaert and Tadjeran  developed explicit and implicit Euler methods for the space-fractional advection-dispersion equation. Liu et al.  developed effective numerical methods for solving several space-time fractional advection-dispersion equations. Liu et al.  used explicit and implicit difference methods to solve space-time fractional advection-dispersion equation and proved the stability and convergence of the methods. More powerful methods, such as implicit MLS meshless method , Adomian decomposition method , homotopy perturbation method , and spectral regularization method , have been used to solve numerical and analytical solutions for various types of fractional advection-dispersion equation. In this paper, we combine time-fractional Black-Scholes equation (1) with conditions satisfied by European and American put options to construct put option pricing models. It is obvious that this fractional derivative model can be reduced to the classical Black-Scholes model if the derivative order α=1. As a generalization, this time-fractional pricing model of European and American put options is solved numerically using the implicit finite difference technique.

2. Numerical Scheme

Numerical and analytical methods  are used to study a great quantity of differential equations. Among them, the finite difference method is a direct and effective numerical algorithm. With this technique, the differential equation is transformed into a difference equation by the discretization of derivatives, and numerical solutions are finally obtained. The approach has been extended successfully to deal with various fractional differential equation [1822, 3843]. In this section, the implicit finite difference mode is given.

Take the change of variable (6)τ=T-t. Equation (1) can be transformed into the following form: (7)τ(α-1)(T-τ)(1-α)αVτα+(rV-rSVS)(T-τ)1-αΓ(2-α)-Γ(1+α)2σ2S22VS2=0.

In order to use finite difference approximation, we start by S[0,Smax] and τ[0,T). Let h=Smax/M, k=T/N (M,NZ+) be the grid sizes in space and time; the computational domain is discretized by a uniform grid (Sm,τn) with Sm=mh (m=0,1,,M) and τn=nk (n=0,1,,N), where SM=Smax is a realistic and practical approximation to infinity. Vmn denotes an approximate solution of (7) in Sm at the time τn.

As pointed out in [12, 13], Jumarie's definition and the so-called Caputo's definition yield the same result when the function is differentiable. So, we take Caputo finite difference approximation  for the modified Riemann-Liouville fractional derivative involved in (7), namely, (8)αVτα=ϱα,kj=1nωj(α)(Vmn-j+1-Vmn-j)+O(k), where ϱα,k=1/Γ(2-α)kα and ωj(α)=j1-α-(j-1)1-α that satisfies (i) ωj(α)>0, (ii) ωj(α)>ωj+1(α) [22, 38].

For spatial derivative, we use the following difference approximation: (9)VS=Vm+1n-Vm-1n2h+O(h),2VS2=Vm+1n-2Vmn+Vm-1nh2+O(h2).

Substituting (8) and (9) to (7) can derive (10)τ(α-1)(T-τ)(1-α)ϱα,kj=1nωj(α)(Vmn-j+1-Vmn-j)+(rVmn-rmhVm+1n-Vm-1n2h)×(T-nk)1-αΓ(2-α)-Γ(1+α)2×σ2(mh)2Vm+1n-2Vmn+Vm-1nh2=0 or (11)τ(α-1)(T-τ)(1-α)ϱα,kj=1nωj(α)(Vmn-j+1-Vmn-j)+(rVmn-rmVm+1n-Vm-1n2)×(T-nk)1-αΓ(2-α)-Γ(1+α)2×σ2m2(Vm+1n-2Vmn+Vm-1n)=0, where the local truncation error is O(k+h2).

From (11), we get the following equality when n=1: (12)k(α-1)(T-k)(1-α)ϱα,k(Vm1-Vm0)+(rVm1-rmVm+11-Vm-112)×(T-k)1-αΓ(2-α)-Γ(1+α)2×σ2m2(Vm+11-2Vm1+Vm-11)=0. After combining like terms, we can derive the following equation: (13)(rm(T-k)1-α2Γ(2-α)-Γ(1+α)2σ2m2)Vm-11+(k(α-1)(T-k)(1-α)ϱα,k+r(T-k)1-αΓ(2-α)+Γ(1+α)σ2m2r(T-k)1-αΓ(2-α))Vm1-(Γ(1+α)2σ2m2+rm(T-k)1-α2Γ(2-α))Vm+11=k(α-1)(T-k)(1-α)ϱα,kVm0,m=1,2,,M-1. For n2, we obtain (14)(rm(T-nk)1-α2Γ(2-α)-Γ(1+α)2σ2m2)Vm-1n+(r(T-nk)1-αΓ(2-α)(nk)(α-1)(T-nk)(1-α)ϱα,k+r(T-nk)1-αΓ(2-α)+Γ(1+α)σ2m2)Vmn-(Γ(1+α)2σ2m2+rm(T-nk)1-α2Γ(2-α))Vm+1n=(nk)(α-1)(T-nk)(1-α)ϱα,k×(Vmn-1-j=2nωj(α)(Vmn-j+1-Vmn-j)),m=1,2,,M-1.

If the minimum limit of variable S is Smin, the corresponding implicit difference approximation can be derived by replacing m of the above results (13) and (14) with (Smin/h)+m.

From the terminal and boundary conditions of the European put option, we can get (15)Vm0=max(K-mh,0),m=0,,M,V0n=Kexp(-rnk),n=1,,N,VMn=0,n=1,,N.

In the case of American put option, we perform the above procedure. At the same time, we should check for the possibility of early exercise after computing Vmn and set (16)Vmn=max(Vmn,K-mh),forn=1,,N. Under this implicit scheme, we do not directly apply (16) at each step but use SOR method to complete this procedure, which was suggested in .

3. Stability and Convergence

In this section, we will analyze the stability and convergence of implicit finite difference scheme (13) and (14) using Fourier analysis involved in . For that, (13) and (14) are rewritten as (17)am,1Vm-11+bm,1Vm1+cm,1Vm+11=Vm0,am,nVm-1n+bm,nVmn+cm,nVm+1n=(1-ω2(α))Vmn-1-j=2n-1(ωj+1(α)-ωj(α))Vmn-j+ωn(α)Vm0, where (18)am,n=(nk)(1-α)(T-nk)(α-1)ϱα,k-1×(rm(T-nk)1-α2Γ(2-α)-Γ(1+α)2σ2m2),bm,n=1+(nk)(1-α)(T-nk)(α-1)ϱα,k-1×(r(T-nk)1-αΓ(2-α)+Γ(1+α)σ2m2),cm,n=-(nk)(1-α)(T-nk)(α-1)ϱα,k-1×(Γ(1+α)2σ2m2+rm(T-nk)1-α2Γ(2-α)).

3.1. Stability Analysis

If Umn is another approximate solution of (13) and (14), we define the round-off error εmn=Vmn-Umn which satisfies the following equations according to (17): (19)am,1εm-11+bm,1εm1+cm,1εm+11=εm0,am,nεm-1n+bm,nεmn+cm,nεm+1n=(1-ω2(α))εmn-1-j=2n-1(ωj+1(α)-ωj(α))εmn-j+ωn(α)εm0,0000n=2,,N,m=1,2,,M-1,(20)ε0n=εMn=0,n=0,,N.

Now, we introduce the grid function in : (21)εn(S)={εmn,whenSm-h2<SSm+h2,m=1,,M-1,0,when0Sh2orSmax-h2<SSmax.

Recall the result of ; defining (22)εn=[ε1n,ε2n,,εM-1n]T,εn2=(m=1M-1h|εmn|2)1/2=(0Smax|εn(S)|2dS)1/2 can get (23)εn22=l=-+|ξn(l)|2, where ξn(l)=(1/Smax)0Smaxεn(S)exp(-I2πlS/Smax)dS.

The solution of (19) and (20) is supposed to have the following form: (24)εmn=ξnexp(Iμmh), where μ=2πl/Smax, I=-1.

Substituting the above expression into (19), we get (25)am,1ξ1exp(Iμ(m-1)h)+bm,1ξ1exp(Iμmh)+cm,1ξ1exp(Iμ(m+1)h)=ξ0exp(Iμmh),am,nξnexp(Iμ(m-1)h)+bm,nξnexp(Iμmh)+cm,nξnexp(Iμ(m+1)h)=(1-ω2(α))ξn-1exp(Iμmh)-j=2n-1(ωj+1(α)-ωj(α))ξn-jexp(Iμmh)+ωn(α)ξ0exp(Iμmh). Simplifying (25), we obtain (26)ϑm,1ξ1=ξ0,ϑm,nξn=(1-ω2(α))ξn-1-j=2n-1(ωj+1(α)-ωj(α))ξn-j+ωn(α)ξ0, where ϑm,n=1+(nk)(1-α)(T-nk)(α-1)ϱα,k-1[(r(T-nk)1-α/Γ(2-α))(1-Imsin(μh))+Γ(1+α)σ2m2(1-  cos(μh))].

Proposition 1.

If ξn is a solution of (26), then |ξn||ξ0|.

Proof.

For n=1, the first equality of (26) gives |ξ1|=(1/|ϑm,1|)|ξ0||ξ0|. If (27)|ξn-1||ξ0|, then using the second equality of (26), we obtain (28)|ξn|1-ω2(α)-(ωn(α)-ω2(α))+ωn(α)|ϑm,n||ξ0|=1|ϑm,n||ξ0||ξ0|.

According to (13) and (14), we can obtain the following conclusion using Proposition 1 and equality (23).

Theorem 2.

The difference scheme (13) and (14) is unconditionally stable.

3.2. Convergence Analysis

Suppose that V(Si,tk) is an exact solution of (7) at grid point (Si,tk) and Vki is the difference solution of (13) and (14); we define the error ϵjk=V(Si,tk)-Vki which satisfies the following equation according to (17): (29)am,1ϵm-11+bm,1ϵm1+cm,1ϵm+11=kαRm1,am,nϵm-1n+bm,nϵmn+cm,nϵm+1n=(1-ω2(α))ϵmn-1-j=2n-1(ωj+1(α)-ωj(α))ϵmn-j+kαRmn,n=2,,N,m=1,2,,M-1,(30)ϵ0n=ϵMn=0,n=1,,N,ϵm0=0,m=0,,M, where |Rmn|C1(k+h2) (n=1,2,,N) and C1 is a positive constant.

For completing the proof of convergence, we recall several results below which came from .

Similar to the stability analysis,  constructed the following grid function: (31)ϵn(S)={ϵmn,whenSm-h2<SSm+h2,m=1,,M-1,0,when0Sh2orSmax-h2<SSmax,Rn(S)={Rmn,whenSm-h2<SSm+h2,m=1,,M-1,0,when0Sh2orSmax-h2<SSmax.

Defining (32)ϵn=[ϵ1n,ϵ2n,,ϵM-1n]T,Rn=[R1n,R2n,,RM-1n]T with the norms (33)ϵn2=(m=1M-1h|ϵmn|2)1/2=(0Smax|ϵn(S)|2dS)1/2,Rn2=(m=1M-1h|Rmn|2)1/2=(0Smax|Rn(S)|2dS)1/2 can obtain (34)ϵn22=l=-+|ζn(l)|2,Rn22=l=-+|ηn(l)|2, where (35)ζn(l)=1Smax0Smaxϵn(S)exp(-I2πlSSmax)dS,ηn(l)=1Smax0SmaxRn(S)exp(-I2πlSSmax)dS.

We can let (36)ϵmn=ζnexp(Iμmh),Rmn=ηnexp(Iμmh), where μ=2πl/Smax, I=-1.

Substituting the above expressions into (29) and simplifying them can get (37)ϑm,1ζ1=kαη1,ϑm,nζn=(1-ω2(α))ζn-1-j=2n-1(ωj+1(α)-ωj(α))ζn-j+kαηn, where ϑm,n=1+(nk)(1-α)(T-nk)(α-1)ϱα,k-1([(r(T-nk)1-α/Γ(2-α))(1-Imsin(μh))+Γ(1+α)σ2m2(1-  cos(μh))].

Proposition 3.

There exists a positive constant C2, so that |ζn|C2nkα|η1|.

Proof.

From , we know (38)Rn2C1Smax(k+h2)(n=1,2,,N),(39)|ηn|C2|η1|(n=1,2,,N). Then the first equality of (37) tells us that (40)|ζ1|1|ϑm,n|kα|η1|C2kα|η1|. If we let (41)|ζn-1|C2(n-1)kα|η1|, using the second equality of (37), we can obtain (42)|ζn|1|ϑm,n||j=2n-1(1-ω2(α))(n-1)+j=2n-1(ωj(α)-ωj+1(α))(n-j)+1|C2kα|η1|1|ϑm,n||j=2n-1(1-ω2(α))(n-1)+j=2n-1(ωj(α)-ωj+1(α))(n-1)+1|C2kα|η1|1|ϑm,n||(1-ωn(α))(n-1)+1|C2kα|η1|nC2kα|η1|.

According to equalities (34) and inequality (38), we can obtain the following conclusion using Proposition 3.

Theorem 4.

The implicit difference scheme (13) and (14) is L2-convergent.

4. Computational Examples

Option pricing model based on the time-fractional differential equation (1) is studied. The implicit difference scheme of (7) is given in Section 2. From the finite difference forms, it is clear that (13) and (14) are just the implicit difference scheme of the classical Black-Scholes equation if we let α=1. This known implicit difference results are given in [16, 17, 36]. In this section, we focus mainly on investigating the results with 0<α<1. To explain the stability and convergence of the implicit numerical schemes, we firstly take Example 1 with the given terminal and boundary condition as an example.

Example 1.

Consider (1) that is subject to the following conditions: (43)V(S,T)=V(S,T)=σ2(2T-Tα)-2ln(S),V(Smin,t)=exp(-r(T-t))(σ2(2t-tα)-2ln(Smin)),V(Smax,t)=exp(-r(T-t))(σ2(2t-tα)-2ln(Smax)).

Under the condition r=σ2, (1) has an analytical solution (44)V(S,t)=exp(-r(T-t))(σ2(2t-tα)-2ln(S)).

Figure 1 shows analytical solution and numerical solution obtained by the implicit difference method at time t=0.5 when r=0.01, σ=0.1, T=1, Smin=0.1, Smax=1, M=19, and N=10. Numerical solution compares well with analytical solution, which proves that the implicit scheme is stable. Under the same parameters, Figure 2 gives the absolute error between numerical solutions and analytical solutions, which illustrates that the numerical results are convergent.

Solution curve with t=0.5. Solid line: numerical solution; Soft dot: analytical solution.

Absolute error between numerical solution and analytical solution. dash-dotted line: t=0.3; dotted line: t=0.5; solid line: t=0.8.

Example 2.

European put option pricing model is based on (1) and the condition (4) under the following parameters: (45)K=50,r=0.01,σ=0.3,T=1,Smax=100,M=20,N=10. Through coding, the European put option values with α=1,1/2,5/7,9/10 are plotted in Figure 3 at t=0.1/0.5/1.

European put option. dash-dotted line: t=0.1; dotted line: t=0.5; solid line: t=1.

Example 3.

Consider American put option pricing model. The following parameters are selected for the present study: (46)K=60,r=0.01,σ=0.4,T=1,Smax=120,M=30,N=10. Figure 4 indicates a price comparison of the American put option at t=0.2/0.6/1 (relaxation parameter and tolerance parameter are 1.2 and 0.001 in the applied process of SOR method).

American put option. dash-dotted line: t=0.2; dotted line: t=0.6; solid line: t=1.

Using Examples 1, 2, and 3, we examine the implementation of the implicit finite difference method for the fractional partial differential equation system. According to Figures 1 and 2 in Example 1, we can confirm that the implicit numerical scheme is stable and convergent. From Figures 3 and 4, we can see that the numerical scheme is very effective. Figures 3 and 4 present numerical simulation of the price of the European and American put options when the order of the time-fractional derivative takes different values. Their visible shapes and development trend are similar to the classical put option pricing model based on the standard Black-Scholes equation, which illustrates the essential characteristics of the European and American put options. For making the figures clear, we choose the small values of M and N and get the similar conclusions if largening the number of steps in time and space. As a generalization of the standard models, these fractional Black-Scholes models are powerful and will be of great interest to researchers in further work.

5. Conclusions

In this work, the finite difference method is employed to solve the time-fractional Black-Scholes equation together with the conditions satisfied by the standard put options. Application of the fractional differential equation to the pricing theory of option is in its beginning stage and needs more further work. This fractional model mentioned in this paper can model the price of other financial derivatives like warrant, swaps, and so on. The successful application of the finite difference method proves that this technique is effective and requires less computational work to solve fractional partial differential equation.

Acknowledgments

The authors would like to express their gratitude to reviewers for the careful reading of the paper and for their constructive comments which greatly improve the quality of this paper. The work is partially supported by the National Natural Science Foundation of China (no. 71171035, no. 71273044, and no. 71271045).

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