Research Article | Open Access
Solution of the Fractional Black-Scholes Option Pricing Model by Finite Difference Method
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.
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 [6–9] 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 . 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 is suggested to be subject to the time-fractional Black-Scholes equation [12, 13] with the following form: where denote risk-free interest rate and volatility, respectively. involved in (1) is a modified Riemann-Liouville fractional derivative [12–15], which is defined using the equality and can be represented as
The value of European put option is taken as a solution of (1) with the following terminal and boundary conditions:
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 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
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 . 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 [16–43] 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 [18–22, 38–43]. In this section, the implicit finite difference mode is given.
Take the change of variable Equation (1) can be transformed into the following form:
In order to use finite difference approximation, we start by and . Let , () be the grid sizes in space and time; the computational domain is discretized by a uniform grid with () and (), where is a realistic and practical approximation to infinity. denotes an approximate solution of (7) in at the time .
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, where and that satisfies (i) , (ii) [22, 38].
For spatial derivative, we use the following difference approximation:
From (11), we get the following equality when : After combining like terms, we can derive the following equation: For , we obtain
From the terminal and boundary conditions of the European put option, we can get
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 and set 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 [39–41]. For that, (13) and (14) are rewritten as where
3.1. Stability Analysis
Now, we introduce the grid function in :
Recall the result of ; defining can get where .
Proposition 1. If is a solution of (26), then .
3.2. Convergence Analysis
Suppose that is an exact solution of (7) at grid point and is the difference solution of (13) and (14); we define the error which satisfies the following equation according to (17): where () and 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:
Defining with the norms can obtain where
We can let where , .
Substituting the above expressions into (29) and simplifying them can get where .
Proposition 3. There exists a positive constant , so that .
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 . This known implicit difference results are given in [16, 17, 36]. In this section, we focus mainly on investigating the results with . 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.
Figure 1 shows analytical solution and numerical solution obtained by the implicit difference method at time when , , , , , , and . 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.
Example 3. Consider American put option pricing model. The following parameters are selected for the present study: Figure 4 indicates a price comparison of the American put option at (relaxation parameter and tolerance parameter are 1.2 and 0.001 in the applied process of SOR method).
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 and 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.
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.
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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