Abstract
In this paper, a high-order compact finite difference method is proposed for a class of temporal fractional subdiffusion equation. A numerical scheme for the equation has been derived to obtain in time and fourth-order in space. We improve the results by constructing a compact scheme of second-order in time while keeping fourth-order in space. Based on the - approximation formula and a fourth-order compact finite difference approximation, the stability of the constructed scheme and its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Applications using two model problems demonstrate the theoretical results.
1. Introduction
The Black-Scholes model, proposed in 1973 by Black and Scholes [1] and Merton [2], gives a theoretical estimate of the price of European-style options. Until Now, some of Black-Scholes models involving the fractional derivatives have emerged. In [3], Wyss priced a European call option by a time-fractional Black-Scholes model. In [4], Liang et al. derive a biparameter fractional Black-Merton-Scholes equation and obtain the explicit option pricing formulas for the European call option and put option, individually. An explicit closed-form analytical solution for barrier options under a generalized time-fractional Black-Scholes model by using eigenfunction expansion method together with the Laplace transform is derived in [5]. In [6], a discrete implicit numerical scheme with a spatially second-order accuracy and a temporally order accuracy is constructed; the stability and convergence of the proposed numerical scheme are analysed using Fourier analysis. In [7], H.Zhang et al. use some numerical technique to price a European double-knock-out barrier option, and then the characteristics of the three fractional Black-Scholes models are analysed through comparison with the classical Black-Scholes model. More recently, a numerical scheme of fourth-order in space and in time is derived in [8]; the solvability and convergence of the proposed numerical scheme are proved rigorously using a Fourier analysis. Some computationally efficient numerical methods have been proposed for solving fractional differential equation, for example, which include finite difference methods, finite element methods, finite volume methods, spectral methods, and meshless methods [9–26].
In this paper, we continue the work of R.H.De Staelen et al. [8]. The class of equations is given bywith the following boundary (barrier) and final conditionsand its initial conditionwhere is the risk free rate, is the dividend rate, and is the volatility of the returns. The functions and are the rebates paid when the corresponding barrier is hit. The terminal playoff of the option is . The fractional derivative in (1) is a Caputo derivative defined asAs described in [8], we consider the transform problem of (1)The rest of the paper is organized as follows: in Section 2, an efficient implicit numerical scheme with second-order accuracy in time and fourth-order accuracy in space is constructed. The analysis of the stability and convergence are presented in Section 3. In Section 4, numerical examples are given to illustrate the accuracy of the presented scheme and to support our theoretical results. Concluding remarks are given in the last section.
2. Construction of the Compact Finite Difference Scheme
In order to simplify the computation and analysis of the following compact finite difference scheme for Black-Scholes model, we use an indirect approach by introducing a suitable transformation.
According to some simple calculations, we transform equation (5) intowhereIt is clear that is a solution of (5) if and only if is a solution of (6).
In order to construct the compact finite difference scheme for the problem (5), we consider the above equivalent form (6).
Let be the time step and be the spatial step, where are positive integers.
Since the grid function , we then define difference operators as follows:
We also define where , and
Lemma 1. It holds (see [27])
In order to discretize (6) into a compact finite difference system, we introduce the following lemmas.
Lemma 2. Assuming , we havewhere .
Proof. From Lemma 2 of [9], we can obtain the proof of lemma.
Lemma 3. Assuming . When , we obtain
Proof. According to some simple calculations, the proof follows from Taylor expansions of the function at the point for and .
Since the above lemmas, we then discretize (6) into a compact finite difference scheme. In order to analyse, we define We also define the grid functions as follows: For the second-order spatial derivative , we adopt the following fourth-order compact approximation (see [28])We consider equation (6) at the point ; we can obtain
From Lemmas 2 and 3, we havewhereWe apply to equation (18); then we havewhereandIf we omit , then we have the compact finite difference scheme:
3. Stability and Convergence of the Proposed Compact Difference Scheme
Theorem 4. The compact difference scheme (23) is uniquely solvable.
Proof. The compact difference scheme (23) can be written in matrix form where The tridiagonal coefficient matrix yields It is easy to see that the tridiagonal coefficient matrix is strictly diagonally dominant. Therefore, the coefficient matrix is nonsingular and hence invertible.
Next, we consider the stability and convergence analysis of the compact difference scheme (23).
Letting , for grid functions , we define the inner product and norm as follows: According to simple calculations, we obtain
In order to analyse, we introduce the discrete inner product and norm: Based on above inner product and norm, we have the following lemmas.
Lemma 5 (see [29]). Suppose , we obtain
Lemma 6 (see [27]). Suppose , we obtain
Lemma 7 (see [9]). Suppose , we obtain
In the next, we then analyse the stability and convergence of the scheme (23).
Theorem 8 (stability). Let be the solution of the compact difference scheme (23) with . Assume that one of the conditions holds for some positive constant .
Then it holds
Proof. We take the inner product of equation (23) with yield Using Lemma 7,When for some positive constant , we have from the Cauchy-Schwarz inequality and Lemmas 6 that By (35) and the Cauchy-Schwarz inequality,Substituting (38) into (35) leads to The above inequality can be rewritten as Since by the definition of , we have from (40) thatLettingand assuming , we obtainand we have the needed estimates.
Letting , we get the following error equation:Since the above error equation (45), we now obtain the following convergence results.
Theorem 9 (convergence). Let denote the value of the solution of (23) at the mesh point and let be the solution of the compact difference scheme (23). Then when , it holdswhere
Proof. It follows from Theorem 8 that Applying (22), we get The estimate (46) is proved.
Remark 10. The constraint condition in Theorems 8 and 9 is only for the analysis of the stability and convergence of the compact difference scheme (23). This condition is easily verifiable for practical problems.
4. Numerical Experiment
For demonstrating the efficiency of the compact difference scheme (23), we make two numerical experiments of it.
Suppose be the value of the solution of the problem (1)–(3) at the mesh point . From (22), we can see thatwhere is a positive constant independent. In order to check this accuracy of the compact difference scheme, we compute the following norm errors:The temporal convergence order and the spatial convergence order are denoted by
Example 1. We first consider a problem, which is governed by equation (1) in with andThe boundary and initial conditions are given by (2) and (3) withIt is easy to check that is the solution of this problem.
For different , we let the spatial step . Table 1 gives the errors and the temporal convergence orders of the computed solution for and different time step . From the table, we can see that the computed solution has the second-order temporal accuracy. For comparison, the corresponding temporal convergence orders given in [8] has only order; thus it is far less accurate than the compact difference scheme (23) given in this paper.
Next, we compute the spatial convergence order of the compact difference scheme (23). Table 2 presents the errors and the spatial convergence orders . The table demonstrates that the compact difference scheme (23) has the fourth-order spatial accuracy.
Example 2. In this example, we test the error and the convergence order of the compact difference scheme (23). Consider equation (1) in the domain with andThe boundary and initial conditions are given by (2) and (3) withIt is clear that is the exact analytical solution of this problem.
Apply the compact difference scheme (23) to solve the above problem. Table 3 presents the errors and the temporal convergence orders ; we can see that the computed solution has the second-order temporal accuracy.
From Table 4, we can obtain the errors and the spatial convergence orders . These numerical results demonstrate that the accuracy of the compact difference scheme (23) is fourth-order.
5. Concluding Remarks
In this paper, a high-order compact finite difference method for a class of time-fractional Black-Scholes equations is presented and analysed. We apply the - approximation formula to the Caputo derivative; then we construct a fourth-order compact finite difference approximation for the spatial derivative. We have analysed the solvability, stability, and convergence of the constructed scheme and provided the optimal error estimates. The constructed scheme has the second-order temporal accuracy and the fourth-order spatial accuracy, which improves the temporal accuracy of the method given in [8].
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported in part by National Natural Science Foundation of China No. 11401363, the Education Foundation of Henan Province No. 19A110030, the Foundation for the Training of Young Key Teachers in Colleges and Universities in Henan Province No. 2018GGJS134.