Numerical Method for American Option Pricing under the Time-Fractional Black–Scholes Model

Sun, Yesen; Gong, Wenxiu; Dai, Hongliang; Yuan, Long

doi:https://doi.org/10.1155/2023/4669161

Mathematical Problems in Engineering

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Abstract Introduction Conclusion Appendix Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 4669161 | https://doi.org/10.1155/2023/4669161

Numerical Method for American Option Pricing under the Time-Fractional Black–Scholes Model

Yesen Sun,¹Wenxiu Gong,²Hongliang Dai,¹and Long Yuan²

Academic Editor: Francisco Ureña

Received09 Dec 2022

Revised14 Jan 2023

Accepted31 Jan 2023

Published20 Feb 2023

Abstract

The fractional Black–Scholes model has had limited applications in financial markets. Instead, the time-fractional Black–Scholes equation has attracted much research interest. However, it is difficult to obtain the analytic expression for American option pricing under the time-fractional Black–Scholes model. This paper will present an operator-splitting method to price the American options under the time-fractional Black–Scholes model. The fractional partial differential complementarity problem (FPDCP) that the American option price satisfied is split into two subproblems: a linear boundary value problem and an algebraic system. A high-order compact (HOC) scheme and a grid stretching (GS) method are considered for the linear boundary problem. Furthermore, numerical results show that the HOC scheme with a GS method gives an accurate numerical solution for American options under the time-fractional Black–Scholes model.

1. Introduction

A financial option is an important hedging instrument, which is one of the most popular and important financial derivatives. It is very important for financial researchers and practitioners to give a fair and reasonable option price. In the field of finance, one of the most classical option pricing models is the Black–Scholes model [1]. The basic assumptions are that the underlying asset price follows geometric Brownian motion and that volatility is a constant. However, the implied volatility depends on strike and maturity. Hence, the assumptions are inconsistent with the behavior of the underlying assets.

A lot of models have been introduced as improvements to the Black–Scholes model. Fractional calculus is one possible way to explain longer term memory effects of stock price variations. With the discovery of the fractional structure of financial markets, geometric Brownian motion has been replaced by fractional Brownian motion which provides a powerful tool to explain longer memory effects of underlying asset prices. Some fractional Black–Scholes models have been proposed for options pricing. Wyss [2] priced a European option by the time-fractional Black–Scholes equation and gave a closed-form solution. Cartea and del-Castillo-Negrete [3] derived the FMLS, KOBol, and CGMY models for pricing exotic options with jumps. Jumarie [4] obtained the time-space fractional Black–Scholes equation based on the fractional Taylor formula and Ito lemma. Liang et al. [5] further introduced a single parameter and a biparameter fractional Black–Scholes model. Chen et al. [6] provided a simplified fractional Black–Scholes version of a model by Liang et al. Farhadi et al. [7] also proposed a time fractional Black–Scholes equation by considering the effect of trend memory.

With the applications of the time fractional Black–Scholes equation in financial theory, different methods have been proposed for the solution of the time-fractional Black–Scholes equation. There are many researchers who present the analytical solution of the fractional Black–Scholes equation, see [2, 4–6, 8, 9]. The common method is Laplace transform coupled with other methods, such as homotopy perturbation methods and homotopy analysis methods or Laplace decomposition methods, see [9]. However, it is difficult to compute the solutions obtained by these methods. Therefore, it is necessary to develop some numerical methods for the fractional Black–Scholes equation.

Song and Wang [10] developed an implicit finite difference scheme for pricing options under Jumarie’s model. Zhang et al. [11] developed a finite difference scheme to the time-fractional Black–Scholes equation. Zhang et al. [12] constructed a discrete implicit numerical scheme for the time-fractional Black–Scholes equation governing European options. However, all of the above numerical methods focus on European option pricing. There are only a few studies on the numerical pricing methods of American options under the time-fractional Black–Scholes equation. It is well known that American option pricing problems can be formulated as a linear complementarity problem (LCP) under geometric Brownian motion. The used methods for the LCP are the penalty method [13, 14], projected SOR method [15, 16], projected LU (PLU) method [17], and operator-splitting method [18–21]. Chen et al. [22] used a predictor-corrector method for pricing American options under a finite moment log-stable model. Chen et al. [23] proposed an operator-splitting method for American options under time fractional Black-Scholes equation.

In this paper, we consider the operator-splitting method for the time-fractional Black–Scholes equation that the American options satisfied. The fractional partial differential complementarity problem (FPDCP) is split into two sub-problems: a linear boundary value problem and an algebraic system. We consider a high-order compact (HOC) scheme with a grid stretching (GS) method for the linear boundary problem. An HOC scheme is widely used in option pricing because of its high order convergence. Tangman et al. [24] discussed the applications of the HOC scheme for pricing American options and European options with stochastic volatility. An HOC finite difference scheme with a uniform mesh that is fourth order accurate in space and second order accurate in time is introduced for option pricing models with stochastic volatility [25]. Düring et al. [26] derived HOC finite difference schemes for option pricing in stochastic volatility models on nonuniform grids. Moreover, HOC finite difference schemes are used to discretize a nonlinear Black–Scholes equation which models transaction costs arising in the hedging of portfolios [27] and a class of parabolic partial differential equations with time and space-dependent coefficients as well as with mixed-second order derivative terms in spatial dimensions [28]. The HOC finite difference scheme with a grid stretching method is also considered for European options under the time-fractional Black–Scholes equation (29). At last, we also show that the HOC scheme with a GS method gives an accurate numerical solution for American options under the time-fractional Black–Scholes model.

The paper is organized as follows: In Section 2, we describe some models for the time-fractional Black–Scholes equation. In Section 3, we first present the GS method for the fractional Black–Scholes equation. Then, we adopt the operator-splitting method for the time-fractional Black–Scholes equation that the American options satisfied. An efficient HOC scheme is constructed at last. In Section 4, the numerical experiments of the convergence and other numerical studies are shown. The conclusions are given in Section 5.

2. Model Description

The assumption of the Black–Scholes option pricing is that the stock price follows the geometric Brownian motion.where and are constants and is the standard Brownian motion. Based on the above assumption, Black and Scholes derived that the price of a European option satisfies the partial differential equation as

With discovery of the fractional structure of financial markets, the fractional Black–Scholes partial differential equations are introduced to extend financial theory.

Caputo’s fractional derivative [30] is defined as

The left-modified Riemann–Liouville derivative is defined by

The right-modified Riemann–Liouville derivative is defined as follows:

Wyss [2] used the right-modified Riemann–Liouville derivative deriving the time-fractional Black–Scholes model.

Jumarie [4] took account of the stochastic processes of a time-fractional order. The stock value satisfies the fractional differential equation aswhere is a normalized Gaussian white noise, i.e., the zero mean and the unit variance. Using the equation and combining with the Ito Lemma and the fractional Taylor’s series of the price of the stock option, Jumarie [31] derived the fractional Black–Scholes equation. Jumarie’s model is given by

Farhadi et al. [7] used the left-modified Riemann–Liouville derivative deriving another time-fractional Black–Scholes model by considering the fractional trend effect. Farhadi’s model is obtained by the following equation:

In order to convert these three models into initial value problems, we use variable substitution .

According to [11], Caputo’s derivative is consistent with the left-modified Riemann–Liouville derivative. Hence, we replace by .

Wyss’ model can be converted into

Similarly, taking the same substitution and using the following chain rule (Corollary 2.3in [4]),

We have

Now, (8) and (9) can be transformed as follows:

The initial and boundary value conditions become

These European option price models are linear initial boundary value problems which can be easily solved. In our paper, we consider the problem of pricing American options under Wyss’ model. Jumarie’s model and Farhadi’s model can be found in Appendixes A and B.

3. Numerical Method

Using the change of the variable , (11) can be rewritten aswhere

The solution region is . To solve (16) numerically, we truncate the unbounded domain into a finite interval . We use the GS method at the strike price which can decrease the error and yield a better convergence. Let be the transformed coordinate, thenwhere , , and is a stretching parameter. Jacobian and Hessian of are

Then, we havewhere

This is the time-fractional Black–Scholes model for European options. Unlike the European options, the American option pricing problem is a free boundary problem. We consider the following linear complementarity problem for American put options. Assuming that is the price of American put options, we have

The initial and boundary value conditions of the FPCDP are

In this section, we present the numerical method for solving (8). Since closed-form solutions for (8) do not exist, a numerical method using a high-order compact scheme with operator splitting is developed for its solution.

Assuming that is a slack function, FPCDP (8) becomes

Let , , , and , where and are the time and space steps, respectively. According to [12], can be approximated by the following difference formula:where

Then, we introduce the operator-splitting method for (24). Here, we introduce an intermediate variable , so (24) will be split into two simple equations. First, we solve the following boundary value problem:

From (27), we can get the intermediate variable . Second, we solve the following algebraic equation:

We have the following solution:

To verify the effectiveness of our method, we add (27) and (28) together and get

For (27), we use the HOC scheme. We rewrite (27) aswhere

Let

Taking Taylor expansion to and , we have

Hence,and

Substituting (13) and (14) into equation (12), we havewhere

We can derive and by differentiating (12) with respect to where

Then, can be given as follows:

Hence, (37) can be written aswhere

Using the Crank–Nicolson difference scheme for (42), we obtain the HOC schemewhere

Let

Then, (44) becomeswhere

4. Numerical Experiments

The numerical performance of the proposed method is described in this section. We first display the nodes distribution with the GS method and compare our option pricing method with other methods. Then, we present the effect of to option prices. Last, we give the order of convergence.

The parameters are

We first display the nodes distribution with the GS method by (18). The result is shown in Figure 1. In Figure 1, we can see that the nodes are not evenly distributed. Figure 2 shows the American put option price with and . Table 1 presents the comparison between the operator-splitting (OS) method based on HOC-GS, projected LU method, and operator splitting [23] of American option prices. The results in Table 1 show the effectiveness of our method.

(a)

(b)

Now, we consider the effect of the parameter . The parameters are set as , and . Figure 3 shows the American option price at different . Table 2 shows the computational time for varying and Figure 4 shows the American option price at different and stock price. Figures 4(a) and 4(b) provide that the American put option price increases in at the at-the-money region and out-of-the-money region. From the two figures, we can see how affects the American put option prices. In Figure 5, we compare the American put option price under Wyss’, Jumarie’s, and Farhadi’s models by our proposed method. In the figure, we can see the similar results between Wyss’ and Jumarie’s models and the pricing of Farhadi’s model is higher than that of the other two models.

(a)

(b)

Next, we check the convergence rates of the HOC-GS scheme. The orders of convergence are for the time step size and for the spatial step size , where , is the numerical solution at the final time step, and is norm or norm.

We check the convergence order for European options. The convergence order in time and space directions for European options is listed in Table 3. Tables 4 and 5 show the orders of convergence in time and space for American put options. In Tables 4 and 5, we can see that convergence orders in time and spatial are oscillation. has an effect on convergence orders. When is bigger, the convergence orders are more stable and about which is better than the method in [23]. The reason of the oscillation is that the two subproblems are coupled which influence convergence orders of the HOC-GS scheme. In addition, the orders oscillate but still appear better than those of the method proposed in [23].

5. Conclusion

This paper gives an efficient method to solve the time-fractional Black–Scholes equation for American option pricing. We first employ the operator-splitting method to solve FPDCP that the American option price satisfied. FPDCP is split into two subproblems: a linear boundary problem and an algebraic system. The two subproblems are easy to solve by numerical methods. For the linear boundary problem, we consider an HOC scheme and a GS method which have a higher order convergence. The numerical results demonstrate the effectiveness of our method. Also, we study the effect of on the option price.

Appendix

A. The HOC Scheme of Jumarie’s Model

For Jumarie’s model (4), we replace the space variable by . Using , we havewhere

Then, we use the GS methodwhere

The FPCDP for American put options is given by

FPCDP (A.1) can be rewritten as follows by introducing :

Using the finite difference formula for and the operator-splitting method, we obtainand

For equation (A.3), we have

We rewrite equation (A.2) aswhere

Similar to Wyss’s model, for equation (A.4), we use the HOC scheme and the Crank–Nicolson difference scheme. We get

Hence,where

B. The HOC Scheme of Farhadi’s Model

For Farhadi’s model (5), we replace the space variable by . Using , we havewhere

Then, we use the GS methodwhere

The FPCDP for American put options is given by

FPCDP (B.1) can be rewritten as follows by introducing :

Using the finite difference formula for and the operator-splitting method, we obtainand

For equation (B.3), we have

We rewrite equation (B.2) aswhere

Similar to Wyss’s model, for equation (B.4), we use the HOC scheme and the Crank–Nicolson difference scheme. We get

Hence,where

Data Availability

The datasets used and analysed during the current study are available from the corresponding author on reasonable request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

YS Sun and WX Gong designed the methodology, developed the models, and analysed the results. HL Dai and L Yuan wrote the formal analysis and conducted validation of the final supplementary experiment. All the authors revised and improved the manuscript.

Acknowledgments

The work was supported by the National Natural Science Foundation of China (12071479) and Shandong Natural Science Foundation (ZR2022QA055, ZR2020MA046).

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