A Novel Fourth-Order Finite Difference Scheme for European Option Pricing in the Time-Fractional Black–Scholes Model

Abstract

This paper addresses the valuation of European options, which involves the complex and unpredictable dynamics of fractal market fluctuations. These are modeled using the $\alpha$-order time-fractional Black–Scholes equation, where the Caputo fractional derivative is applied with the parameter $\alpha$ ranging from 0 to 1. We introduce a novel, high-order numerical scheme specifically crafted to efficiently tackle the time-fractional Black–Scholes equation. The spatial discretization is handled by a tailored finite point scheme that leverages exponential basis functions, complemented by an L1-discretization technique for temporal progression. We have conducted a thorough investigation into the stability and convergence of our approach, confirming its unconditional stability and fourth-order spatial accuracy, along with $(2-\alpha )$-order temporal accuracy. To substantiate our theoretical results and showcase the precision of our method, we present numerical examples that include solutions with known exact values. We then apply our methodology to price three types of European options within the framework of the time-fractional Black–Scholes model: (i) a European double barrier knock-out call option; (ii) a standard European call option; and (iii) a European put option. These case studies not only enhance our comprehension of the fractional derivative’s order on option pricing but also stimulate discussion on how different model parameters affect option values within the fractional framework.

1. Introduction

Options rank as some of the most frequently traded financial instruments globally. The precise valuation of these instruments has been a central theme in financial research since the seminal Black–Scholes (B-S) model was formulated by Black, Scholes [1] and Merton [2] in 1973. Although the B-S model is foundational in financial theory and has been instrumental in valuing a variety of equity options, including European and American styles, and in creating portfolios that replicate option values, it has notable limitations. It struggles to accommodate rapid market fluctuations and price jumps that happen within brief periods [3].

Fractional calculus has gained prominence in stochastic modeling and financial economics since it was recognized that asset price dynamics often exhibit fractal properties. Pioneering works such as Wyss’s [4] fractional adaptation of a time-fractional B-S model for pricing European call options highlighted the potential of fractional derivatives to better reflect real-world market behavior. More recently, there has been a burgeoning interest in applying fractional derivatives and integrals due to their exceptional capability in accounting for past influences through their non-local attributes [5]. In [6], Jumarie derived the time and space fractional B-S models for stock exchange dynamics and gave an optimal fractional Merton’s portfolio. Liang et al. [7] advanced the field by deriving both single and bi-parameter fractional Black–Scholes–Merton differential equations, assuming that the stock price evolution adheres to a fractional Ito process. In this study, we focus specifically on the time-fractional adaptation of the B-S model

$${\displaystyle \frac{{\partial }^{\alpha }\mathcal{V}(S,\tau )}{\partial {\tau }^{\alpha }}}+{\displaystyle \frac{{\sigma }^2}{2}}{S}^2{\displaystyle \frac{{\partial }^2\mathcal{V}(S,\tau )}{\partial S^2}}+rS{\displaystyle \frac{\partial \mathcal{V}(S,\tau )}{\partial S}}-r\mathcal{V}(S,\tau )=0,(S,\tau )\in {\mathbb{R}}^{+}\times (0,T),0<\alpha <1,$$

(1)

subject to the constraints of the boundary conditions and the terminal state requirement

$$\left\{\begin{array}{c}\mathcal{V}(0,\tau )=\widetilde{{\varphi }_1}\left(\tau \right),\mathcal{V}(\infty ,\tau )=\widetilde{{\varphi }_2}\left(\tau \right),\hfill \\ \mathcal{V}(S,T)=\eta \left(S\right).\hfill \end{array}\right.$$

(2)

Here, S denotes the stock’s current price, $\tau$ represents the present time, and $\mathcal{V}$ signifies the value of a European option price, which depends on S and $\tau$. The term T refers to the expiration date of the contract, $r=r\left(x\right)$ stands for the risk-free interest rate, and $\sigma =\sigma \left(x\right)$ represents the volatility inherent in the returns of the underlying asset. Further, ${\displaystyle \frac{{\partial }^{\alpha }\mathcal{V}(S,\tau )}{\partial {\tau }^{\alpha }}}$ is the modified Reimann–Liouville derivative defined as

$${\displaystyle \frac{{\partial }^{\alpha }\mathcal{V}(S,\tau )}{\partial {\tau }^{\alpha }}}={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{d}{d\tau }}{\int }_{\tau }^T{\displaystyle \frac{\mathcal{V}(S,\xi )-\mathcal{V}(S,T)}{{(\tau -\xi )}^{\alpha }}}d\xi ,0<\alpha <1.$$

(3)

Setting $x=\log S,\tau =T-t$ in (1) and (2), we derive

$$\begin{array}{c}\hfill {\displaystyle -\frac{{\partial }^{\alpha }\mathcal{V}(e^x,T-t)}{\partial t^{\alpha }}+\frac{{\sigma }^2}{2}\frac{{\partial }^2\mathcal{V}(e^x,T-t)}{\partial x^2}+(r-\frac{{\sigma }^2}{2})\frac{\partial \mathcal{V}(e^x,T-t)}{\partial x}-r\mathcal{V}(e^x,T-t)=0,}\\ \hfill (x,t)\in {\mathbb{R}}^{+}\times (0,T),\end{array}$$

(4)

with boundary and initial conditions

$$\left\{\begin{array}{c}\mathcal{V}(e^{-\infty },T-t)={\varphi }_1(T-t),\mathcal{V}(e^{\infty },T-t)={\varphi }_2(T-t),t\in (0,T),\hfill \\ \mathcal{V}(e^x,T)=\eta \left(e^x\right)=v\left(x\right),x\in {\mathbb{R}}^{+}.\hfill \end{array}\right.$$

(5)

Denoting $u(x,t)=\mathcal{V}(e^x,T-t)$, the system (4) and (5) can be simplified to

$$\left\{\begin{array}{c}{\displaystyle {}_0{}^{C}D_t^{\alpha }u(x,t)=\frac{{\sigma }^2}{2}\frac{{\partial }^{2}u(x,t)}{\partial x^2}+\left(r-\frac{{\sigma }^2}{2}\right)\frac{\partial u(x,t)}{\partial x}-ru(x,t),(x,t)\in \mathbb{R}\times (0,T)}\hfill \\ \\ u(-\infty ,t)={\varphi }_1\left(t\right),u(\infty ,t)={\varphi }_2\left(t\right),t\in (0,T)\hfill \\ \\ u(x,0)=v\left(x\right),x\in \mathbb{R}.\hfill \end{array}\right.$$

(6)

Furthermore, the fractional derivative employed in Equation (6) is specifically the Caputo fractional derivative. This type of derivative is commonly used when dealing with initial value problems involving fractional order differential equations, as it allows for classical initial conditions to be imposed. The Caputo derivative is defined as

$${}_0{}^{C}D_t^{\alpha }u(x,t)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{d}{dt}}{\int }_0^t{\displaystyle \frac{u(x,s)-u(x,0)}{{(t-s)}^{\alpha }}}ds={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_0^t{\displaystyle \frac{du(x,s)}{ds}}{(t-s)}^{-\alpha }ds.$$

Analytical solutions for fractional differential equations are often elusive, prompting researchers to obtain solutions via approximation methods when tackling the time-fractional B-S model. For instance, Zhang et al. [8] proposed a second-order accurate finite difference method coupled with a fast bi-conjugate gradient-stabilized algorithm to minimize storage requirements and numerically solve a time-fractional B-S model. Tian et al. [9] presented three compact difference schemes tailored to the time-fractional B-S model for European option pricing. They employed an exponential transformation to eliminate the convection term in the B-S equation and achieved fourth-order spatial accuracy via the Padé approximation.

Samuel et al. [10] put forth a robust numerical method grounded in the extension of the Crank–Nicolson finite difference approach for solving the time-fractional B-S model characterizing stock market dynamics. Roul et al. [11] developed a fourth-order compact finite difference method utilizing quintic B-spline basis functions, while Abdi et al. [12] constructed high-order accurate compact difference schemes in space. Golbabai et al. [13,14] presented the utilization of radial basis functions within a meshless framework for numerically addressing the time-fractional B-S equation. Their work revealed a convergence rate of $2-\alpha$ in relation to the time variable.

Akram et al. [15] utilized the extended cubic B-spline to generalize the collocation method for the time-fractional B-S European option pricing model. Their approach’s distinctive characteristic lies in transforming such problems into systems of algebraic equations suitable for computer implementation, with a spatial convergence rate of 2. A distinct space-time spectral approach was introduced, employing Jacobi polynomials for temporal discretization and Fourier-type basis functions for spatial discretization [16]. Zhang et al. [17] derived a numerical scheme with second-order spatial accuracy, which was subsequently enhanced by De Staelen et al. [18] through the development of a fourth-order spatially accurate scheme.

Fractional calculus, with its non-local properties, is essential for modeling memory effects and has led to the development of efficient numerical methods for solving fractional differential equations [19,20,21,22].These methods include finite difference [23], finite element [24], finite volume [25], spectral [26,27], local discontinuous Galerkin finite elements [28], and meshless methods [29]. Recently, we have focused on tailored finite point schemes (TFPSs) for solving a variety of fractional diffusion equations, including those that describe convection-dominated diffusion and anisotropic subdiffusion [30,31]. TFPS, first introduced by Han et al. [32] for singularly perturbed elliptic equations and later extended to other models [33,34], uses local basis functions that satisfy the homogeneous equation with constant coefficients. This approach allows for remarkable accuracy even at coarser grid resolutions by closely matching the intrinsic properties of the solution. TFPS has proven effective in capturing boundary and interface layers within fractional diffusion problems, prompting its application in our study to formulate a TFPS approach for the time-fractional Black–Scholes Model.

In this paper, we introduce a novel high-order tailored finite TFPS specifically crafted for solving the time-fractional Black–Scholes equation with high accuracy. Our approach differs from the compact finite difference methods referenced in [11,12,35], which rely on high-order Taylor series expansions, by starting with the exact solution of the local grid cell equation and constructing linear relationships among nodes based on their positional arrangements. This method inherently aligns with the exact solution, enabling the development of a finite difference scheme with high-order accuracy. We introduce two key innovations: first, the development of a TFPS that uses exponential basis functions for spatial discretization and an L1-discretization for time-stepping, which we have shown to be unconditionally stable with fourth-order spatial accuracy and $(2-\alpha )$-order temporal accuracy through a thorough analysis; and second, an in-depth exploration of the impact of the fractional derivative order on option pricing, along with a discussion on the influence of different model parameters on option prices within the context of this model.

The structure of this paper is outlined as follows: Section 2 presents our TFPS, which achieves fourth-order accuracy in spatial discretization for the time-fractional B-S model. Section 3 delves deeply into the theoretical aspects, analyzing the stability and convergence of the proposed method. Section 4 presents a comprehensive set of numerical experiments, offering empirical evidence that demonstrates the efficiency and reliability of our method. This paper closes with a summary in Section 5.

2. TFPS for a Time-Fractional B-S Model

For the numerical resolution of model (6), we restrict the unbounded domain of the variable x to a finite interval $L=[a,b]$ and, without loss of generality, introduce a source term $f(x,t)$ on the right-hand side of the equation. Consequently, we obtain

$$\left\{\begin{array}{c}{\displaystyle {}_0{}^{C}D_t^{\alpha }u(x,t)=\frac{{\sigma }^2}{2}\frac{{\partial }^{2}u(x,t)}{\partial x^2}+\left(r-\frac{{\sigma }^2}{2}\right)\frac{\partial u(x,t)}{\partial x}-ru(x,t)+f(x,t),(x,t)\in (a,b)\times (0,T),}\hfill \\ \\ u(a,t)={\varphi }_1\left(t\right),u(b,t)={\varphi }_2\left(t\right),t\in (0,T),\hfill \\ \\ u(x,0)=v\left(x\right),x\in (a,b).\hfill \end{array}\right.$$

(7)

2.1. The Main Idea of TFPS for Convection–Diffusion Equations

Our starting point consists of the convection–diffusion equations in their steady-state configuration, which can be expressed as follows:

$$\begin{array}{c}{\displaystyle Lu\equiv -u_{xx}+B\left(x\right)u_x=g.}\hfill \end{array}$$

(8)

Suppose the computational domain is $L=[a,b]$. Let $h=(b-a)/I$, $x_i=ih$ with $i=0,1,\cdots ,I$. On $L_i$, $B_i=B\left(x_i\right)$, Equation (8) can be approximated by

$$-u_{xx}+B_i{u}_x={\tilde{g}}_i,$$

(9)

where ${\tilde{g}}_i={\displaystyle \frac{1}{|L_i|}}{\int }_{L_i}g\left(x\right)dx$ is an approximation to ${g\left(x\right)|}_{L_i}$.

The framework of the TFPS

Let

$$u\left(x\right)={\displaystyle \frac{x}{B_i}}{\tilde{g}}_i+w\left(x\right)e^{{\displaystyle \frac{B_{i}x}{2}}}.$$

(10)

Then, w satisfies

$$-w_{xx}+{\displaystyle \frac{B_i^2}{4}}w=0.$$

(11)

In the subsequent part, we construct a discretization scheme for (11), and then we can obtain the TFPS for (9).

Step one: To build finite difference scheme for (11). Similarly to what has been carried out in [33,34], the exact solution of (11) can be expressed as the special form $w\left(x\right)=e^{\lambda x}$; then, $\lambda$ is the solution to

$$-{\lambda }^2+{\displaystyle \frac{B_i^2}{4}}=0.$$

(12)

We can obtain $\lambda =\pm {\displaystyle \frac{B_i}{2}}$. The general solutions of (11) belong to the linear space

$$\begin{array}{cc}\hfill W_i=& \left\{c_{1,i}{e}^{{\displaystyle \frac{B_i}{2}}(x-x_i)}+c_{2,i}{e}^{-{\displaystyle \frac{B_i}{2}}(x-x_i)},\forall c_{1,i},c_{2,i}\in R\right\}.\hfill \end{array}$$

(13)

The TFPS aims at devising a discrete representation of the variable w, structured as follows:

$${\kappa }_1{w}_{i+1}+{\kappa }_2{w}_{i-1}+{\kappa }_0{w}_i=0.$$

(14)

To accurately deduce the coefficients ${\kappa }_l$, $l=0,1,2$, we employ the two basis functions $e^{-{\lambda }_{i}x}$, $e^{{\lambda }_{i}x}$, which are specifically chosen to fulfill Equation (14) precisely. A simple calculation gives

$$\left\{\begin{array}{c}{\kappa }_1{e}^{-{\lambda }_{i}h}+{\kappa }_2{e}^{{\lambda }_{i}h}+{\kappa }_0=0,\hfill \\ {\kappa }_1{e}^{{\lambda }_{i}h}+{\kappa }_2{e}^{-{\lambda }_{i}h}+{\kappa }_0=0.\hfill \end{array}\right.$$

(15)

For any nonzero constant ${\kappa }_0\in \mathbb{R}$, ref. (15) has the unique solution

$${\kappa }_1={\kappa }_2=-{\displaystyle \frac{{\kappa }_0}{e^{{\lambda }_{i}h}+e^{-{\lambda }_{i}h}}}.$$

(16)

Thus, the discretized scheme derived from (11) can be formulated as

$$w_i-{\displaystyle \frac{1}{e^{{\lambda }_{i}h}+e^{-{\lambda }_{i}h}}}(w_{i+1}+w_{i-1})=0.$$

(17)

Step two: To build TFPS for (9).

Substituting Equation (10) into Equation (17) results in the following expression

$$\begin{array}{c}{\displaystyle u_i-\frac{e^{-{\lambda }_{i}h}{u}_{i+1}+e^{{\lambda }_{i}h}{u}_{i-1}}{e^{{\lambda }_{i}h}+e^{-{\lambda }_{i}h}}=\frac{\frac{h}{B_i}(e^{{\lambda }_{i}h}-e^{-{\lambda }_{i}h})}{e^{{\lambda }_{i}h}+e^{-{\lambda }_{i}h}}{\tilde{g}}_i.}\hfill \end{array}$$

(18)

By replacing ${\displaystyle \frac{h}{B_i}}(e^{{\lambda }_{i}h}-e^{-{\lambda }_{i}h})$ with $h^2$, we can obtain the TFPS for (9)

$$\begin{array}{c}{\displaystyle \frac{2\cosh \left({\lambda }_{i}h\right)u_i-e^{-{\lambda }_{i}h}{u}_{i+1}-e^{{\lambda }_{i}h}{u}_{i-1}}{h^2}={\tilde{g}}_i.}\hfill \end{array}$$

(19)

Applying expansion methods where exponentials are expanded in terms of their argument, we obtain a modified differential equation that corresponds to the aforementioned finite difference scheme and is given as

$$\begin{array}{c}\hfill -u_{xx}+B_i{u}_x={\tilde{g}}_i+2E_x{h}^2+O\left(h^4\right),\end{array}$$

(20)

where

$$\begin{array}{cc}\hfill E_x=& -{\displaystyle \frac{1}{3!}}{\left({\displaystyle \frac{B_i}{2}}\right)}^3{u}_x{|}_{x=x_i}+{\displaystyle \frac{1}{2!2!}}{\left({\displaystyle \frac{B_i}{2}}\right)}^2{u}_{xx}{|}_{x=x_i}-{\displaystyle \frac{1}{3!}}\left({\displaystyle \frac{B_i}{2}}\right)u_{3x}{|}_{x=x_i}+{\displaystyle \frac{1}{4!}}{u}_{4x}{|}_{x=x_i},\hfill \end{array}$$

(21)

indicating that the accuracy of this scheme is of the second order.

2.2. Fourth-Order TFPS for Convection–Diffusion Equations

The scheme (19) clearly exhibits diagonal dominance. As the most straightforward instance within the TFPS, this $h^2$-accurate scheme serves as our foundation. Our objective is to make subtle refinements to the core Formula (19) in order to enhance its accuracy to the fourth order. Considering that Formula (20) deviates from the baseline Equation (9) by a mere second-order small quantity, we propose that the fourth-order formulation maintains the same structure as scheme (19), with the sole distinction being the inclusion of second-order corrections to the convective coefficient and the source term. We have

$$\begin{array}{cc}& {\displaystyle \frac{2}{h^2}}\cosh \left({\lambda }_{pi}h\right)u_i={\displaystyle \frac{1}{h^2}}\left(e^{-{\lambda }_{pi}h}{u}_{i+1}+e^{{\lambda }_{pi}h}{u}_{i-1}\right)+{\tilde{g}}_{pi},\hfill \end{array}$$

(22)

where

$$\begin{array}{ccc}& {\lambda }_{pi}={\lambda }_i+\Delta B_i·h^2\hfill & \hfill {\tilde{g}}_{pi}={\tilde{g}}_i+2\Delta g_i·h^2.\end{array}$$

Define the modified coefficient $B_{pi}=B_i+2\Delta B_i·h^2$. This is used in an equivalent discrete differential equation

$$-u_{xx}+B_{pi}{u}_x={\tilde{g}}_{pi}+O\left(h^4\right).$$

(23)

It can be seen that an $h^4$ accuracy is achieved if only

$$\Delta B_i{u}_x=\Delta g_i+E_x+O\left(h^2\right).$$

(24)

From (20), we have

$$\begin{array}{cc}& u_{xx}{|}_{x=x_i} =B_i{u}_x{|}_{x=x_i}-{\tilde{g}}_i+O\left(h^2\right),\hfill \\ & u_{3x}{|}_{x=x_i} =(B_i^2+B_{ix})u_x{|}_{x=x_i}-B_i{\tilde{g}}_i-{\tilde{g}}_{ix}+O\left(h^2\right),\hfill \\ & u_{4x}{|}_{x=x_i} =(B_{ixx}+3B_i{B}_{ix}+B_i^3)u_x{|}_{x=x_i}-(B_i^2+2B_{ix}){\tilde{g}}_i-B{\tilde{g}}_{ix}-{\tilde{g}}_{ixx}+O\left(h^2\right).\hfill \end{array}$$

(25)

By substituting the system represented by Equation (25) into Equation (23), we derive expressions for the perturbation values

$$\begin{array}{c}\hfill \Delta B_i={\displaystyle \frac{1}{24}}\left(B_i{B}_{ix}+B_{ixx}\right)+O\left(h^2\right),\\ \hfill \Delta g_i={\displaystyle \frac{1}{24}}\left[\left({\displaystyle \frac{B_i^2}{2}}+B_{ix}\right){\tilde{g}}_i-B_i{\tilde{g}}_{ix}+{\tilde{g}}_{ixx}\right]+O\left(h^2\right).\end{array}$$

(26)

To calculate the derivatives in Equation (26), we utilize central difference schemes of second-order precision

$$\begin{array}{cc}\hfill {\left(\Delta B\right)}_i=& {\displaystyle \frac{1}{24h^2}}\left[\left(1-{\displaystyle \frac{B_i}{2}}h\right)B_{i-1}-2B_i\right.\left.+\left(1+{\displaystyle \frac{B_i}{2}}h\right)B_{i+1}\right],\hfill \\ \hfill {\left(\Delta g\right)}_i=& {\displaystyle \frac{1}{24h^2}}\left\{\left(1+{\displaystyle \frac{B_i}{2}}h\right){\tilde{g}}_{i-1}+\left[-2+{\displaystyle \frac{B_i^2}{2}}{h}^2\right.\right.\left.\left.+\left({\displaystyle \frac{B_{i+1}-B_{i-1}}{2}}\right)h\right]{\tilde{g}}_i+\left(1-{\displaystyle \frac{B_i}{2}}h\right){\tilde{g}}_{i+1}\right\}.\hfill \end{array}$$

(27)

An $h^4$-accurate TFPS can be derived for Equation (8)

$$\begin{array}{cc}& {\displaystyle \frac{2}{h^2}}\cosh \left({\lambda }_{pi}h\right)u_i-{\displaystyle \frac{1}{h^2}}\left(e^{-{\lambda }_{pi}h}{u}_{i+1}-e^{{\lambda }_{pi}h}{u}_{i-1}\right)\hfill \\ & =\left(1+{\displaystyle \frac{1}{12}}\left[-2+{\displaystyle \frac{B_i^2}{2}}{h}^2+\left({\displaystyle \frac{B_{i+1}-B_{i-1}}{2}}\right)h\right]\right){\tilde{g}}_i\hfill \\ & + {\displaystyle \frac{1}{12}}\left(1+{\displaystyle \frac{B_i}{2}}h\right){\tilde{g}}_{i-1}+{\displaystyle \frac{1}{12}}\left(1-{\displaystyle \frac{B_i}{2}}h\right){\tilde{g}}_{i+1}.\hfill \end{array}$$

(28)

2.3. Temporal Discretization

Now, we proceed to address the numerical resolution of the time-fractional problem presented in system (7). To accomplish this, our initial step involves transforming (7) into the form of (8). The equivalent formulation for (7) reads as follows:

$$-u_{xx}+(1-{\displaystyle \frac{2r}{{\sigma }^2}})u_x={\displaystyle \frac{2}{{\sigma }^2}}\left(f-{}_0{}^{C}D_t^{\alpha }u-ru\right).$$

(29)

Let $B\left(x\right)=1-{\displaystyle \frac{2r}{{\sigma }^2}}$, and replace g in (29) by ${\displaystyle \frac{2}{{\sigma }^2}}\left(f-{}_0{}^{C}D_t^{\alpha }u-ru\right)$, then we can adopt the scheme to give the approximation scheme for (29). Given a grid $t_n=n\Delta t$, $n=0,1,2,\cdots ,N$, with $\Delta t={\displaystyle \frac{T}{N}}$, and N is a positive integer. The values of $u(x_i,t)$ at grid point $t_n$ are denoted by $u_i^n=u(x_i,t_n)$. Therefore, we obtain

$$\begin{array}{cc}& {\displaystyle \frac{2}{h^2}}\cosh \left({\lambda }_{pi}h\right)u_i^n-{\displaystyle \frac{1}{h^2}}\left(e^{-{\lambda }_{pi}h}{u}_{i+1}^n-e^{{\lambda }_{pi}h}{u}_{i-1}^n\right)\hfill \\ & =\left(1+{\displaystyle \frac{1}{12}}\left[-2+{\displaystyle \frac{B_i^2}{2}}{h}^2+\left({\displaystyle \frac{B_{i+1}-B_{i-1}}{2}}\right)h\right]\right){\left[{\displaystyle \frac{2}{{\sigma }^2}}\left(f-{}_0{}^{C}D_t^{\alpha }u-ru\right)\right]}_i^n\hfill \\ & + {\displaystyle \frac{1}{12}}\left(1+{\displaystyle \frac{B_i}{2}}h\right){\left[{\displaystyle \frac{2}{{\sigma }^2}}\left(f-{}_0{}^{C}D_t^{\alpha }u-ru\right)\right]}_{i-1}^n\hfill \\ & + {\displaystyle \frac{1}{12}}\left(1-{\displaystyle \frac{B_i}{2}}h\right){\left[{\displaystyle \frac{2}{{\sigma }^2}}\left(f-{}_0{}^{C}D_t^{\alpha }u-ru\right)\right]}_{i+1}^n+O\left(h^4\right).\hfill \end{array}$$

(30)

The Caputo fractional time derivative present in (30) is approximated via the L1 scheme,

$${}_0{}^{C}D_t^{\alpha }u\left(x_i,t_n\right)={\displaystyle \frac{\Delta t^{-\alpha }}{\Gamma (2-\alpha )}}\left[{\omega }_0^{\alpha }{u}_i^n-\sum _{j=1}^{n-1}\left({\omega }_{n-j-1}^{\alpha }-{\omega }_{n-j}^{\alpha }\right)u_i^j-{\omega }_{n-1}^{\alpha }{u}_i^0\right]+O\left(\Delta t^{2-\alpha }\right).$$

(31)

Lemma 1

([21,26]). Suppose $y\left(t\right)\in C^2\left[0,t_k\right]$. Let

$$\begin{array}{cc}\hfill \overline{R}\left(y\left(t_k\right)\right):=& {\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_0^{t_k}{\displaystyle \frac{y^{\prime }\left(s\right)}{{\left(t_k-s\right)}^{\alpha }}}ds\hfill \\ & - {\displaystyle \frac{\Delta t^{-\alpha }}{\Gamma (2-\alpha )}}\left[{\omega }_0^{\alpha }y\left(t_k\right)-\sum _{j=1}^{k-1}\left({\omega }_{k-j-1}^{\alpha }-{\omega }_{k-j}^{\alpha }\right)y\left(t_j\right)-{\omega }_{k-1}^{\alpha }y\left(t_0\right)\right],\hfill \end{array}$$

(32)

then

$$\left|\overline{R}\left(y\left(t_k\right)\right)\right|\le {\displaystyle \frac{1}{\Gamma (2-\alpha )}}\left[{\displaystyle \frac{1-\alpha }{12}}+{\displaystyle \frac{2^{2-\alpha }}{2-\alpha }}-\left(1+2^{-\alpha }\right)\right]\underset{0\le t\le t_k}{\max }\left|y^{″}\left(t\right)\right|\Delta t^{2-\alpha },$$

where $\alpha \in (0,1)$ and ${\omega }_j^{\alpha }={(j+1)}^{1-\alpha }-j^{1-\alpha }$.

Lemma 2

([30]). For the coefficients {${\omega }_k^{\alpha }$} defined in Equation (31), it holds that

$$\begin{array}{cc}& {\omega }_k^{\alpha }>0,k=0,1,...,n,\hfill \\ & 1={\omega }_0^{\alpha }>{\omega }_1^{\alpha }>\cdots >{\omega }_n^{\alpha },{\omega }_n^{\alpha }\to 0,asn\to \infty ,\hfill \\ & \sum _{k=1}^{n-1}({\omega }_{n-k-1}^{\alpha }-{\omega }_{n-k}^{\alpha })+{\omega }_{n-1}^{\alpha }=1.\hfill \end{array}$$

(33)

Lemma 3.

Let $\Delta t={\displaystyle \frac{T}{N}}$, $t_n=n\Delta t$ $\left(n=1,2,\cdots ,N\right)$; then, we have

$$\begin{array}{c}{\displaystyle \frac{\Delta t^{\alpha }\Gamma (2-\alpha )}{{\omega }_{n-1}^{\alpha }}\le {\left(t_n\right)}^{\alpha }\Gamma (1-\alpha ).}\hfill \end{array}$$

(34)

Proof. 

Assuming the function $g\left(x\right)$ is defined as $g\left(x\right)=x^{1-\alpha }$, $x\ge 1$; through the application of the Mean Value Theorem for differentiation, we deduce that

$$n^{1-\alpha }-{(n-1)}^{1-\alpha }\ge (1-\alpha )n^{-\alpha }.$$

Thus

$$\begin{array}{c}{\displaystyle \frac{\Delta t^{\alpha }\Gamma (2-\alpha )}{{\omega }_{n-1}^{\alpha }}\le \frac{\Delta t^{\alpha }\Gamma (2-\alpha )}{(1-\alpha )n^{-\alpha }}={(n\Delta t)}^{\alpha }\Gamma (1-\alpha )={\left(t_n\right)}^{\alpha }\Gamma (1-\alpha ).}\hfill \end{array}$$

(35)

□

2.4. Fourth Order TFPS for the Time-Fractional Black–Scholes Model

Denote $\Delta {\tilde{B}}_i=-2+{\displaystyle \frac{B_i^2}{2}}{h}^2+\left({\displaystyle \frac{B_{i+1}-B_{i-1}}{2}}\right)h$, $\tilde{\Gamma }={\displaystyle \frac{\Delta t^{-\alpha }}{\Gamma (2-\alpha )}}$. At this stage, we are equipped to outline the implicit version of the TFPS approach to (7). $U_i^n$ is the numerical solution of $u_i^n$, omitting the truncation errors $O(\Delta t^{2-\alpha }+h^4)$ in (30). To begin with, we proceed by solving for the numerical solutions $\left\{U_i^1\right\}$ through the following steps:

$$\begin{array}{cc}& -\left[{\displaystyle \frac{1}{h^2}}{e}^{{\lambda }_{pi}h}-{\displaystyle \frac{1}{12}}\left(1+{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i-1}^2}}\left(\tilde{\Gamma }+r_{i-1}\right)\right]U_{i-1}^1\hfill \\ & +\left[{\displaystyle \frac{2}{h^2}}\cosh \left({\lambda }_{pi}h\right)+\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_i\right){\displaystyle \frac{2}{{\sigma }_i^2}}\left(\tilde{\Gamma }+r_i\right)\right]U_i^1\hfill \\ & -\left[{\displaystyle \frac{1}{h^2}}{e}^{-{\lambda }_{pi}h}-{\displaystyle \frac{1}{12}}\left(1-{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i+1}^2}}\left(\tilde{\Gamma }+r_{i+1}\right)\right]U_{i+1}^1\hfill \\ & ={\displaystyle \frac{1}{12}}\left(1+{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i-1}^2}}\left(\tilde{\Gamma }{U}_{i-1}^0+f_{i-1}^1\right)+\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_i\right){\displaystyle \frac{2}{{\sigma }_i^2}}\left(\tilde{\Gamma }{U}_i^0+f_i^1\right)\hfill \\ & + {\displaystyle \frac{1}{12}}\left(1-{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i+1}^2}}\left(\tilde{\Gamma }{U}_{i+1}^0+f_{i+1}^1\right),1\le i\le I.\hfill \end{array}$$

(36)

Then, the numerical solutions $\left\{U_i^n\right\}$ are computed via the following procedure

$$\begin{array}{cc}& -\left[{\displaystyle \frac{1}{h^2}}{e}^{{\lambda }_{pi}h}-{\displaystyle \frac{1}{12}}\left(1+{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i-1}^2}}\left(\tilde{\Gamma }+r_{i-1}\right)\right]U_{i-1}^n\hfill \\ & +\left[{\displaystyle \frac{2}{h^2}}\cosh \left({\lambda }_{pi}h\right)+\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_i\right){\displaystyle \frac{2}{{\sigma }_i^2}}\left(\tilde{\Gamma }+r_i\right)\right]U_i^n\hfill \\ & -\left[{\displaystyle \frac{1}{h^2}}{e}^{-{\lambda }_{pi}h}-{\displaystyle \frac{1}{12}}\left(1-{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i+1}^2}}\left(\tilde{\Gamma }+r_{i+1}\right)\right]U_{i+1}^n\hfill \\ & ={\displaystyle \frac{1}{12}}\left(1+{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i-1}^2}}\left(\tilde{\Gamma }\sum _{l=1}^{n-1}\left({\omega }_{n-j-1}^{\alpha }-{\omega }_{n-j}^{\alpha }\right)u_{i-1}^j+\tilde{\Gamma }{\omega }_{n-1}^{\alpha }{U}_{i-1}^0+f_{i-1}^n\right)\hfill \\ & +\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_i\right){\displaystyle \frac{2}{{\sigma }_i^2}}\left(\tilde{\Gamma }\sum _{j=1}^{n-1}\left({\omega }_{n-j-1}^{\alpha }-{\omega }_{n-j}^{\alpha }\right)u_i^j+\tilde{\Gamma }{\omega }_{n-1}^{\alpha }{U}_i^0+f_i^n\right)\hfill \\ & + {\displaystyle \frac{1}{12}}\left(1-{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i+1}^2}}\left(\tilde{\Gamma }\sum _{l=1}^{n-1}\left({\omega }_{n-j-1}^{\alpha }-{\omega }_{n-j}^{\alpha }\right)u_{i+1}^j+\tilde{\Gamma }{\omega }_{n-1}^{\alpha }{U}_{i+1}^0+f_{i+1}^n\right)1\le i\le I,\hfill \end{array}$$

(37)

with the initial and boundary conditions

$$\begin{array}{cc}& U_i^0=v\left(x_i\right),i=0,1,\cdots ,I,\hfill \\ & U_0^n={\varphi }_1\left(t_n\right),U_I^n={\varphi }_2\left(t_n\right),n=1,2,\cdots ,N.\hfill \end{array}$$

Theorem 1.

The TFPS (36) and (37) has a unique solution for sufficiently small h.

Proof. 

As the TFPS scheme embodied by Equations (36) and (37) constitutes a linear method, it is necessary to verify that its coefficient matrix exhibits diagonal dominance as $h\to 0$. Given that the functions $B\left(x\right)$, $\sigma \left(x\right)$, $r\left(x\right)$ are all continuous, and considering $h\to 0$, the distinction between subscripts ($i-1$, i or $i+1$) can be disregarded for our purposes. Let us assume that ${\lambda }_{pi}={\displaystyle \frac{B_i}{2}}$. By applying Taylor expansions to the exponential functions $e^{{\lambda }_{pi}h}$, $e^{-{\lambda }_{pi}h}$, we obtain the following insights:

$$K_{i-1}=\left[{\displaystyle \frac{1}{h^2}}{e}^{{\lambda }_{pi}h}-{\displaystyle \frac{1}{12}}\left(1+{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i-1}^2}}\left(\tilde{\Gamma }+r_{i-1}\right)\right]>0,$$

$$K_{i+1}=\left[{\displaystyle \frac{1}{h^2}}{e}^{{\lambda }_{pi}h}-{\displaystyle \frac{1}{12}}\left(1-{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i-1}^2}}\left(\tilde{\Gamma }+r_{i-1}\right)\right]>0,$$

$$K_i=\left[{\displaystyle \frac{2}{h^2}}\cosh \left({\lambda }_{pi}h\right)+\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_i\right){\displaystyle \frac{2}{{\sigma }_i^2}}\left(\tilde{\Gamma }+r_i\right)\right]>0.$$

Then

$$\begin{array}{cc}\hfill |K_{i-1}| + |K_{i-1}|& =K_{i-1}+K_{i+1}\hfill \\ & ={\displaystyle \frac{2}{h^2}}\cosh \left({\lambda }_{pi}h\right)-{\displaystyle \frac{2}{12}}{\displaystyle \frac{2}{{\sigma }_i^2}}\left(\tilde{\Gamma }+r_i\right)\hfill \\ & <{\displaystyle \frac{2}{h^2}}\cosh \left({\lambda }_{pi}h\right)+\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_i\right){\displaystyle \frac{2}{{\sigma }_i^2}}\left(\tilde{\Gamma }+r_i\right)=\left|K_i\right|.\hfill \end{array}$$

Consequently, due to the strict diagonal dominance of the coefficient matrix A, we can confirm the existence and uniqueness of the solution in the fourth-order TFPS. □

Remark 1.

The TFPS (36) and (37) maintains the three-point stencil and ensures the strict diagonal dominance of the coefficient matrix A. To solve the linear equations associated with a tridiagonal matrix, several rapid computational methods can be utilized. Among these, the Thomas algorithm is a direct and efficient method for solving tridiagonal systems.

3. Theoretical Analysis of the Fourth-Order TFPS

3.1. The Local Truncation Error

Denote

$$\begin{array}{cc}& {\displaystyle \frac{2}{h^2}}\cosh \left({\lambda }_{pi}h\right)u_i^n-{\displaystyle \frac{1}{h^2}}\left(e^{-{\lambda }_{pi}h}{u}_{i+1}^n-e^{{\lambda }_{pi}h}{u}_{i-1}^n\right)\hfill \\ & =-{\displaystyle \frac{u_{i-1}^n-2u_i^n+u_{i+1}^n}{h^2}}+{\displaystyle \frac{(1-e^{-{\lambda }_{pi}h})(u_{i+1}^n-u_i^n)+(e^{{\lambda }_{pi}}-1)(u_i^n-u_{i-1}^n))}{h^2}}\hfill \\ & =-{\delta }_x^2{u}_i^n+B_i{\delta }_x{u}_i^n.\hfill \end{array}$$

Utilizing Taylor series expansions, we can demonstrate that for the genuine solution $u(x,t)$ of the system comprised of system (36) and (37), which is assumed to be sufficiently smooth, the following relationship holds true:

$${\left[I+{\displaystyle \frac{1}{12}}\left(\left({\displaystyle \frac{B_i^2}{2}}+B_{ix}\right)-B_i{\delta }_x+{\delta }_x^2\right)\right]}^{-1}\left(B_i{\delta }_x-{\delta }_x^2\right)u_i^n={\left[{\displaystyle \frac{2}{{\sigma }^2}}\left(f-{}_0{}^{C}D_t^{\alpha }u-ru\right)\right]}_i^n+O\left(h^4\right).$$

Applying Lemma 1, we obtain the following result:

$$\begin{array}{cc}& {\left[I+{\displaystyle \frac{1}{12}}\left(\left({\displaystyle \frac{B_i^2}{2}}+B_{ix}\right)-B_i{\delta }_x+{\delta }_x^2\right)\right]}^{-1}\left(B_i{\delta }_x-{\delta }_x^2\right)u_i^n\hfill \\ & -{\displaystyle \frac{2}{{\sigma }_i}}\left\{f_i^n-{\displaystyle \frac{\Delta t^{-\alpha }}{\Gamma (2-\alpha )}}\left[{\omega }_0^n{u}_i^n-\sum _{j=1}^{n-1}\left({\omega }_{n-j-1}-{\omega }_{n-j}\right)u_i^j-{\omega }_{n-1}^n{u}_i^0\right]-r_i{u}_i^n\right\}\hfill \\ & =O\left(\Delta t^{2-\alpha }\right)+O\left(h^4\right).\hfill \end{array}$$

(38)

Thus, the local truncation error for the TFPS scheme defined by systems (36) and (37) is estimated to be

$$R_i^n=O\left(\Delta t^{2-\alpha }\right)+O\left(h^4\right).$$

This estimation is valid across the indices $1\le i\le I$ and $1\le n\le N$.

3.2. Stability Analysis for Constant Coefficients

In this section, we examine the stability of the TFPS through a Fourier analysis approach. Initially, we introduce a lemma that will play a crucial role in our subsequent discussion.

Lemma 4

(Discrete Gronwall’s inequality [19]). Assume that y is a non-negative constant and $a_n$ and $b_n$ sequences. If the inequality

$$a_n\le y+\sum _{0\le k<n}{b}_k{a}_k,n\ge 0,$$

holds, then it follows that

$$a_n\le y\exp \left(\sum _{0\le j<n}{b}_j\right),n\ge 0.$$

For the current stability analysis based on the Fourier method, we restrict ourselves to equations with constant coefficients. Considering that B, $\sigma$ and r are all constants at this point, we remove their indices n in the coefficient. Let ${\overline{U}}_i^n$ denote the solution of (36) and (37) with an initial condition $U_i^0$. Define the perturbation term as

$${\epsilon }_i^n=U_i^n-{\overline{U}}_i^n,1\le j\le I,0\le n\le N,$$

then, we have

$$\begin{array}{cc}& -\left[{\displaystyle \frac{1}{h^2}}{e}^{{\lambda }_{pi}h}-{\displaystyle \frac{1}{12}}\left(1+{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i-1}^2}}\left(\tilde{\Gamma }+r_{i-1}\right)\right]{\epsilon }_{i-1}^n\hfill \\ & +\left[{\displaystyle \frac{2}{h^2}}\cosh \left({\lambda }_{pi}h\right)+\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_i\right){\displaystyle \frac{2}{{\sigma }_i^2}}\left(\tilde{\Gamma }+r_i\right)\right]{\epsilon }_i^n\hfill \\ & -\left[{\displaystyle \frac{1}{h^2}}{e}^{-{\lambda }_{pi}h}-{\displaystyle \frac{1}{12}}\left(1-{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i+1}^2}}\left(\tilde{\Gamma }+r_{i+1}\right)\right]{\epsilon }_{i+1}^n\hfill \\ & ={\displaystyle \frac{1}{12}}\left(1+{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i-1}^2}}\left(\tilde{\Gamma }\sum _{l=1}^{n-1}\left({\omega }_{n-j-1}^{\alpha }-{\omega }_{n-j}^{\alpha }\right){\epsilon }_{i-1}^j+\tilde{\Gamma }{\omega }_{n-1}^{\alpha }{\epsilon }_{i-1}^0\right)\hfill \\ & +\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_i\right){\displaystyle \frac{2}{{\sigma }_i^2}}\left(\tilde{\Gamma }\sum _{j=1}^{n-1}\left({\omega }_{n-j-1}^{\alpha }-{\omega }_{n-j}^{\alpha }\right){\epsilon }_i^j+\tilde{\Gamma }{\omega }_{n-1}^{\alpha }{\epsilon }_i^0\right)\hfill \\ & +{\displaystyle \frac{1}{12}}\left(1-{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i+1}^2}}\left(\tilde{\Gamma }\sum _{l=1}^{n-1}\left({\omega }_{n-j-1}^{\alpha }-{\omega }_{n-j}^{\alpha }\right){\epsilon }_{i+1}^j+\tilde{\Gamma }{\omega }_{n-1}^{\alpha }{\epsilon }_{i+1}^0\right)1\le i\le I.\hfill \end{array}$$

(39)

We interpolate ${\epsilon }_i^n$ from the discrete mesh point $x_i$ onto the control volume ($x_i-{\displaystyle \frac{1}{2}}h$, $x_i+{\displaystyle \frac{1}{2}}h)$ using piecewise constant functions, thereby yielding a continuous function ${\epsilon }^n\left(x\right)$ defined over the interval $\left(a,b\right)$. Since we are dealing with Dirichlet boundary conditions, we have ${\epsilon }^n\left(x\right)=0$ at the boundaries; hence, we can extend it periodically throughout the entire domain. Consequently, the Fourier series representation of ${\epsilon }^n\left(x\right)$ is given by

$${\epsilon }^n\left(x\right)={\displaystyle \frac{1}{\sqrt{b-a}}}\sum _{m=-\infty }^{\infty }{\rho }^n\left(m\right)e^{i2\pi m\left(x-a\right)/\left(b-a\right)}.$$

This series allows us to express the perturbation function ${\epsilon }^n\left(x\right)$ in terms of its frequency components ${\rho }^n\left(m\right)$ across the periodic domain. For any vector $w={\left(w_1,w_2,\cdots ,w_I\right)}^T\in R^I$, we define the discrete $l^2$ norm as $\parallel w\parallel ={\left(h{\sum }_{j=1}^I{w}_j^2\right)}^{1/2}$, and with this,

Lemma 5.

The discrete Fourier coefficients are

$${\rho }^n\left(m\right)={\displaystyle \frac{1}{\sqrt{b-a}}}{\int }_a^b{e}^{-i2\pi m\left[\left(\xi -a\right)/\left(b-a\right)\right]}{\epsilon }^n\left(\xi \right)d\xi .$$

(40)

Furthermore, we can apply the discrete version of Parseval’s identity for the Fourier transform, which states that

$${\int }_a^b{\left|{\epsilon }^n\left(\xi \right)\right|}^{2}d\xi =h\sum _{i=1}^I{\left|{\epsilon }_i^n\right|}^2=\sum _{m=-\infty }^{\infty }{\left|{\rho }^n\left(m\right)\right|}^2.$$

In order to investigate the stability of our scheme relative to initial perturbations, we may introduce the simplifying expression ${\epsilon }_i^n={\displaystyle \frac{1}{\sqrt{b-a}}}{\rho }^n{e}^{i\varpi ih}$, where $\varpi =2\pi /\left(b-a\right)$, and subsequently substitute this into Equation (39). This leads us to derive

$$\begin{array}{cc}& \{\left[{\displaystyle \frac{2}{h^2}}\cosh \left({\lambda }_{pi}h\right)+\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_i\right){\displaystyle \frac{2}{{\sigma }_i^2}}\left(\tilde{\Gamma }+r_i\right)\right]-\left[{\displaystyle \frac{2}{h^2}}\cosh \left({\lambda }_{pi}h\right)-{\displaystyle \frac{1}{6}}{\displaystyle \frac{2}{{\sigma }_i^2}}\left(\tilde{\Gamma }+r_i\right)\right]\cos \left(\varpi h\right)\hfill \\ & + i\left[{\displaystyle \frac{2}{h^2}}\sinh \left({\lambda }_{pi}h\right)-{\displaystyle \frac{1}{12}}{\displaystyle \frac{2}{{\sigma }_i^2}}{B}_{i}h\left(\tilde{\Gamma }+r_i\right)\right]\sin \left(\varpi h\right)\}{\rho }^n\hfill \\ & =\left[\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_i\right){\displaystyle \frac{2}{{\sigma }_i^2}}+{\displaystyle \frac{1}{6}}{\displaystyle \frac{2}{{\sigma }_i^2}}\cos \left(\varpi h\right)-i{\displaystyle \frac{1}{12}}{\displaystyle \frac{2}{{\sigma }_i^2}}{B}_{i}h\sin \left(\varpi h\right)\right]\hfill \\ & ·\left[\tilde{\Gamma }\sum _{j=1}^{n-1}\left({\omega }_{n-j-1}^{\alpha }-{\omega }_{n-j}^{\alpha }\right){\rho }^j+\tilde{\Gamma }{\omega }_{n-1}^0{\rho }^0\right],1\le i\le I,1\le n\le N.\hfill \end{array}$$

(41)

To establish the unconditional stability of the scheme under any initial conditions, we must initially validate the subsequent lemma.

Lemma 6.

Let it be assumed that ${\rho }^n(1\le n\le N)$ is determined by the relation (40). Then, for any $\alpha \in (0,1)$, the following inequality holds

$$\left|{\rho }^n\right|\le \left|{\rho }^0\right|,n=1,2,\cdots ,N.$$

Proof. 

We shall substantiate that $\left|{\rho }^j\right|\le \left|{\rho }^0\right|(j=1,2,\cdots ,N)$ using mathematical induction. With reference to (41), we aim to demonstrate the validity of the following inequality:

$$\begin{array}{cc}& \left|\left[\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_i\right){\displaystyle \frac{2}{{\sigma }_i^2}}+{\displaystyle \frac{1}{6}}{\displaystyle \frac{2}{{\sigma }_i^2}}\cos \left(\varpi h\right)-i{\displaystyle \frac{1}{12}}{\displaystyle \frac{2}{{\sigma }_i^2}}{B}_{i}h\sin \left(\varpi h\right)\right]\tilde{\Gamma }\right|\hfill \\ & \le |\left[{\displaystyle \frac{2}{h^2}}\cosh \left({\lambda }_{pi}h\right)+\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_i\right){\displaystyle \frac{2}{{\sigma }_i^2}}\left(\tilde{\Gamma }+r_i\right)\right]-\left[{\displaystyle \frac{2}{h^2}}\cosh \left({\lambda }_{pi}h\right)-{\displaystyle \frac{1}{6}}{\displaystyle \frac{2}{{\sigma }_i^2}}\left(\tilde{\Gamma }+r_i\right)\right]\cos \left(\varpi h\right)\hfill \\ & + i\left[{\displaystyle \frac{2}{h^2}}\sinh \left({\lambda }_{pi}h\right)-{\displaystyle \frac{1}{12}}{\displaystyle \frac{2}{{\sigma }_i^2}}{B}_{i}h\left(\tilde{\Gamma }+r_i\right)\right]\sin \left(\varpi h\right)|.\hfill \end{array}$$

(42)

Denote ${\varpi }_1=\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_{i}h\right){\displaystyle \frac{2}{{\sigma }_i^2}}+{\displaystyle \frac{1}{6}}{\displaystyle \frac{2}{{\sigma }_i^2}}\cos \left(\varpi h\right)$, ${\varpi }_2$ = ${\displaystyle \frac{1}{12}}{\displaystyle \frac{2}{{\sigma }_i^2}}{B}_{i}h\sin \left(\varpi h\right)$, ${\varpi }_3$ = ${\displaystyle \frac{2}{h^2}}\cosh \left({\lambda }_{pi}h\right)$$(1-\cos (\theta \left)\right)$, ${\varpi }_4={\displaystyle \frac{2}{h^2}}\sinh \left({\lambda }_{pi}h\right)\sin \left(\theta \right)$. Assume that the inequality we are referring to is given by (42), which takes the form

$$\left|{\varpi }_1\tilde{\Gamma }-i{\varpi }_2\tilde{\Gamma }\right|\le \left|{\varpi }_3+{\varpi }_1(\tilde{\Gamma }+r_i)+i({\varpi }_4-{\varpi }_2(\tilde{\Gamma }+r_i))\right|$$

(43)

and note that $r_i\ge 0$. To prove (43), it is sufficient that we prove that

$${\varpi }_1{\varpi }_3\ge {\varpi }_2{\varpi }_4.$$

(44)

As (44) is equivalent to

$$\left(\cosh \left({\lambda }_{pi}h\right)\left[{\displaystyle \frac{5}{6}}+{\displaystyle \frac{B_i^2{h}^2}{24}}+{\displaystyle \frac{1}{6}}\cos \left(\varpi h\right)\right]\right)\ge \sinh \left({\lambda }_{pi}h\right){\displaystyle \frac{B_{i}h}{12}}(1+\cos \left(\varpi h\right)).$$

Note that

$$\left(\cosh \left({\lambda }_{pi}h\right)[{\displaystyle \frac{5}{6}}+{\displaystyle \frac{B_i^2{h}^2}{24}}+{\displaystyle \frac{1}{6}}\cos \left(\varpi h\right)]\right)\ge \cosh \left({\lambda }_{pi}h\right){\displaystyle \frac{B_{i}h}{3}},$$

and

$$\sinh \left({\lambda }_{pi}h\right){\displaystyle \frac{B_{i}h}{12}}(1+\cos \left(\varpi h\right))\le \sinh \left({\lambda }_{pi}h\right){\displaystyle \frac{B_{i}h}{6}}.$$

The given inequality

$$\sinh \left({\lambda }_{pi}h\right){\displaystyle \frac{B_{i}h}{6}}\le \cosh \left({\lambda }_{pi}h\right){\displaystyle \frac{B_{i}h}{3}}$$

naturally holds true as $h\to 0$. This, in turn, ensures the validity of (44). As a result, (42) is also proven true. It is easy to see that $\left|{\rho }^1\right|\le \left|{\rho }^0\right|$. Assume that $\left|{\rho }^m\right|\le \left|{\rho }^0\right|$ holds for $m=1,2,\cdots ,n-1$; then using (41) and Lemma 2, we can derive

$$\begin{array}{cc}\hfill \left|{\rho }^n\right|& \le \sum _{i=1}^{n-1}\left({\omega }_{n-j-1}^{\alpha }-{\omega }_{n-l}^{\alpha }\right)\left|{\rho }^j\right|+{\omega }_{n-1}^{\alpha }\left|{\rho }^0\right|\hfill \\ & \le \left[\sum _{l=1}^{n-1}\left({\omega }_{n-l-1}^{\alpha }-{\omega }_{n-1}^{\alpha }\right)+{\omega }_{n-1}^{\alpha }\right]\left|{\rho }^0\right|\hfill \\ & =\left|{\rho }^0\right|,\hfill \end{array}$$

and the mathematical induction is completed. □

Theorem 2.

The TFPS (36) and (37) is unconditionally stable with initial values for $\alpha \in (0,1)$.

Proof. 

Using Lemma 6, we obtain

$$\begin{array}{cc}\hfill {∥U^n-{\overline{U}}^n∥}^2& =h\sum _{i=1}^I{\left|{\epsilon }_i^n\right|}^2={\displaystyle \frac{h}{b-a}}\sum _{i=1}^I{\left|{\rho }^n{e}^{i\varpi ih}\right|}^2\hfill \\ & ={\displaystyle \frac{h}{b-a}}\sum _{i=1}^I{\left|{\rho }^n\right|}^2\le {\displaystyle \frac{h}{b-a}}\sum _{i=1}^I{\left|{\rho }^0\right|}^2={∥{\epsilon }^0∥}^2\hfill \\ & ={∥U^0-{\overline{U}}^0∥}^2,n=1,2,\cdots ,N,\hfill \end{array}$$

and the proof is finished. □

3.3. Convergence Analysis of TFPS

In this section, we delve into the investigation of the convergence properties of the TFPS. Let ${\overline{\epsilon }}_i^n$ be defined as

$${\overline{\epsilon }}_i^n=u_i^n-U_i^n,1\le i\le I,1\le n\le N.$$

(45)

By substituting Equation (45) in Equation (37), and noticing that ${\overline{\epsilon }}_i^0=0$, we have

$$\begin{array}{cc}& -\left[{\displaystyle \frac{1}{h^2}}{e}^{{\lambda }_{pi}h}-{\displaystyle \frac{1}{12}}\left(1+{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i-1}^2}}\left(\tilde{\Gamma }+r_{i-1}\right)\right]{\overline{\epsilon }}_{i-1}^n\hfill \\ & +\left[{\displaystyle \frac{2}{h^2}}\cosh \left({\lambda }_{pi}h\right)+\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_{i}h\right){\displaystyle \frac{2}{{\sigma }_i^2}}\left(\tilde{\Gamma }+r_i\right)\right]{\overline{\epsilon }}_i^n\hfill \\ & -\left[{\displaystyle \frac{1}{h^2}}{e}^{-{\lambda }_{pi}h}-{\displaystyle \frac{1}{12}}\left(1-{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i+1}^2}}\left(\tilde{\Gamma }+r_{i+1}\right)\right]{\overline{\epsilon }}_{i+1}^n\hfill \\ & ={\displaystyle \frac{1}{12}}\left(1+{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i-1}^2}}\left(\tilde{\Gamma }\sum _{l=1}^{n-1}\left({\omega }_{n-j-1}^{\alpha }-{\omega }_{n-j}^{\alpha }\right){\overline{\epsilon }}_{i-1}^j\right)\hfill \\ & +\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_{i}h\right){\displaystyle \frac{2}{{\sigma }_i^2}}\left(\tilde{\Gamma }\sum _{j=1}^{n-1}\left({\omega }_{n-j-1}^{\alpha }-{\omega }_{n-j}^{\alpha }\right){\overline{\epsilon }}_i^j\right)\hfill \\ & + {\displaystyle \frac{1}{12}}\left(1-{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i+1}^2}}\left(\tilde{\Gamma }\sum _{l=1}^{n-1}\left({\omega }_{n-j-1}^{\alpha }-{\omega }_{n-j}^{\alpha }\right){\overline{\epsilon }}_{i+1}^j\right)\hfill \\ & + {\displaystyle \frac{1}{12}}\left(1+{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i-1}^2}}\tilde{\Gamma }{R}_{i-1}^n+\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_{i}h\right){\displaystyle \frac{2}{{\sigma }_i^2}}\tilde{\Gamma }{R}_i^n\hfill \\ & + {\displaystyle \frac{1}{12}}\left(1-{\displaystyle \frac{B_i}{2}}h\right){\displaystyle \frac{2}{{\sigma }_{i+1}^2}}\tilde{\Gamma }{R}_{i+1}^{n}1\le i\le I.\hfill \end{array}$$

(46)

where $R_i^n=O\left(\Delta t^{2-\alpha }+h^4\right)$. In line with the analogous reasoning employed in the stability analysis, we proceed to express both ${\overline{\epsilon }}^n\left(x\right)$ and $R^n\left(x\right)$ in terms of their corresponding Fourier series expansions as follows:

$$\begin{array}{cc}& {\xi }^n\left(m\right)={\displaystyle \frac{1}{\sqrt{b-a}}}{\int }_a^b{e}^{-i2\pi m\left[\left(\xi -a\right)/\left(b-a\right)\right]}{\overline{\epsilon }}^n\left(x\right)dx,\hfill \\ & {\mu }^n\left(m\right)={\displaystyle \frac{1}{\sqrt{b-a}}}{\int }_a^b{e}^{-i2\pi m\left[\left(\xi -a\right)/\left(b-a\right)\right]}{R}^n\left(x\right)dx.\hfill \end{array}$$

(47)

Employing Parseval’s equality in conjunction with the definition of the $l^2$ norm, we arrive at

$$\begin{array}{cc}& {∥{\overline{\epsilon }}^n∥}_2={\left(\sum _{i=1}^{I-1}h{\left|{\overline{\epsilon }}_i^n\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}={\left(\sum _{i=-\infty }^{\infty }{\left|{\xi }^n\left(m\right)\right|}^2\right)}^{{\displaystyle \frac{1}{2}}},\hfill \\ & {\parallel R^n\parallel }_2={\left(\sum _{i=1}^{I-1}h{\left|R_i^n\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}={\left(\sum _{i=-\infty }^{\infty }{\left|{\mu }^n\left(m\right)\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

(48)

where

$${\overline{\epsilon }}^n={\left({\overline{\epsilon }}_1^n,{\overline{\epsilon }}_2^n,\dots ,{\overline{\epsilon }}_I^n\right)}^T,R^n={\left(R_1^n,R_2^n,\dots ,R_I^n\right)}^T.$$

Now, assume that

$${\overline{\epsilon }}_i^n={\displaystyle \frac{1}{\sqrt{b-a}}}{\xi }^n{e}^{i\varpi ih},R_i^n={\displaystyle \frac{1}{\sqrt{b-a}}}{\mu }^n{e}^{i\varpi ih},$$

(49)

where $\varpi =2\pi /\left(b-a\right)$. Inserting the expressions from Equation (49) into Equation (46), followed by a series of simplifications, we obtain

$$\begin{array}{cc}& \{\left[{\displaystyle \frac{2}{h^2}}\cosh \left({\lambda }_{pi}h\right)+\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_{i}h\right){\displaystyle \frac{2}{{\sigma }_i^2}}\left(\tilde{\Gamma }+r_i\right)\right]-\left[{\displaystyle \frac{2}{h^2}}\cosh \left({\lambda }_{pi}h\right)-{\displaystyle \frac{1}{6}}{\displaystyle \frac{2}{{\sigma }_i^2}}\left(\tilde{\Gamma }+r_i\right)\right]\cos \left(\varpi h\right)\hfill \\ & +i\left[{\displaystyle \frac{2}{h^2}}\sinh \left({\lambda }_{pi}h\right)-{\displaystyle \frac{1}{12}}{\displaystyle \frac{2}{{\sigma }_i^2}}{B}_{i}h\left(\tilde{\Gamma }+r_i\right)\right]\sin \left(\varpi h\right)\}{\xi }^n\hfill \\ & =\left[\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_{i}h\right){\displaystyle \frac{2}{{\sigma }_i^2}}+{\displaystyle \frac{1}{6}}{\displaystyle \frac{2}{{\sigma }_i^2}}\cos \left(\varpi h\right)-i{\displaystyle \frac{1}{12}}{\displaystyle \frac{2}{{\sigma }_i^2}}{B}_{i}h\sin \left(\varpi h\right)\right]·\left[\tilde{\Gamma }\sum _{j=1}^{n-1}\left({\omega }_{n-j-1}^{\alpha }-{\omega }_{n-j}^{\alpha }\right){\xi }^j\right]\hfill \\ & +\left[\left(1+{\displaystyle \frac{1}{12}}\Delta {\tilde{B}}_{i}h\right){\displaystyle \frac{2}{{\sigma }_i^2}}+{\displaystyle \frac{1}{6}}{\displaystyle \frac{2}{{\sigma }_i^2}}\cos \left(\varpi h\right)-i{\displaystyle \frac{1}{12}}{\displaystyle \frac{2}{{\sigma }_i^2}}{B}_{i}h\sin \left(\varpi h\right)\right]\tilde{\Gamma }{\mu }^{n}1\le i\le I,1\le n\le N.\hfill \end{array}$$

(50)

Lemma 7.

Let ${\xi }^n(n=1,2,\dots ,N)$ represent the solution corresponding to Equation (50). Consequently, there exists a positive constant K, such that

$$\left|{\xi }^n\right|\le K\left|{\mu }^1\right|,n=1,2,\dots ,N.$$

Proof. 

$\forall 1\le n\le N$ and by virtue of convergence of series (48), there exists a positive constant $K_n$, so that

$$\left|{\mu }^n\right|\le K_n\left|{\mu }^1\right|,n=1,\dots ,N.$$

(51)

Let $K^{\prime }=\max \left\{K_1,K_2,\dots ,K_N\right\}$. According to (50) and (51), we have

$$\left|{\xi }^n\right|\le \sum _{k=1}^{n-1}\left|{\omega }_{n-k-1}^{\left(\alpha \right)}-{\omega }_{n-k}^{\left(\alpha \right)}\right|\left|{\xi }^k\right|+K^{\prime }\left|{\mu }^1\right|,$$

where $\left|{\omega }_{n-k-1}^{\left(\alpha \right)}-{\omega }_{n-k}^{\left(\alpha \right)}\right|$ and $\left|{\xi }^k\right|$ are both non-negative. As a result of applying Lemma 4,

$$\begin{array}{cc}\hfill \left|{\xi }^n\right|& \le K^{\prime }\left|{\mu }^1\right|\exp \left(\sum _{k=1}^n\left|{\omega }_{m-k-1}^{\left(\alpha \right)}-{\omega }_{m-k}^{\left(\alpha \right)}\right|\right)\hfill \\ & \le K^{\prime }\left|{\mu }^1\right|\exp \left(1\right)\le K\left|{\mu }^1\right|,n=1,\dots ,N.\hfill \end{array}$$

This concludes the demonstration of the lemma. □

Theorem 3.

The TFPS is convergent in the $l^2$ norm, and the order of convergence is $O\left(\Delta t^{2-\alpha }+h^4\right)$.

Proof. 

Utilizing Lemma 7 alongside the expressions from Equation (48), we obtain the following

$$\begin{array}{cc}\hfill {∥{\overline{\epsilon }}^n∥}_2^2& =\sum _{i=1}^{I}h{\left|{\overline{\epsilon }}_i^n\right|}^2\hfill \\ & =h\sum _{i=1}^I{\left|{\xi }^n{e}^{i\left(\varpi ih\right)}\right|}^2=h\sum _{i=1}^I{\left|{\xi }^n\right|}^2\le Kh\sum _{i=1}^I{\left|{\mu }^1\right|}^2\hfill \\ & =Kh\sum _{i=1}^I{\left|{\mu }^1{e}^{i\left(\varpi ih\right)}\right|}^2=K{∥R^1∥}_2^2=K{\left(\Delta t^{2-\alpha }+h^4\right)}^2.\hfill \end{array}$$

This completes the proof. □

4. Numerical Example

In this section, we showcase the accuracy and convergence rate of TFPS introduced in Section 2 through two test cases featuring known exact solutions. Additionally, we deploy this scheme to compute the prices of various European options under the auspices of a time-fractional B-S model, which constitutes a particularly significant and intriguing model within financial markets. To test the order of convergence for the $l^2$ norm, we progressively refine the mesh by halving both the spatial step size h to $h/2$ and the time step size $\Delta t$ to $\Delta t/2^{{\displaystyle \frac{4}{2-\alpha }}}$. The number of time steps N is then determined by rounding the quantity $1/\Delta t$ to the nearest integer.

Example 1.

Consider the following time-fractional B-S model subject to homogeneous boundary conditions

$$\left\{\begin{array}{c}{\displaystyle {}_0{}^{C}D_t^{\alpha }u(x,t)-\frac{1}{2}{\sigma }^2\frac{{\partial }^{2}u(x,t)}{\partial x^2}+(\frac{1}{2}{\sigma }^2-r)\frac{\partial u(x,t)}{\partial x}+ru(x,t)=f(x,t),(x,t)\in (0,1)\times (0,1)}\hfill \\ \\ u(0,t)=0,u(1,t)=0,\hfill \\ \\ u(x,0)=x^4(1-x),\hfill \end{array}\right.$$

(52)

where the source term

$$f={\displaystyle \frac{6t^{3-\alpha }}{\Gamma (4-\alpha )}}{x}^4(1-x)-(t^3+1)\left[{\displaystyle \frac{1}{2}}{\sigma }^2(12x^2-20x)+(r-{\displaystyle \frac{1}{2}}{\sigma }^2)\left(4x^3-5x^4\right)-r{x}^4(1-x)\right]$$

is chosen so that the exact solution of (52) is $u=(t^3+1)x^4(1-x)$. Here we take the parameter values as $r=0.05,\sigma =0.25$.

Table 1. The $l^2$ errors of TFPS for different $\alpha$.

$\mathit{h}\setminus \mathit{\alpha }$	0.2	0.4	0.6	0.8
1/8	$3.4125\times {10}^{-5}$	$7.0396\times {10}^{-5}$	$1.4746\times {10}^{-4}$	$3.0195\times {10}^{-4}$
1/16	$2.2659\times {10}^{-6}$	$4.6722\times {10}^{-6}$	$9.4263\times {10}^{-6}$	$1.8636\times {10}^{-5}$
1/32	$1.4949\times {10}^{-7}$	$3.0227\times {10}^{-7}$	$5.9892\times {10}^{-7}$	$1.1727\times {10}^{-6}$
1/64	$9.7921\times {10}^{-9}$	$1.9339\times {10}^{-8}$	$3.7665\times {10}^{-8}$	$7.3760\times {10}^{-8}$
1/128	$6.0322\times {10}^{-10}$	$1.2181\times {10}^{-9}$	$2.3575\times {10}^{-9}$	$4.6090\times {10}^{-9}$

presents the errors in the $l^2$ norm for varying fractional order $\alpha$. The corresponding convergence rates are displayed in Figure 1. The observed convergence rates of the TFPS corroborate the findings put forth by Theorem 3. The numerical solutions $u_n(x,t)$ produced by the TFPS exhibit striking agreement with the corresponding exact solutions $u(x,t)$ as depicted in Figure 2.

Example 2.

Consider the following time-fractional B-S model subject to homogeneous boundary conditions

$$\left\{\begin{array}{c}{\displaystyle {}_0{}^{C}D_t^{\alpha }u(x,t)-\frac{1}{2}{\sigma }^2\frac{{\partial }^{2}u(x,t)}{\partial x^2}+(\frac{1}{2}{\sigma }^2-r)\frac{\partial u(x,t)}{\partial x}+ru(x,t)=f(x,t),(x,t)\in (0,1)\times (0,1)}\hfill \\ \\ u(0,\tau )={(t+1)}^2,u(1,\tau )=3{(t+1)}^2,\hfill \\ \\ u(x,0)=x^3+x^2+1.\hfill \end{array}\right.$$

(53)

The exact solution of (53) is chosen as $u(x,t)={(t+1)}^2\left(x^3+x^2+1\right)$, the source term is defined as

$$\begin{array}{c}{\displaystyle f=\left(\frac{2t^{2-\alpha }}{\Gamma (3-\alpha )}+\frac{2t^{1-\alpha }}{\Gamma (2-\alpha )}\right)\left(x^3+x^2+1\right)-}\hfill \\ {\displaystyle {(t+1)}^2\left[\frac{1}{2}{\sigma }^2(6x+2)+(r-\frac{1}{2}{\sigma }^2)\left(3x^2+2x\right)-r\left(x^3+x^2+1\right)\right].}\hfill \end{array}$$

(54)

Here, the parameters’ values are $r=0.5,\sigma =\sqrt{2}$.

Table 2. The $l^2$ errors of TFPS for different $\alpha$ values.

$\mathit{h}\setminus \mathit{\alpha }$	$0.2$	$0.4$	$0.6$	$0.8$
1/8	$4.2022\times {10}^{-4}$	$1.3499\times {10}^{-4}$	$3.2947\times {10}^{-3}$	$7.1945\times {10}^{-3}$
1/16	$3.1218\times {10}^{-5}$	$9.3103\times {10}^{-5}$	$2.1515\times {10}^{-4}$	$4.5596\times {10}^{-4}$
1/32	$2.1563\times {10}^{-6}$	$6.0613\times {10}^{-6}$	$1.3687\times {10}^{-5}$	$2.8758\times {10}^{-5}$
1/64	$1.4548\times {10}^{-7}$	$3.8865\times {10}^{-7}$	$8.6022\times {10}^{-7}$	$1.7922\times {10}^{-6}$
1/128	$9.0944\times {10}^{-9}$	$2.4479\times {10}^{-8}$	$5.4094\times {10}^{-8}$	$1.1118\times {10}^{-7}$

presents $l^2$ norm errors for different fractional orders $\alpha$, while Figure 3 showcases the corresponding convergence rates. These convergence rates obtained with the TFPS align consistently with the theoretical predictions established by Theorem 3. The observed convergence rates of the TFPS corroborate the findings put forth by Theorem 3. The excellent correspondence between the numerical solutions $u_n(x,t)$ generated by the TFPS and the exact solutions $u(x,t)$ is visually demonstrated in Figure 4.

Example 3.

In this example, we implement the TFPS to calculate the prices of several distinct European options within the framework of a time-fractional B-S model.

$$\left\{\begin{array}{c}{\displaystyle \frac{{\partial }^{\alpha }u(x,t)}{\partial t^{\alpha }}-\frac{1}{2}{\sigma }^2{x}^2\frac{{\partial }^{2}u(x,t)}{\partial x^2}-(r-D)S\frac{\partial u(x,t)}{\partial x}-ru(x,t)=0,(x,t)\in \left(B_d,B_u\right)\times (0,T)}\hfill \\ \\ {\displaystyle u\left(B_d,t\right)=p\left(t\right),C\left(B_u,t\right)=q\left(t\right),}\hfill \\ \\ u(x,0)=v\left(x\right).\hfill \end{array}\right.$$

(55)

Case 1: In the scenario where the payoff function is given by $v\left(x\right)=\max \{x-K,0\}$ and $p\left(t\right)=q\left(t\right)=0$, the model (55) describes a time-fractional B-S model for pricing a European double barrier knock-out call option. The option price curves for varying $\alpha$ are depicted in Figure 5a, with parameters $\sigma =0.45, r=0.03, T=1$ (year), $B_d=3, B_u=15, K=10$ and the dividend yield $D=0.01$. The parameters values are selected according to the paper [12,17]. The attributes of Figure 5 are similar to those described in [12,17] perfectly. Observing Figure 5a, we notice that the time-fractional B-S model generates lower option prices when S is beneath a critical threshold near the strike price K, thereby illustrating the presence of fat tails. Conversely, the model provides higher prices for in-the-money options $(x>K)$. Thus, compared to the standard B-S process, the time-fractional B-S Model more accurately captures the essence of sudden jumps or substantial movements in asset prices. The option price curves for varying $\sigma$ at $\alpha =0.2$ are depicted in Figure 5b. The model generates lower options prices when $(x<K)$ but provides higher options prices when $(x>K)$ at smaller $\sigma$.

Case 2: For the European call option, the initial and boundary conditions are set as $v\left(x\right)=\max \{x-K,0\},p\left(t\right)=D=0$ and $q\left(t\right)=B_u-K\exp (-r(T-t))$. The corresponding call option price curves for different $\alpha$ values at $t=1$ are illustrated in Figure 6a,b, with the parameters specified as $r=0.05,\sigma =0.25,B_d=0.1,B_u=100,K=50$ and $T=1$ (year).

Case 3: In parallel, the European put option is characterized by the conditions $v\left(x\right)=\max \{K-x,0\},q\left(t\right)=D=0$ and $p\left(t\right)=K\exp (-r(T-t\left)\right)$. The associated put option price curves for different $\alpha$ at $t=1$ are plotted in Figure 6c,d, again with the parameters $r=0.05,\sigma =0.25,B_d=0.1,B_u=100,K=50$ and $T=1$ (year).

Applying the TFPS, the numerical solution of case 2 and case 3 for different values of parameters are plotted in Figure 6, Figure 7 and Figure 8. The following points summarize the findings from the numerical solutions of call and put option prices under various conditions:

• Impact of Time-Fractional Derivatives: The numerical solution of the call option and put option price for varying $\alpha$ are depicted in Figure 6. Figure 6 illustrates how time-fractional derivatives notably affect option prices, particularly around the at-the-money region $S\approx K$. However, their impact diminishes for options that are either deeply out-of-the-money or deeply in-the-money.
• Volatility Influence: The effect of the stock price’s volatility on option prices when $\alpha =0.4r=0.05,K=50$, and $t=1$ for different $\sigma$ is shown in Figure 7a,b. Reflecting real-world dynamics, higher volatility leads to higher potential gains, as evidenced by the larger option prices when the stock price is close to the exercise price.
• Risk-Free Interest Rates: The role of risk-free interest rates on option pricing with $\alpha =0.4$, $\sigma =0.4$, $K=50$, and $t=1$ across different values of r (0.005, 0.05, 0.1, 0.2, 0.3, 0.4) is portrayed in Figure 7c,d. These figures indicate that higher interest rates tend to increase call option prices while decreasing put option prices.
• Strike Price Effect: The change in option price due to the strike price is presented in Figure 7e,f. This illustrates that higher strike prices result in lower call option values, whereas increasing strike prices lead to escalating put option prices.
• Expiration Date Impact:Figure 8 explores how the expiration date influences call and put option prices. When the stock price significantly exceeds the strike price, a call option featuring a shorter expiration period assumes greater value compared to an option with a longer tenure; conversely, a put option endowed with a longer expiration horizon is more beneficial. In instances where the stock price significantly undershoots the strike, a call option possessing a longer expiration and a put option with a shorter expiry prove more favorable.

All these empirical outcomes align with actual market behaviors, providing a strong basis for understanding and predicting option pricing dynamics in real-world financial markets.

5. Conclusions

In this study, we developed a TFPS for solving the time-fractional B-S model pertinent to European option pricing. The temporal variable was discretized via the L1 scheme, while the spatial variable was treated using the TFPS approach. The stability and convergence properties of the TFPS were thoroughly examined. Theoretical results established the stability of the proposed TFPS method. Furthermore, we demonstrated that the method exhibits $2-\alpha$-order accuracy with respect to the time dimension and fourth-order accuracy in the spatial domain.

Numerical experiments were performed to validate the effectiveness and accuracy of the novel technique. The results unequivocally demonstrated that our proposed numerical method excels in precisely pricing European options, with convergence rates aligning with theoretical expectations. As a practical demonstration, we applied our methodology to price three distinct categories of European options under the time-fractional Black–Scholes (B-S) model.

Given that the financial payoff function for options at the strike price is frequently nonsmooth, the Temporal Fractional Partial Differential Equation Solver (TFPS) may not attain the desired fourth-order spatial convergence. To mitigate this numerical limitation, we can employ piecewise uniform meshes and/or local mesh refinement techniques without compromising the efficiency of our rapid temporal discretization method. Addressing this challenge will be a key focus of our future research. Looking ahead, we anticipate extending the higher-order TFPS and parallel-in-time method [36] to multidimensional B-S models in the coming future.