The impact of the Chebyshev collocation method on solutions of the time-fractional Black–Scholes

This paper presents a numerical solution of the temporal-fractional Black–Scholes equation governing European options (TFBSE-EO) in the finite domain so that the temporal derivative is the Caputo fractional derivative. For this goal, we firstly use linear interpolation with the (2-α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2-\alpha)$$\end{document}-order in time. Then, the Chebyshev collocation method based on the second kind is used for approximating the spatial derivative terms. Applying the energy method, we prove unconditional stability and convergence order. The precision and efficiency of the presented scheme are illustrated in two examples.


Introduction
The Black-Scholes model (BSM) is a mathematical equation for pricing and options treaty. In special, the model computes the transformation over time of financial mechanisms (such as stocks or futures). It presumes that they will have a lognormal distribution of prices. Using this presumption and factoring in other great variables, the model concludes the value of a call option. In 1973, pricing options have experienced a lot of consideration that the first time Black and Scholes [1] and Merton [2] proposed BSM for them. Although this model is very favorite, it has some deficiencies like lacking the "volatility smile" [3] in the actual marketplaces. A strong tool for generalization of this model is the use of fractional derivatives and integrals due to its non-local nature [4]. The concept and applications of fractional calculus have greatly expanded over the nineteenth and twentieth centuries, and numerous authors have provided descriptions for fractional derivatives. One of the most popular fractional derivatives is the left and right Caputo derivative of any arbitrary order, real or complex, which is defined as respectively. In addition, taking → n , leads to lim →n a D t u(x, t) = n u(x,t) t n and lim →n t D b u(x, t) = n u(x,t) t n . There are many ways to solve fractional differential equations such as finite difference methods [5,6], finite element methods [7], finite volume methods [8], spectral methods [9,10], and meshless methods [11]. In recently, many authors applied the various methods for solving these equations [12][13][14].
The detection of the fractional form of the stochastic process led to the discovery of fractional calculus into financial theory. In 2000, a time-fractional BSM was presented by Wyss [15] for the first time. In this article, we consider the presented equation by Wyss, which is as follows with the following initial and boundary conditions where = 1 2 2 > 0 , = r − and r > 0 are the known constant and f(x, t) is the source term. When > 0 , = 0 , r ≠ 0 and > 0 , < 0 , r = 0 , the model (1) is a reaction-diffusion and time-fractional advection-diffusion model, respectively.
There is even more method for determining the analytical process as [16][17][18][19]. Because it is difficult to receive an accurate solution, different approaches to estimate them are presented in [20][21][22][23][24][25]. In recent years, a finite-difference schema in second-order accurate and an implicit finite difference model with first-order is used for solving TFBSE-EO in [26] and [27], respectively. In addition, an explicit-implicit numerical method is proposed by Bhowmik In 2014 to solve the partial integrodifferential equation [28] which is the base of the option pricing hypothesis. In [29], Chen evaluated to price American options with using a predictor-corrector. In 2016, Zhang proposed a discrete implicit numerical scheme for pricing American options [30].
The outline of this paper is organized as below: In Sect. 2, by using linear interpolation for the time variable we get the semi-discrete design of the TFBSE-EO. So, we will consider the stability and convergence analysis. In Sect. 3, we apply a Chebyshev collocation method of the second kind for approximating the spatial fractional derivative for obtaining the full scheme. Finally, the fourth section contains two numerical examples to show the efficiency of the method.

The investigated analysis of the temporal-discrete scheme
In this section, we first present some notations and define the functional space with the L 2 -norm, ‖u(x)‖ 2 = ⟨u(x), u(x)⟩ 1 2 , as following that has used in our paper.
where L 2 (Ω) is the space of measurable function with the square of Lebesgue integrable in Ω. It is necessary to prove the following lemma to find the convergence order and denote the unconditional stability of the semi-discrete scheme.
be the solution and the initial condition of the semi-discrete (6), respectively, then we have where C is a positive value.
Proof Multiplying relation (6) by ∈ H 2 Ω and then integrating, we conclude The second and third terms on the left side of the above relation can be written as ⟨ Proof Suppose the approximation solution and the initial condition of Eq. (6) be U M ∈ H 2 Ω and U 0 , respectively. Then, the error j = u j − U j , j = 0, 1, … , M satisfies as following Using the Lemma 2 and the above relationship, we get This completes the condition of unconditional stability. ◻ Theorem 2 Suppose U k and u k for k = 1, 2, … , M be the approximate solution and the exact of relation (6) and (1), respectively with initial condition U 0 = (x) . Then, we have the following error evaluation where C is a positive constant.
We can indeed inscribe the following error equation where R M ≤ CO( 2− ) . By considering Lemma 2 and ‖ 0 ‖ = 0 , we get Therefore, the theorem is proved. ◻

Approximation in space: the Chebyshev collocation of the second kind
In the current section, we get the full scheme of Eq. (6) by using the shifted Chebyshev polynomials of the second kind i (x), i = 0, 1, … , N . The SCPSK is defined by the analytical str ucture of the Jacobi polynomials i.e.
The closed-form fractional derivative for the −order is written as where ⌈ ⌉ is the ceiling of value and i,k is determined by Therefore, an expansion of u(x, t j ) around the space variable is determined only by using the first N + 1-terms of SCPSK in interval [0, 1] as below where {c j i } N i=0 is the unknown coefficients and define as follows [32] By applying the linearity properties of the fractional derivative and substituting Eq. (12) in (11), we get the following relation Substituting Eqs. (12) and (14) in (6)

Numerical investigation
The introduced procedure of the Chebyshev collocation of the second kind in this section has a simple implementation for approaching TFBSE-EO. The numerical results in this section confirmed our claim. We then compare the achieved option prices from our scheme with those of [33] and [25] that our method gives better results. In addition, the accuracy and efficiency of the developed method demonstrated that they are obtained by the relation so that E i+1 and E i are errors matching to mesh size 2M and M, respectively.  The results are demonstrated in Table 1 that hold up the theoretical results proved in Theorem 2, that is, a convergence order in time is O(2 − ) at T = 1 with N = 7 . In Table 2, we test the presented method for the computation of the European put option with the methods of [33] and [25]. We conclude that our results are better than others with a highly low space size. In Fig. 1, we plot the approximate solution (left panel) and absolute error (right panel) at T = 1 for N = 5 and M = 400.

Example 2
In this example, we consider the following TFBSE-EO with nonhomogeneous boundary conditions where the dependent parameters can be considered as = 1, = r − , r = 0.5 and = 0.7 . The source term f(x, t) is achieved from the exact solution u(x, t) = (t + 1) 2 (x 3 + x 2 + 1).
The results of this problem are presented in Tables 3, 4. In Table 3, we obtain a convergence order that hold up the theoretical results proved in Theorem 2. In Table 4, we list comparisons between the present method with compact finite difference method [33] and radial basis functions based on finite difference scheme [25] that computational results confirm that our method gives better results.

Conclusion
The TFBSE-EO can be considered as a generalization of the classical Black-Scholes model. Due to the "globality" characteristic of the model's fractional-order derivative, the numerical solutions of this model are more complicated than the integer-order model. Accordingly, in this Table 1 The L ∞ and L 2 errors and computational order with N = 7 for Example 1 at T = 1  Table 2 Comparison of error and order convergence with the methods of [33] and [25] for Example 1 at T = 1 Method of [33] Method of [25] The current method for N = 150 and = 0.7 for N = 150 and = 0.7 for N = 10 and = 0.7  article, a numerical model to solve TFBSE-EO is proposed. First, the time discretization via the linear interpolation with the (2 − )-order is described. Then, we explain how to obtain the approximated solution by using the Chebyshev collocation method based on the second kind. In addition, the unconditional stability of the time-discrete model is proved by using the energy method, and also it is shown that the convergence order of the time-discrete is O( 2− ) . To demonstrate the convergence order and precision of the numerical process, we have chosen two numerical examples with exact solutions so that the numerical result has denoted the accuracy of the current model.   Table 4 Comparison of error and order convergence with the methods of [33] and [25] for Example 2 at T = 1 Method of [33] Method of [25] The current method for N = 150 and = 0.7 for N = 150 and = 0.7 for N = 7 and = 0.7