Numerical solution of the time-fractional Black-Scholes equation for European double barrier option with time-dependent parameters under the CEV model

Barrier options are considered the most widely used financial derivatives, which are massively traded in the financial markets due to its cheaper price in comparison with other standard options. Also, these options are family-path-dependent options, because their value depends directly on the movement of the underlying asset value during the option contract. Because the accurate order differential equation is unable the effect of trend memory in financial market, in this paper, we consider the fractional order differential equation. In order that our problem is closer to the real market model, we assume that interest rate, dividend yield and volatility are as function. The main purpose of this paper is to determine the price of European double-knock-out barrier option under the time-fractional Black-Scholes model with a fractional order . In here, the underlying asset follows the constant elasticity of variance (CEV) model. Such problems do not have exact solution in closed form, so using a finite difference method we find a suitable numerical solution by introducing implicit difference scheme. In the continuation, we investigate unconditional stability and convergence the proposed scheme by using Fourier analysis. We finally show the efficiency of the proposed difference scheme and its numerical convergence order by mentioning some numerical examples. In addition, we study the effect of the important model parameters (,  and ) on long memory in form table and figure.