Analyzing the Black-Scholes equation with fractional coordinate derivatives using an implicit discrete design

An option is a financial contract or a derivative security entitling the owner to trade a certain quantity of a particular asset having a certain cost on or before a certain date. Therefore, in the last few years, not only mathematicians but also financial engineers have paid a great deal of attention to pricing options. Applying the fractal structure in the processes of stochasticity led to both fractional calculus (FC) and fractional partial differential equations (FPDEs) being associated with the stochastic models in financial theory. Thus, the beginning of the 20th century witnessed the use of stochastic processes to model the financial market. By studying the price behavior of assets, a model was presented, which is known as the Black-Scholes equation. The main focus of the present paper is the time-fractional Black-Scholes (TFB-S) model. The difficulty or impossibility of providing an analytical solution for the aforesaid equation has made numerical solutions more helpful or even the only option. In this work, using the Crank-Nicolson scheme, a numerical solution with an implicit discrete design is demonstrated. We use the Fourier analysis method to investigate the stability of the implicit discrete design and demonstrate that the proposed method is unconditionally stable. The truncation error is checked. We also show that the numerical scheme suggested to solve the TFBS model is convergent. This method is the second order in space and 2 – β order in time, where 0 < β < 1 is the order of the time-fractional derivative. Finally, the accuracy as well as the efficiency considered for the method are evaluated by providing three examples and comparing them with previous works. Finally, the method’s accuracy and efficiency are assessed through three examples, with results compared to previous studies. Additional advantages of the method include its high computational speed, ease of implementation, and the reliability of obtaining an approximate solution, supported by stability proof.