Numerically Pricing Nonlinear Time-Fractional Black–Scholes Equation with Time-Dependent Parameters Under Transaction Costs

One of the assumptions of the classical Black–Scholes (B–S) is that the market is frictionless. Also, the classical B–S model cannot show the memory effect of the stock price in the financial markets. Previously, Ankudinova and Ehrhardt (Comput Math Appl 56:799–812, 2008) priced a European option under the classical B–S model with transaction costs when dividends are paid on assets during that period. But due to the importance of the trend memory effect in financial pricing, we extend Ankudinova’s and Ehrhardt’s study under the fractional B–S model when the price change of the underlying asset with time follows a fractal transmission system. The option price is governed by a time-fractional B–S equation of order 0<α<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 0<\alpha <1 $$\end{document}. The main objective of this study is to obtain a numerical solution to determine the European option price with transaction costs based on the implicit difference scheme. This difference scheme is unconditionally stable and convergent and is shown stability and convergence by Fourier analysis. Numerical results and comparisons demonstrate that the introduced difference scheme has high accuracy and efficiency.