A compact quadratic spline collocation method for the time-fractional Black–Scholes model

A compact quadratic spline collocation (QSC) method for the time-fractional Black–Scholes model governing European option pricing is presented. Firstly, after eliminating the convection term by an exponential transformation, the time-fractional Black–Scholes equation is transformed to a time-fractional sub-diffusion equation. Then applying L1-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L1 - 2$$\end{document} formula for the Caputo time-fractional derivative and using a collocation method based on quadratic B-spline basic functions for the space discretization, we establish a higher accuracy numerical scheme which yields 3-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3-\alpha $$\end{document} order convergence in time and fourth-order convergence in space. Furthermore, the uniqueness of the numerical solution and the convergence of the algorithm are investigated. Finally, numerical experiments are carried out to verify the theoretical order of accuracy and demonstrate the effectiveness of the new technique. Moreover, we also study the effect of different parameters on option price in time-fractional Black–Scholes model.