An efficient wavelet method for the time-fractional Black–Scholes equations

A European option is one of the common types of options in financial markets, which can be modeled by a time-fractional parabolic PDE, known as the time-fractional Black–Scholes equation (BSE). In this article, we propose an effective numerical scheme by applying Müntz–Legendre wavelets (MLW) for the solution of the given BSE. Different from classical wavelets (such as Legendre and Chebyshev), the MLW have an extra parameter representing the fractional order. Therefore, they provide more reliable results for certain fractional calculus problems. The convergence analysis of the method is provided in detail. Several test examples are given to illustrate the advantages of MLW over other classical wavelets and the high accuracy of this technique compared to existing methods in the literature.