Abstract: Aiming at the pricing problem of American options under the time-fractional Black-Scholes model, we  proposed an effective numerical solution method.  Firstly, the linear complementarity model satisfied by the American option was transformed into a nonlinear parabolic problem on a bounded domain by using the variable substitution and penalty method. Secondly,  the semi-implicit finite difference method was used to solve the problem, and the error results of the method and the non negativity proof of the solution were given. Finally, numerical experiments were used to verify the correctness and effectiveness of the proposed method.

Key words: time-fractional American option, Caputo fractional derivative, penalty method, semi-implicit finite difference method