An efficient adaptive wavelet method for pricing time-fractional American option variational inequality

Based on the time-fractional  Black-Scholes pricing model, the evaluation of an American-style option problem  can be formulated as  a   free boundary problem.  It is equivalent  to a time-fractional parabolic variational inequality. Due to the time-fractional derivative involved in  the problem, increasing the computational cost for large final times has been expected in  the numerical   solution for this problem. In this paper, we want to propose a new adaptive numerical method to solve   this problem   accurately, with low computational cost. The presented method is based on interpolating wavelets family. An adaptive scheme in time discretization with an adaptive wavelet collocation method for space discretization has been used for  the given problem. We show  that combination of interpolating wavelet basis and finite difference method, makes an accurate structure to design an optimal adaptive mesh  for this problem. The presented computational mesh  by this method can prevent growing of computational cost by time. The  performance of the proposed method has been tested by means of some numerical experiments. We show that,  in comparison with the full grid algorithms, the presented adaptive algorithm can capture the priori unknown free boundary and is able to find the value of American put option price with high accuracy and reasonable CPU time.