Document Type : Research Paper
Department of Mathematics, University of Garmsar, Garmsar, Iran.
Department of Mathematics Education, Farhangian University, P.O. Box 14665-889, Tehran, Iran.
Faculty of Management and Accounting, College of Farabi, Tehran University, Qom, Iran.
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.