IN THE FINANCIAL DERIVATIVE MARKETS, THE GREAT IMPORTANCE SHOULD BE GIVEN TO THE EVALUATION OF THE PRICING OF THE OPTIONS. DUE TO THIS, THE problem OF PRICING AMERICAN OPTIONS UNDER THE JUMP-DIFFUSION MODEL IS CONSIDERED. THEREFORE, FIRST, THE PRICING AMERICAN OPTIONS WAS SHOWN AS A FREE boundary value problem FOR A PARTIAL INTEGRO-DIFFERENTIAL EQUATION , THEN AN EFFICIENT NUMERICAL METHOD FOR SOLVING THE FREE boundary value problem IS DISCUSSED.