A Black-Scholes type model for American options will be considered where the underlying asset price experiences Brownian motion with random jumps. The mathematical problems is an obstacle problem for a linear one-dimensional diffusion equation with a functional source term. The problem is time discretized and solved at each time level iteratively with a Riccati method. Some numerical experiments for a call and put with multiple jumps are presented. Convergence of the iteration at a given time level will be discussed for the simpler problem of a European put where there is no free boundary.