In this presentation, the Crandall-Douglas scheme for the heat equation will be applied to the computation of both European and American stock option prices. This method has the advantage of being second-order accurate in time and fourth-order accurate in space. For European options, discretization of the initial-boundary value problem leads to a system of linear equations to be solved at each time step. For American options, discretization of the free boundary problem leads to a linear complementarity problem to be solved at each time step. Numerical examples for both types of stock options will be presented.