In this article, we propose a computational method for the numerical solution of Black-Scholes PDEs arising in option pricing. First, we discretize the time-domain by uniform mesh and apply the Crank-Nicolson method to approximate the time variable. Then, we use the streamline-diffusion finite element method (SDFEM) for the spatial derivative on different nonuniform meshes. The proposed method is of second-order convergent in both variables. For comparison purposes, we use the backward-Euler scheme for the time derivative, which will be of first-order convergent. Numerical experiments are carried out to verify theoretical results.