This paper outlines a method for calculating option prices using the Green function and Gaussian quadrature. Multidimensional formulas for analytical transition probabilities are presented. We derive an algorithm that intuitively can be described as a generalized multinomial algorithm, or as an approximating Markov chain. For speed, we change variables in a special way and sample points using Gaussian quadrature. The efficiency of the method is assessed by comparing the numerical results to that of other methods, such as finite differences and binomial lattices. We address the American put problem, basket options, and the effects of stochastic volatility.