In this article, we propose an efficient pricing method for Asian options with early–exercise features. It is based on a two–dimensional integration and a backward recursion of the Fourier coefficients, in which several numerical techniques, like Fourier cosine expansions, Clenshaw–Curtis quadrature and the Fast Fourier transform (FFT) are employed. Rapid convergence of the pricing method is illustrated by an error analysis. Its performance is further demonstrated by various numerical examples, where we also show the power of an implementation on the Graphics Processing Unit (GPU).