Pricing Early-Exercise and Discrete Barrier Options by Fourier-Cosine Series Expansions

We present a pricing method based on Fourier-cosine expansions for early-exercise and discretely-monitored barrier options. The method works well for exponential Levy asset price models. The error convergence is exponential for processes characterized by very smooth transitional probability density functions. The computational complexity is $O((M-1) N \log{N})$ with $N$ a (small) number of terms from the series expansion, and $M$, the number of early-exercise/monitoring dates.