Fast and accurate pricing of barrier options under Lévy processes

We suggest two new fast and accurate methods, Fast Wiener-Hopf method (FWH-method) and Iterative Wiener-Hopf method (IWH-method), for pricing barrier options for a wide class of Lévy processes. Both methods use the Wiener-Hopf factorization and Fast Fourier Transform algorithm. Using an accurate albeit relatively slow finite-difference algorithm developed in Levendorski\v{i} et al (2006) (FDS-method), we demonstrate the accuracy and fast convergence of the two methods for processes of finite variation. We explain that the convergence of the methods must be better for processes of infinite variation, and, as a certain supporting evidence, demonstrate with numerical examples that the results obtained by two methods are in extremely good agreement. Finally, we use FDS, FWH and IWH-methods to demonstrate that Cont and Volchkova method (CV--method), which is based on the approximation of small jumps by an additional diffusion, may lead to sizable relative errors, especially near the barrier and strike. The reason is that CV--method presumes that the option price is of class $C^2$ up to the barrier, whereas for processes without Gaussian component, it is typically not of class $C^1$ at the barrier