Pricing Barrier Options Using Integral Transforms

Barrier options are a class of exotic options that are traded in over-the-counter markets worldwide. These options are particularly attractive for their lower cost compared to vanilla options. However, the closed form analytical solutions for the partial differential equations modeling these options are not easy to obtain and therefore one usually seeks numerical approaches to find them. In this paper, we consider two types of exotic options, namely a single barrier European down-and-out call and a double barrier European knock-out call options. Like some other standard and nonstandard options, these barrier options also have non-smooth payoffs at the exercise price. This non-smooth payoff is the main cause of the reduction in accuracy when the classical numerical methods, for example, lattice method, Monte Carlo method, or other methods based on finite difference and finite elements are used to solve such problems. In fact, the same happens when one uses the spectral method which is known to preserve the exponential accuracy. In order to retain this high-order accuracy, in this paper we propose a spectral decomposition method which approximates the unknown solution by rational interpolants on each sub-domain. The resulting semi-discrete problem is solved by a contour integral method. Our numerical results affirm that the proposed approach is very robust and gives very reliable results.