A transform approach to compute prices and greeks of barrier options driven by a class of Lévy processes

In this paper we propose a transform method to compute the prices and greeks of barrier options driven by a class of Lévy processes. We derive analytical expressions for the Laplace transforms in time of the prices and sensitivities of single barrier options in an exponential Lévy model with hyper-exponential jumps. Inversion of these single Laplace transform yields rapid, accurate results. These results are employed to construct an approximation of the prices and sensitivities of barrier options in exponential generalised hyper-exponential models. The latter class includes many of the Lévy models employed in quantitative finance such as the variance gamma (VG), KoBoL, generalised hyperbolic, and the normal inverse Gaussian (NIG) models. Convergence of the approximating prices and sensitivities is proved. To provide a numerical illustration, this transform approach is compared with Monte Carlo simulation in the cases that the driving process is a VG and a NIG Lévy process. Parameters are calibrated to Stoxx50E call options.