  In this paper we propose a transform method to compute the prices and greeks
of barrier options driven by a class of Levy processes. We derive analytical
expressions for the Laplace transforms in time of the prices and sensitivities
of single barrier options in an exponential Levy 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 (GHE) Levy models. The latter class includes many of the Levy
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 Levy
process. Parameters are calibrated to Stoxx50E call options.
