Barrier options and touch- and-out options under regular Lévy processes of exponential type

We derive explicit formulas for barrier options of European type and touchand-out options assuming that under a chosen equivalent martingale measure the stock returns follow a L evy process from a wide class, which contains Brownian Motions (BM), Normal Inverse Gaussian Processes (NIG), Hyperbolic Processes (HP) and Truncated L evy Processes (TLP), and any nite mixture of independent BM, NIG, HP and TLP. In contrast to the Gaussian case, for a barrier option, a rebate must be speci ed not only at a barrier but for all values of the stock the other side of the barrier, the reason being that trajectories of a non-Gaussian L evy process are discontinuous. We consider options with the constant or exponentially decaying rebate, and options which pay a xed rebate when the rst barrier has been crossed but the second one has not. We obtain pricing formulas by solving corresponding boundary problems for the generalized Black-Scholes equation. We use the connection between the resolvent and the in nitesimal generator of the process, the representation theorem for analytic semigroups, the Wiener-Hopf factorization method and the theory of pseudo-di erential operators.