Explicit solutions of the exit problem for a class of Lévy processes; applications to the pricing of double-barrier options

Lewis and Mordecki have computed the Wiener–Hopf factorization of a Lévy process whose restriction of the Lévy measure on ]0,+∞[]0,+∞[ has a rational Laplace transform. This allowed them to compute the distribution of (Xt,inf0≤s≤tXs)(Xt,inf0≤s≤tXs). For the same class of Lévy processes, we compute the distribution of (Xt,inf0≤s≤tXs,sup0≤s≤tXs) and also the behavior of this triple at certain stopping times, such as the time of first exit of an interval containing the origin. Some applications to the pricing of double-barrier options with or without rebate are described.