Explicit solutions for the exit problem for a class of L\'evy processes. Applications to the pricing of double barrier options

TL;DR

This paper derives explicit solutions for the exit problem of a specific class of Lévy processes, enabling precise calculations of their path extrema and first exit times, with applications to double barrier option pricing.

Contribution

It extends existing results by computing the joint distribution of the process, its infimum, and supremum, and analyzes their behavior at key stopping times for a particular Lévy process class.

Findings

Explicit formulas for the joint distribution of process, infimum, and supremum.

Analysis of the process behavior at first exit times from intervals.

Applications demonstrated in double barrier option pricing.

Abstract

Lewis and Mordecki have computed the Wiener-Hopf factorization of a L\'evy process whose restriction on $]0,+\infty[$ of their L\'evy measure has a rational Laplace transform. That allows to compute the distribution of $(X_t,\inf_{0\leq s\leq t}X_s)$. For the same class of L\'evy processes, we compute the distribution of $ (X_t,\inf_{0\leq s\leq t}X_s,\sup_{0\leq s\leq t} X_s)$ and also the behavior of this triple at certain stopping time, like the first exit time of an interval containing the origin. Some applications to the pricing of double barrier options with or without rebate are evocated.