Perpetual Barrier Options in Jump-Diffusion Models

Abstract

We present a closed form solution to the perpetual American double barrier call option problem in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial irregular optimal stopping problem to an integro-differential free-boundary problem and solving the latter by using continuous and smooth fit. The obtained solution of the nontrivial free-boundary problem gives the possibility to observe some special analytic properties of the value function at the optimal stopping boundaries.American double barrier options, optimal stopping problem, jump-diffusion model, integro-differential free-boundary problem, continuous and smooth fit, Ito-Tanaka-Meyer formula