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.

Keywords:

• American double barrier options
• Optimal stopping problem
• Jump-diffusion model
• Integro-differential free-boundary problem
• Continuous and smooth fit
• Itô–Tanaka–Meyer formula

MSC (2000)::

• Primary 60G40
• 34K10
• 91B28. Secondary 60J60
• 60J75

JEL Classification::

• G13

Acknowledgements

The results of the paper were presented at the Symposium on Optimal Stopping with Applications held at the University of Manchester in January 2006. The author is grateful to the organizers and participants for their interest and useful comments. The author is indebted for both anonymous referees for their valuable suggestions, which helped to revise the paper.

Notes

†This research was supported by the Deutsche Forschungsgemeinschaft through SFB 649 Economic Risk.