On the structure of discounted optimal stopping problems for one-dimensional diffusions

Abstract

We connect two approaches for solving discounted optimal stopping problems for one-dimensional time-homogeneous regular diffusion processes on infinite time intervals. The optimal stopping rule is assumed to be the first exit time of the underlying process from a region restricted by two constant boundaries. We provide an explicit decomposition of the reward process into a product of a gain function of the boundaries and a uniformly integrable martingale inside the continuation region. This martingale plays a key role for stating sufficient conditions for the optimality of the first exit time. We also consider several illustrating examples of rational valuation of perpetual American strangle options.

Keywords:

• optimal stopping problem
• diffusion process
• first exit time
• free-boundary problem
• martingale approach of Beibel and Lerche
• perpetual American strangle options

2000 Mathematics Subject Classification::

• primary 60G40, 60G44, 60J25
• secondary 91B28, 62L10, 35R35

Acknowledgements

The authors are grateful to the Editor for his encouragement to prepare the revised version and the Associate Editor and two referees for their useful suggestions, which have helped to improve the presentation of the paper. The authors thank Mihail Zervos for several helpful discussions. The paper was partially written when the first author was visiting Albert-Ludwigs-Universität Freiburg (Germany) in July 2008 and in April 2009. The hospitality at the Abteilung für Mathematische Stochastik and financial support from the European Science Foundation (ESF) through the Short Visit Grant No. 2316 of the programme Advanced Mathematical Methods for Finance (AMaMeF) are gratefully acknowledged.

Notes

1. [email protected]

^†Supported by ESF AMaMeF Short Visit Grant No. 2316.

Additional information

Notes on contributors

Hans Rudolf Lerche

1 1. [email protected]