Filling the gap between American and Russian options: adjustable regret

Abstract

We study several infinite-horizon optimal multiple-stopping problems for (geometric) Brownian motion. In finance, they naturally span between the American and Russian option formulations in terms of price and reduced regret. In statistics, they are continuous-time examples of best-choice problems with multiple rights. We find explicit formulas for the value functions and describe completely optimal exercise strategies whenever one exists. We also conjecture a new characterization of the value function for the open problem of the Russian option for arithmetic Brownian motion with drift.

Keywords:

• Optimal multiple-stopping
• Russian option
• Best-choice problems
• Brownian motion

AMS Subject Classification::

• Primary: 60G40
• Secondary: 60J65
• 62L15

Acknowledgements

This work was initiated while both authors were attending the Symposium on Optimal Stopping with Applications in Manchester, United Kingdom between 22 and 27 January 2006. They are very grateful to the organizers for this stimulating event.

Notes

^¶ [email protected]

Additional information

Notes on contributors

Michael Ludkovski

¶ ¶ [email protected]