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The generalized perpetual American exchange-option problem

Part of: Sequential methods Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Shek-Keung Tony Wong
Shek-Keung Tony Wong*
Affiliation:
Kyoto University
*
∗ Postal address: Daiwa Financial Engineering Laboratory, Graduate School of Economics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto, 606-8501, Japan. Email address: [email protected]
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Abstract

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This paper revisits a general optimal stopping problem that often appears as a special case in some finance applications. The problem is essentially of the same form as the investment-timing problem of McDonald and Siegel (1986) in which the underlying processes are two correlated geometric Brownian motions (GBMs) with drifts less than the discount rate. By contrast, we attempt to analyze the underlying optimal stopping problem to its full generality without imposing any restriction on the drifts of the GBMs. By extending the first passage time approach of Xia and Zhou (2007) to the current context, we manage to obtain a complete and explicit characterization of the solution to the problem on all possible drift domains. Our analysis leads to a new and interesting observation that the underlying optimal stopping problem admits a two-sided optimal continuation region on some certain parameter domains.

Keywords

Optimal stopping problemfree boundary approachfirst passage time approach

MSC classification

Primary: 62L15: Optimal stopping
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G51: Processes with independent increments; L'evy processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 1 , March 2008 , pp. 163 - 182
Copyright
Copyright © Applied Probability Trust 2008 

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