Hostname: page-component-7bb8b95d7b-pwrkn Total loading time: 0 Render date: 2024-09-17T21:00:13.959Z Has data issue: false hasContentIssue false
  • English
  • Français

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: tony.wong@z05.mbox.media.kyoto-u.cu.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel.CrossRefGoogle Scholar
Fakeev, A. (1971). Optimal stopping of a Markov process. Theory Prob. Appl. 16, 694–696.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. (1998). Methods of Mathematical Finance. Springer, New York.Google Scholar
McDonald, R. and Siegel, D. (1986). The value of waiting to invest. Quart. J. Econom. 101, 707–728.CrossRefGoogle Scholar
McKean, H. P. (1965). A free boundary problem for the heat equation arising from a problem in mathematical economics. Industrial Manag. Rev. 6, 32–39.Google Scholar
Moerbeke, P. L. (1976). On optimal stopping and free boundary problems. Archive Rational Mech. Anal. 60, 101–148.CrossRefGoogle Scholar
Øksendal, B. (2003). Stochastic Differential Equations: An Introduction with Applications, 6th edn. Springer, New York.CrossRefGoogle Scholar
Shiryaev, A. N. (1999). Essentials of Stochastic Finance. Facts, Models, Theory. World Scientific, Singapore.CrossRefGoogle Scholar
Xia, J. and Zhou, X. Y. (2007). Stock loans. Math. Finance 17, 307–317.CrossRefGoogle Scholar