Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T05:26:25.392Z Has data issue: false hasContentIssue false
  • English
  • Français

Stopping at the maximum of geometric Brownian motion when signals are received

Part of: Stochastic processes Hamilton-Jacobi theories, including dynamic programming

Published online by Cambridge University Press:  14 July 2016

X. Guo  and
J. Liu
X. Guo*
Affiliation:
Cornell University
J. Liu*
Affiliation:
Yale University
*
∗Postal address: School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, USA. Email address: [email protected]
∗∗Postal address: School of Public Health, Yale University, New Haven, CT 06520, USA.
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.

Consider a geometric Brownian motion Xt(ω) with drift. Suppose that there is an independent source that sends signals at random times τ1 < τ2 < ⋯. Upon receiving each signal, a decision has to be made as to whether to stop or to continue. Stopping at time τ will bring a reward Sτ, where St = max(max0≤u≤tXu, s) for some constant s ≥ X0. The objective is to choose an optimal stopping time to maximize the discounted expected reward E[e−rτiSτi | X0 = x, S0 = s], where r is a discount factor. This problem can be viewed as a randomized version of the Bermudan look-back option pricing problem. In this paper, we derive explicit solutions to this optimal stopping problem, assuming that signal arrival is a Poisson process with parameter λ. Optimal stopping rules are differentiated by the frequency of the signal process. Specifically, there exists a threshold λ* such that if λ>λ*, the optimal stopping problem is solved via the standard formulation of a ‘free boundary’ problem and the optimal stopping time τ* is governed by a threshold a* such that τ* = inf{τn: Xτn≤a*Sτn}. If λ≤λ* then it is optimal to stop immediately a signal is received, i.e. at τ* = τ1. Mathematically, it is intriguing that a smooth fit is critical in the former case while irrelevant in the latter.

Keywords

Optimal stopping problemsmooth fitvariational inequality

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 49L20: Dynamic programming method
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 3 , September 2005 , pp. 826 - 838
Copyright
© Applied Probability Trust 2005 

References

Benes, V. E., Shepp, L. and Witsenhauser, H. S. (1980). Some solvable stochastic control problems. Stochastics 4, 39–83.CrossRefGoogle Scholar
Chernoff, H. (1961). Sequential tests for the mean of a normal distribution. In Proc. 4th Berkeley Symp. Statist. Prob. Vol. 1, University of California Press, Berkeley, CA, pp. 79–91.Google Scholar
Dubins, L. E., Shepp, L. and Shiryaev, A. (1993). Optimal stopping rules and maximal inequalities for Bessel processes. Theory Prob. Appl. 38, 226–261.CrossRefGoogle Scholar
Duffie, D. and Harrison, M. (1993). Arbitrage pricing of perpetual lookback options. Ann. Appl. Prob. 3, 641–651.CrossRefGoogle Scholar
Dupuis, P. and Wang, H. (2002). Optimal stopping with random intervention time. Adv. Appl. Prob. 34, 141–157.CrossRefGoogle Scholar
Guo, X. (2001). An explicit solution to an optimal stopping time with regime switching. J. Appl. Prob. 38, 454–481.CrossRefGoogle Scholar
Guo, X. and Shepp, L. (2001). Some optimal stopping problems with nontrivial boundaries in pricing exotic options. J. Appl. Prob. 39, 1–12.Google Scholar
Jacka, S. D. (1991). Optimal stopping and best constants for Doob-like inequalities: the case p=1. Ann. Prob. 19, 1798–1821.Google Scholar
Jacka, S. D. (1991). Optimal stopping and the American put. Math. Finance 1, 1–14.Google Scholar
Karatzas, I. and Shreve, S. E. (1998). Brownian Motion and Stochastic Calculus (Graduate Texts Math. 113). Springer, New York.Google Scholar
Krylov, N. V. (1980). Controlled Diffusion Process. Springer, New York.CrossRefGoogle Scholar
McKean, H. P. (1965). Appendix: A free boundary problem for the heat equation arising from a problem in mathematical economics. Industrial Manag. Rev. 6, 32–39.Google Scholar
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.CrossRefGoogle Scholar
Rogers, L. C. G. and Zane, O. (2000). A simple model of liquidity effects. In Advances in Finance and Stochastics, eds. Sandmann, K. and Schoenbucher, P., Springer, Berlin, pp. 161–176.Google Scholar
Shepp, L. and Shiryaev, A. N. (1993). The Russian option: reduced regret. Ann. Appl. Prob. 3, 631–640.CrossRefGoogle Scholar
Shepp, L. A. and Shiryaev, A. N. (1994). A new look at the “Russian Option”. Theory Prob. Appl. 39, 103–119.CrossRefGoogle Scholar
Van Moerbeke, P. L. J. (1976). On optimal stopping and free boundary problems. Archive Rational Mech. Anal. 60, 101–148.Google Scholar