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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.
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Abstract

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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 

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