Consider a geometric Brownian motion X
t
(ω) with drift. Suppose that there is an independent source that sends signals at random times τ
1 < τ
2 < ⋯. 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 S
τ
, where S
t
= max(max0≤u≤t
X
u
, s) for some constant s ≥ X
0. The objective is to choose an optimal stopping time to maximize the discounted expected reward E[e−r
τ
i
S
τ
i
| X
0 = x, S
0 = 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{τ
n
: X
τ
n
≤a
*
S
τ
n
}. If λ≤λ* then it is optimal to stop immediately a signal is received, i.e. at τ
* = τ
1. Mathematically, it is intriguing that a smooth fit is critical in the former case while irrelevant in the latter.
Investing and Stopping
