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(max 0≤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 rewardwhere 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.
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.
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