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

Abstract

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.

Bounded Variation Control of Itô Diffusions with Exogenously Restricted Intervention Times

“…In [18], the authors study utility maximization when the stock price can be observed and traded only at the jump times of N , corresponding to the quotes on the market. Related studies on optimal stopping of diffusions can be found in [6], [12], and [8]. In [6], the authors consider a perpetual American call with underlying geometric Brownian motion when the process can be stopped only at the jump times of N .…”

Optimal Stopping Times with Different Information Levels and with Time Uncertainty

“…. , τ n , • • • ), where τ i is an independent Poisson process with rate λ; see for instance [7,20,9,14].…”

Periodic strategies in optimal execution with multiplicative price impact

“…See, for example, the discussion in the introduction of Schaefer and Szimayer (), motivated by rational inattention (see Sims, ) in the macroeconomics literature. See also the discussions given in Boyarchenko and Levendorskii () and Lempa () for real option problems with random intervention times, and Dupuis and Wang () and Guo & Liu () for applications to optimal stopping problems and Bermudan look‐back option pricing. In this regard, an important motivation for considering the Poissonian interarrival model is its potential applications to approximate the constant interarrival time cases.…”