Optimal Stopping with Randomly Arriving Stopping Opportunities

Abstract

We develop simulation-based methods to solve general optimal stopping problems with opportunities to stop that arrive randomly. Such problems occur naturally in a wide variety of applications with market frictions, such as limited liquidity, incomplete information and transaction costs. Our approach employs random time scales to map the original problem to an integer-time stopping problem with (possibly) infinite horizon. We introduce an infinite-horizon policy iteration algorithm to generate lower-biased estimates of the value function and establish a martingale dual representation to generate upper-biased estimates. Furthermore, we extend a family of backward dynamic programming methods that includes least-squares Monte Carlo. We illustrate the performance of our methods, their general applicability and the managerial implications of randomly arriving opportunities to stop in three examples: a stopped jump-diffusion that can be analyzed in closed form, a random-exercise version of a multi-dimensional max-call contract, and a real options problem in an illiquid market.