Timing in the Presence of Directional Predictability: Optimal Stopping of Skew Brownian Motion

TL;DR

This paper studies optimal stopping problems with skew Brownian motion, revealing how directional predictability influences timing strategies, with key findings on the exercise region structure and effects of skewness.

Contribution

It provides new insights into optimal stopping with skew Brownian motion, showing waiting is optimal at the skew point and how skewness affects timing incentives.

Findings

Waiting is optimal at the skew point for many payoffs.

Exercise regions can be disconnected due to skewness.

Higher skewness delays optimal investment timing.

Abstract

We investigate a class of optimal stopping problems arising in, for example, studies considering the timing of an irreversible investment when the underlying follows a skew Brownian motion. Our results indicate that the local directional predictability modeled by the presence of a skew point for the underlying has a nontrivial and somewhat surprising impact on the timing incentives of the decision maker. We prove that waiting is always optimal at the skew point for a large class of exercise payoffs. An interesting consequence of this finding, which is in sharp contrast with studies relying on ordinary Brownian motion, is that the exercise region for the problem can become unconnected even when the payoff is linear. We also establish that higher skewness increases the incentives to wait and postpones the optimal timing of an investment opportunity. Our general results are explicitly illustrated for a piecewise linear payoff.