Valuation of Employee Stock Options (ESOs) by means of Mean-Variance Hedging

Abstract

We consider the problem of ESO valuation in continuous time. In particular, we consider models that assume that an appropriate random time serves as a proxy for anything that causes the ESO's holder to exercise the option early, namely, reflects the ESO holder's job termination risk as well as early exercise behaviour. In this context, we study the problem of ESO valuation by means of mean-variance hedging. Our analysis is based on dynamic programming and uses PDE techniques. We also express the ESO's value that we derive as the expected discounted payoff that the ESO yields with respect to an equivalent martingale measure, which does not coincide with the minimal martingale measure or the variance-optimal measure. Furthermore, we present a numerical study that illustrates aspects or our theoretical results.