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

TL;DR

This paper develops a continuous-time valuation model for Employee Stock Options (ESOs) using mean-variance hedging, incorporating early exercise and job termination risks through a proxy random time, and employs PDE techniques and numerical analysis.

Contribution

It introduces a novel ESO valuation framework that accounts for early exercise and termination risks via a proxy time, using PDE and mean-variance hedging methods.

Findings

Derived ESO value as expected discounted payoff under an equivalent martingale measure.

Presented a numerical study illustrating the theoretical valuation model.

Showed the valuation differs from minimal martingale and variance-optimal measures.

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.