ESO Valuation with Job Termination Risk and Jumps in Stock Price

TL;DR

This paper develops a comprehensive framework for valuing employee stock options considering job termination risk and stock jumps, providing both numerical methods and closed-form solutions under specific conditions.

Contribution

It introduces a novel valuation model that incorporates job termination risk and jumps in stock prices, with analytical solutions for certain cases and improved numerical techniques.

Findings

Higher job termination risk leads to earlier exercise by ESO holders.

The model accurately captures the impact of stock jumps on ESO valuation.

Closed-form formulas are derived for perpetual ESOs under geometric Brownian motion.

Abstract

Employee stock options (ESOs) are American-style call options that can be terminated early due to employment shock. This paper studies an ESO valuation framework that accounts for job termination risk and jumps in the company stock price. Under general L\'evy stock price dynamics, we show that a higher job termination risk induces the ESO holder to voluntarily accelerate exercise, which in turn reduces the cost to the company. The holder's optimal exercise boundary and ESO cost are determined by solving an inhomogeneous partial integro-differential variational inequality (PIDVI). We apply Fourier transform to simplify the variational inequality and develop accurate numerical methods. Furthermore, when the stock price follows a geometric Brownian motion, we provide closed-form formulas for both the vested and unvested perpetual ESOs. Our model is also applied to evaluate the probabilities of understating ESO expenses and contract termination.