ESO Valuation with Job Termination Risk and Jumps in Stock Price

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 Lévy 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.

Penentuan Harga Opsi Saham Karyawan Dengan Memperhitungkan Risiko Pemutusan Hubungan Kerja Menggunakan Levy-exponential Process

“…Jika selama vesting period karyawan meninggalkan perusahaan maka akan mengakibatkan hilangnya nilai opsi (opsi menjadi tidak berharga). Setelah vesting period, ketika karyawan meninggalkan perusahaan, OSK akan kedaluwarsa meskipun karyawan dapat memilih untuk melakukan exercise terlebih dahulu jika opsinya menguntungkan (Leung, T., & Wan, H. , 2015).…”

A Top-Down Approach for the Multiple Exercises and Valuation of Employee Stock Options

“…We assume that ζ and ξ are mutually independent. This approach of modeling job termination by an exogenous random variable is also used by Jennergren and Naslund (1993), Carpenter (1998), Carr and Linetsky (2000), Hull and White (2004), Sircar and Xiong (2007), Leung and Sircar (2009b), Carmona et al (2011), and Leung and Wan (2015), among others. In our model, using two different exponential times allows us to account for the varying level of job termination risk during and after the vesting period.…”

“…In the literature, Leung and Wan (2015) apply a Fourier time-stepping (FST) method it to compute the cost of an American-style ESO when the company stock is driven by a Levy process. This FST method has been applied more broadly by Jackson et al (2008) to solve partial-integro differential equations (PIDEs) that arise in options pricing problems.…”

“…Another approach is to an optimal exercise time that maximizes the expected discounted payoff under some risk-neutral pricing measure (Leung and Wan (2015)). Instead of the risk neutrality assumption, a number of related studies incorporate the employee's risk preferences and hedging constraints and derive the optimal exercise strategy by solving a utility maximization problem.…”

The Value of Being Lucky: Option Backdating and Nondiversifiable Risk

“…The agent's partial hedging/control problem is considered in tandem with American option exercise, resulting in mixed control/stopping. Applications to executive stock options include Detemple & Sundaresan (1999), Oberman & Zariphopolou (2003), Leung and Sircar (2009), Henderson (2007), Grasselli andHenderson (2009), Carpenter, Stanton &Wallace (2010), Henderson, Sun & Whalley (2014), and Leung & Wan (2015). Optimal stopping problems are solved via a free boundary problem and the associated Hamilton-Jacobi-Bellman equation.…”