ESO Valuation with Job Termination Risk and Jumps in Stock Price
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.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Sluggish news reactions: A combinatorial approach for synchronizing stock jumps
Stock prices often react sluggishly to news, producing gradual jumps and jump delays. Econometricians typically treat these sluggish reactions as microstructure effects and settle for a coarse sampling grid to guard agai…
Hedging in a market with jumps - an FBSDE approach
We propose a model for hedging in a market with jumps for a large investor. The dynamics of the stock prices and the value process is governed by forward-backward SDEs driven by Teugels martingales. Unlike known FBSDE ma…
Do Investors Hedge Against Green Swans? Option-Implied Risk Aversion to Wildfires
Measuring beliefs about natural disasters is challenging. Deep out-of-the-money options allow investors to hedge at a range of strikes and time horizons, thus the 3-dimensional surface of firm-level option prices provide…
Equity Tail Risk in the Treasury Bond Market
This paper quantifies the effects of equity tail risk on the US government bond market. We estimate equity tail risk with option-implied stock market volatility that stems from large negative price jumps, and we assess i…
Measures of Model Risk in Continuous-time Finance Models
Measuring model risk is required by regulators on financial and insurance markets. We separate model risk into parameter estimation risk and model specification risk, and we propose expected shortfall type model risk mea…
parameter estimation