Valuation of Employee Stock Options using the Exercise Multiple Approach and Life Tables

Employee stock options (ESOs) are highly exotic derivatives including various forms of call options and performance shares. Much effort in the academic literature has been devoted to modelling employee risk aversion and early exercise of ESOs and less attention has been paid to the effects of employee attrition during the lifetime of the ESO. We show that under the exercise multiple approach proposed by Hull and White (2004), an employee stock option can be decomposed into a gap call option and two partial time barrier options. Analytic formulae for these are derived using the Method of Images developed in Buchen (2001); Konstandatos (2003, 2008) and European exotic bivariate power options (Kyng, 2011). We propose an actuarial approach to incorporate employee attrition into the valuation method. Using exit probabilities obtained from empirically determined multiple decrement tables or life tables we model stock price independent causes of involuntary exercise or forfeiture of ESOs. This allows us to construct a portfolio of analytically tractable ESOs to obtain a valuation which correctly accounts for employee attrition. This is an alternative to the approaches used by Gerber et al. (2012) and Cvitanic et al. (2008) which attempt to model employee attrition without an empirically determined set of exit probabilities.