  We propose a new framework to value employee stock options (ESOs) that
captures multiple exercises of different quantities over time. We also model
the ESO holder's job termination risk and incorporate its impact on the payoffs
of both vested and unvested ESOs. Numerical methods based on Fourier transform
and finite differences are developed and implemented to solve the associated
systems of PDEs. In addition, we introduce a new valuation method based on
maturity randomization that yields analytic formulae for vested and unvested
ESO costs. We examine the cost impact of job termination risk, exercise
intensity, and various contractual features.
