Valuing executive stock options: A quadratic approximation

This paper develops a continuous-time model for valuing executive stock options (ESOs) with features of early exercise, delayed vesting and forfeiture. Applying the quadratic approximation established for valuing American options into ESOs, we obtain an explicit formula for the fair ESO value at its grant date. We show that the approximation formula is consistent with the exact results for two special cases either with no dividend or infinite maturity, and also that the perpetual value for the latter case gives an upper bound of the ESO value. To see the performance of the formula, we numerically examine it with benchmark results generated by a binomial-tree model for some particular cases. Numerical experiments show that there is a complementary relation between the vesting and trading periods with respect to exit rate of ESO holders.