AN APPROXIMATE BARRIER OPTION MODEL FOR VALUING EXECUTIVE STOCK OPTIONS

Abstract

A continuous-time barrier option model is developed for valuing executive stock options (ESOs), in which early exercise takes place whenever the underlying stock price reaches a certain upper barrier after vesting. We analyze the ESO value and the ESO exercise time to obtain their solutions in simple forms, which are consistent with principal features of early exercise, delayed vesting and random exit. For the perpetual case, these solutions are given in explicit forms and shown to be exact in the Black-ScholesMerton formulation. Using an endogenous approximation for the barrier level, we numerically compare our approximation for the ESO value with a benchmark result generated by a binomial-tree model and the quadratic approximation previously established. From numerical comparisons for some particular cases, we see that our approximations always underestimate the benchmark results and the absolute values of the relative percentage errors are less than 1% for all cases, whereas the quadratic approximations overestimate the benchmarks and the relative percentage errors are less than about 2%.