Executive Stock Options and Concavity of the Option Price

Abstract

Accounting for grants of executive stock options (ESOs) now requires that they be treated as an expense and valued at their fair values at the time of issue. But unlike traded options, maturity dates for ESOs are uncertain. They can not be exercised until a vesting period has passed, but after that, exercise may take place over a wide range of dates. Because the Black-Scholes model is nonlinear in time to expiration, simply putting the expected value of the date of exercise into the formula as the option maturity will produce a bias. It is commonly believed that this bias is positive, i.e., an option priced at the expected exercise date will be worth more than the mean value of a set of options exercised at dates uniformly distributed over the exercise period. Boyle and Scott discuss this problem and show, among other things, that there will be a bias, but it can go in either direction as a function of the other model parameters. The way to eliminate the bias is to value the option within a framework, such as a lattice model, in which the exercise decision is modeled specifically. The true expected life for accounting purposes should then be the implied time to maturity, that is, the maturity input that makes the Black-Scholes equation produce the same value as the lattice model.