Perpetual Call Options with Non-Tradability

Abstract

We explicitly solve an optimal stopping problem related to the exercise of a perpetual American call option when the option holder cannot trade the underlying asset. We prove the verification theorem for the solution proposed. We derive the moment generating function of the optimal exercise time and also the elasticity of the option value with respect to stock price. The class of admissible utility functions that we solve for contains the CRRA family with some parametric restrictions. This theoretical framework provides the exact exercise boundary and the value of perpetual real options for a self interested manager whose incentives are not aligned with those of the shareholders. It can also serve as an approximation to the valuation of executive stock options.

Keywords: Non-Tradability, Incomplete Markets, Real Options, Executive Stock Options, Optimal Stopping