Expected Optimal Exercise Time of a Perpetual American Option : A Closed Form Solution

Using martingale methods, we find that the expected optimal exercise time of a perpetual, dividend paying American call option contract is: the ratio of the time independent stopping boundary to the risk adjusted drift of the stock price process. This ratio is an analytical expression. Of independent interest is the computational simplicity of our derivation. Specifically, we use only the optional sampling theorem of martingale theory and elementary algebra. In contrast, the non-martingale approach requires tedious integration and solution of an ordinary differential equation