A class of options with stochastic lives and an extension of the Black–Scholes formula

Certain options have a fixed date of maturity but may be cancelled prematurely. This can happen for a stock option in case of a merger or for an executive stock option in case the executive leaves his/her present job. The differential equation is given which governs the value of an option with a stochastic life. Solutions can be obtained through integration in certain cases. The main result is an extension of the Black–Scholes formula to options where the time to expiration is stochastic.