 American call option

An American call option permits buying the underlying at strike price $X$ at any chosen stopping time from the current date $t$ through maturity $T$. Under standard complete-market pricing and integrability hypotheses,The conditional expectation uses information available at time $t$, and the optimization allows only future exercise times. A European call option restricts exercise to $T$. The no early exercise of a call without dividends result applies with nonnegative interest under its stated assumptions; dividends or negative rates can change that conclusion. The discounted value is the Snell envelope of discounted exercise rewards, as in American option.