= American call option
{c}
{title2=$V_t=\operatorname*{ess\,sup}_{t\le\tau\le T}\mathbb E_Q[e^{-\int_t^\tau r_sds}(S_\tau-X)^+\mid\mathcal F_t]$}
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,
$$
V_t=\operatorname*{ess\,sup}_{t\le\tau\le T}\mathbb E_Q\!\left[e^{-\int_t^\tau r_sds}(S_\tau-X)^+\mid\mathcal F_t\right].
$$
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>.
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