American call option (${\sup }_{\tau \le T}{\mathbb{E}}_Q[e^{-{\int }_t^{\tau }{r}_{s}ds}(S_{\tau }-X{)}^{+}]$)

An American call permits purchase of the underlying at strike $X$ at a chosen stopping time no later than maturity. Under standard pricing and integrability hypotheses its value is the displayed optimal stopping value. A European call option restricts exercise to the final time. Dividends and interest on the deferred strike affect the incentive to exercise early.

For a single known cash dividend at $t_1$, nonnegative interest excludes exercise strictly before the last cum-dividend opportunity. Immediately before the dividend, continuation has lower bound $S_{t_1-}-D-XB(t_1,T)$, whereas positive immediate exercise pays $S_{t_1-}-X$. The displayed inequality makes continuation strictly better. With no further dividends, it therefore excludes all early exercise. If the inequality fails, exercise just before the dividend may be worthwhile, but is not automatically optimal because remaining option time value also matters.

For a non-dividend-paying stock and nonnegative deterministic interest, the European call lower bound is $C_E\ge S-XB$, where $B\le 1$ is the zero-coupon bond price. This is at least the in-the-money exercise payoff $S-X$; the call also preserves the possibility of avoiding payment of the strike. Positive interest makes the comparison strict when exercise would have positive payoff. At zero interest there can be indifference, and with negative interest the statement can fail: a deterministic stock $S_u=S_0{e}^{ru}$ with $r<0$ can give a larger payoff by immediate exercise.