No early exercise of a call without dividends

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.