= Solution
{c}
A <European call option> has nonnegative <financial payoff>, so its price is nonnegative under <no-arbitrage>. A second <arbitrage> comparison buys one call and a <zero-coupon bond> paying $K$, and shorts one share. Its terminal value is $(S_T-K)^++K-S_T=(K-S_T)^+\geq0$, so its initial cost cannot be negative. Consequently
$$
\boxed{C_t^e\geq\max(S_t-Ke^{-r(T-t)},0).}
$$
The holder of an <American call option> can choose to wait until expiry, so $C_t^a\geq C_t^e$. Since $r>0$ and $t<T$, an in-the-money call has $C_t^a\geq S_t-Ke^{-r(T-t)}>S_t-K$: immediate exercise sacrifices value. An out-of-the-money exercise pays zero and cannot improve the holder's position. This proves <no early exercise of a call without dividends>: waiting is always at least as good, and is strictly better for positive immediate exercise payoff. In a degenerate market with zero continuation value an out-of-the-money holder can be indifferent; the <Black-Scholes model> with positive volatility gives strictly positive continuation value before expiry.
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