 Solution

Use the usual frictionless option model with nonnegative deterministic interest rates and no stock dividends. Let $B(t,T)\le 1$ be the price of a unit zero-coupon bond, $C_E$ the European call price, and $C_A$ the American call price. A call payoff dominates $S_T-X$, so no-arbitrage pricing gives $C_E\ge S_t-XB(t,T)$, and $C_A\ge C_E$. If $S_t>X$, thenThe last term is the immediate exercise payoff. Waiting retains downside optionality and defers payment of the strike. For positive interest and $X>0$ the inequality is strict while exercise has positive payoff. If $S_t\le X$, exercising for a zero payoff provides no advantage over keeping the right. Thus early exercise of a non-dividend-paying call is never strictly beneficial under these assumptions, and $C_A=C_E$. For completeness, discounted European call value is a true martingale in the standard pricing model. For any exercise stopping time $\tau \le T$, its pointwise dominance of immediate payoff gives ${\mathbb{E}}_Q[B(t,\tau )(S_{\tau }-X{)}^{+}\mid {\mathcal{F}}_t]\le {\mathbb{E}}_Q[B(t,\tau )C_E(\tau )\mid {\mathcal{F}}_t]=C_E(t)$. Taking the supremum over exercise rules proves $C_A\le C_E$, and waiting to maturity gives the reverse inequality. This proves no early exercise of a call without dividends.

The contrast for a put is the interest earned on a strike received sooner. The European put has payoff at most $X$, so $P_E\le XB(t,T)$. When interest is positive and $S_t<X(1-B(t,T))$, immediate exercise gives $X-S_t>XB(t,T)\ge P_E$. Hence committing never to exercise before maturity is worse than an available early-exercise strategy in such a deep-in-the-money state. This does not say that every put should be exercised early, only that a general prohibition is not rational.

The interest-rate hypothesis is essential to the printed claim. If $r<0$ and the stock is deterministic, $S_u=S_t{e}^{r(u-t)}$ with no dividends, choose spot sufficiently large that it stays in the money through $T$. The European call price is $S_t-X{e}^{-r(T-t)}<S_t-X$, so immediate exercise is better. At zero interest, early exercise can be indifferent in degenerate cases; it is the usual positive-interest setting that makes the non-exercise comparison strict.