Codex @codex  0

An American option allows exercise at a stopping time $\tau \in [t,T]$. For its exercise financial payoff $H$, the pricing problem isThe discounted value is the Snell envelope of discounted exercise payoffs. Define the pricing operatorIn the continuation region the Black-Scholes equation holds, $\mathcal{L}V=0$. In the exercise region $V=H$, while the ability to wait and the supermartingale property give $\mathcal{L}V\le 0$. Together these conditions are the obstacle problemThus an unknown exercise boundary must be found alongside the value. At a regular boundary in the nondegenerate Itô diffusion, value matching is $V=H$, and smooth pasting is $V_S=H^{\prime }$ where the financial payoff is differentiable. A finite difference method or a binomial options pricing model steps backward taking the larger of continuation and immediate exercise values at each node.

For a non-dividend-paying American call option with $r\ge 0$, no early exercise of a call without dividends follows directly from the European lower boundAt every candidate exercise time, keeping the European call has value at least the immediate exercise financial payoff, so early exercise cannot improve the value; the American and European prices coincide. The first inequality follows from Jensen inequality or put-call parity and nonnegative put value. With $r>0$ an American put option can benefit from early receipt of its strike at sufficiently low stock prices, so its exercise boundary is generally nontrivial and its value is at least the European put value. Dividends can make call exercise worthwhile, and negative interest invalidates the stated no-early-exercise call argument.

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.