= Solution
An <American option> allows exercise at a <stopping time> $\tau\in[t,T]$. For its exercise <financial payoff> $H$, the pricing problem is
$$
\boxed{V(t,S)=\sup_{\tau\in[t,T]}\mathbb E_Q[e^{-r(\tau-t)}H(S_\tau)\mid S_t=S].}
$$
The discounted value is the <Snell envelope> of discounted exercise payoffs. Define the pricing operator
$$
\mathcal LV=V_t+\frac12\sigma^2S^2V_{SS}+rSV_S-rV.
$$
In the continuation region the <Black-Scholes equation> holds, $\mathcal LV=0$. In the exercise region $V=H$, while the ability to wait and the <supermartingale> property give $\mathcal LV\le0$. Together these conditions are the <obstacle problem>
$$
\boxed{\max\{H-V,\mathcal LV\}=0,\qquad V(T,S)=H(S).}
$$
Thus 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 $V_S=H'$ 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 $r\ge0$, <no early exercise of a call without dividends> follows directly from the European lower bound
$$
C_E(t,S)\ge\max\{S-Ke^{-r(T-t)},0\}\ge(S-K)^+.
$$
At 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.
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