American Options

A European option can be exercised only at its expiry date. An American option can be exercised at any time up to expiry. That extra freedom sounds like a small footnote, but it transforms the problem: the holder must now decide not just whether the option is valuable, but when to pull the trigger — turning pricing into an optimal stopping problem.

Pricing as optimal stopping

Because the holder may exercise at any stopping time \tau \le T, the fair (risk-neutral) value is the best such choice — the supremum over all stopping rules of the expected discounted payoff:

V_0 = \sup_{\tau \le T} \mathbb{E}^{\mathbb{Q}}\!\left[ e^{-r\tau}\, \text{payoff}(S_\tau) \right].

Crucially, \tau must be a genuine stopping time — the decision to exercise can depend only on information seen so far, never on a peek at tomorrow's price. There is no formula as tidy as Black–Scholes; instead the value is computed by backward induction (the "Snell envelope"): at each moment, the option is worth the larger of exercising now (its intrinsic value) and holding on (its continuation value). That comparison traces out an early-exercise boundary — a free boundary in the pricing PDE.

Exercisable at any time up to expiry, so pricing is an optimal-stopping problem: V_0 = \sup_{\tau \le T} \mathbb{E}^{\mathbb{Q}}[e^{-r\tau}\,\text{payoff}].
An American option is worth at least the matching European one — the right to exercise early can only add value.
At each instant its value is \max(\text{exercise now},\ \text{continue}).

When early exercise does — and doesn't — pay

The most famous result is a surprise: for an American call on a stock that pays no dividends, it is never optimal to exercise early. Holding the option is always at least as good as exercising, so an American call equals its European twin. But an American put genuinely can be worth exercising early — when it's deep in the money, taking the cash now (and earning interest on it) can beat waiting. So the American put is strictly more valuable than the European put, while (dividend-free) the calls agree.

Exercising a call early throws away two things. First, you pay the strike sooner than you must, so you lose the interest you could have earned on that cash in the meantime. Second, you surrender the option's insurance: if the stock later crashes below the strike, an unexercised call simply expires worthless, but an exercised one has already locked in the shares at a loss. Holding keeps the upside and the downside protection and your cash earning interest — so, with no dividend tempting you to grab the shares, you always wait. (Add a juicy dividend and the calculus flips: capturing it can make early exercise worthwhile.)

An American option is always ≥ the European one — never cheaper. The gap is the early-exercise premium, which can be zero (the dividend-free call).
The exercise decision must be a stopping time: "exercise at the price's peak" is not allowed — it requires seeing the future. Optimality is about the best rule using only past information.