American Options and Early Exercise

American options add one right to the European contract: exercise any time before expiry. Pricing that right is a clean application of backward induction, and reasoning about when it's worth using is one of the most reliable derivatives interview questions in existence.

The famous asymmetry

Never exercise an American call early on a non-dividend stock. The proof is worth performing, not citing. Exercising early pays $S - K$ now. But the alive call is worth at least $S - Ke^{-rT}$ (it dominates a forward struck at $K$ — model-free bound from the no-arbitrage lesson), and

$$S - Ke^{-rT} > S - K$$

whenever rates are positive. Alive beats exercised, always: paying the strike later is cheaper, and killing the option destroys its remaining optionality. Selling the option always beats exercising it. Hence American calls on non-dividend stocks are worth exactly their European twins — the early-exercise right is worthless.

Puts break the symmetry. Exercising a deep-in-the-money put pays $K - S$ now — cash you can invest at $r$. If the stock has fallen near zero, the put's remaining upside is capped (price can't drop below zero), while interest on $K$ ticks along. Somewhere below a critical stock price $S^*$, taking the money wins. So American puts carry a genuine early-exercise premium, and there's an exercise boundary: exercise when $S < S^*$, hold above.

Dividends restore the call's case. A dividend is value leaking out of the stock on a known date; an in-the-money call holder may rationally exercise just before the ex-dividend date to capture it. The complete decision rule interviewers want: compare the dividend gained against the time value surrendered — exercise iff the dividend exceeds the remaining time value of the option.

The compact summary that answers 80% of follow-ups: early exercise is about carry. Calls die early to catch income leaving the stock (dividends); puts die early to start income on the strike (interest). No income motive, no early exercise.

Pricing on the tree

The binomial lesson's backward induction gains one line. At every node:

$$V = \max\Big(\underbrace{\text{exercise value}}_{K - S \text{ or } S - K}, \; \underbrace{e^{-r\Delta t}\left[qV_{up} + (1-q)V_{down}\right]}_{\text{continuation value}}\Big)$$

That max(exercise, continue) is precisely the take-or-skip pattern from the dynamic-programming lesson — American option pricing is DP, with the exercise boundary emerging as the set of nodes where exercise wins. This is also why American options lack a Black–Scholes-style closed form: the boundary must be found, not assumed. (Related trivia with content: on indexes and FX you'll meet "Bermudan" options — exercisable on specific dates — sitting between European and American, priced the same way with the max applied only on exercise dates.)

Consequences worth knowing

• Put-call parity degrades to inequalities for American options ($S - K \le C - P \le S - Ke^{-rT}$): the clean equality was a European-only privilege.
• An American option is never worth less than its European twin or its intrinsic value — free extra rights can't have negative value; any quote violating that is arbitrage.
• Almost all single-stock listed options are American; index options (SPX) are European — so this lesson is about the contracts people actually trade.

The interview version

"Rates are 5%. Deep ITM American put, stock at \2, strike \100, a year left. What do you do and why?" — Exercise: collect \98 now and earn ~\4.90 of interest; the put can gain at most $2 more if the stock hits zero, and time value at that depth is negligible. "Same but a call?" — Never early (no dividend): sell it if you want out. If you can also sketch the tree's max(exercise, continue) and say "the boundary is where these cross," the topic is closed.