American-option pricing
An American option differs from a European one in a single but decisive way: the holder may exercise it at ANY time up to expiry, not only at the final date. Pricing it is therefore not a one-shot expectation but an optimal stopping problem: the fair value is the supremum, over all stopping times tau bounded by the maturity T, of the discounted expected payoff under the risk-neutral measure, V_0 = sup_{tau <= T} E*[ e^(-r tau) g(S_tau) ]. The holder chooses tau to maximise; the seller must hedge against the holder's best choice. This is the canonical place where the Snell envelope, the free-boundary problem, and risk-neutral pricing all meet in one applied object.
Mechanically there are three equivalent views. (1) Snell envelope: discretise time and run backward induction, U_t = max( g(S_t), e^(-r dt) E*[ U_{t+dt} given F_t ] ); the price is U_0 and you exercise wherever U = g. (2) Free boundary: in the Black-Scholes diffusion model V(t,s) solves the variational inequality min{ -V_t - L_BS V + r V, V - g } = 0, where L_BS is the Black-Scholes operator; the early-exercise boundary s*(t) is the free boundary, joined by smooth fit (V and V_s continuous across it). (3) Decomposition: the American price equals the European price plus an early-exercise premium, an integral of the cash flows accruing while the option is in the exercise region. For the American call on a non-dividend-paying stock, a classic and honest fact: it is never optimal to exercise early, so its price equals the European call's; the early-exercise feature is worthless without dividends.
American pricing is where theory becomes computation: no closed form exists in general, so practitioners use binomial trees (the discrete Snell envelope), finite-difference solvers for the variational inequality, or Monte Carlo with regression (Longstaff-Schwartz) to estimate the continuation value E*[ U_{t+dt} given F_t ]. An honest caveat: the put on a dividend-free stock, and the call on a dividend-paying stock, genuinely have early exercise, and the boundary s*(t) has no elementary formula; also the equivalence of the three views relies on the model being complete and arbitrage-free, so that the risk-neutral expectation is the unique no-arbitrage price — in incomplete markets pricing American claims becomes a more delicate (sub/superhedging) story.
An American put on a non-dividend stock: as the stock S falls, the payoff (K - S)^+ rises and time value erodes, so below a critical level s*(t) the holder should exercise to capture K - S now rather than wait. On a binomial tree this shows up as nodes where (K - S)^+ exceeds the discounted continuation value. The American put is strictly worth more than the European put — the gap is the early-exercise premium.
The American put has genuine early exercise below a critical price; the American call on a dividend-free stock does not.
Common error: thinking American always beats European. For a call on a non-dividend stock, early exercise is never optimal, so the two prices coincide — the extra right is worthless without dividends.