American Option

The defining feature is not the payoff but the timing choice. At every moment before expiry the holder compares two quantities: the exercise value available immediately, which is the intrinsic value of the contract, and the continuation value, which is what the still-alive option is worth if left unexercised. The option's fair price is the greater of the two, so it can never trade below intrinsic value without creating an arbitrage. The gap between the American price and the price of the matching European contract is the early-exercise premium, and it is the market's valuation of that timing right.

Because the right is optional, it can only add value or add nothing. Where immediate exercise is never optimal under any future path, the continuation value dominates everywhere and the two styles converge to the same price.

For a call on an asset that pays no cash flow before expiry, exercising early is dominated. Exercising surrenders the remaining time value and pays the strike sooner than necessary, forfeiting interest on that cash, while the holder gains nothing that could not be obtained by selling the option instead. Such a call is therefore priced as if it were European.

Two situations restore economic content. A dividend or other cash distribution reduces the underlying price on the ex-date, so a call holder may prefer to own the share and capture the payment when it exceeds the time value given up. For puts the logic is reversed: exercising delivers the strike in cash now, and that cash earns interest, which is why a deep in-the-money put on a low-yielding underlying can be worth exercising well before expiry.

No simple closed-form solution exists for the general American case, so valuation is normally lattice-based. In a binomial model the terminal payoffs are computed first, then at each earlier node the risk-neutral expected value of the two successor nodes is discounted one period to give the continuation value, and that figure is compared with the intrinsic value at that node. The node value is set to the larger of the two, and the comparison is repeated backward to time zero. Any node where intrinsic wins marks an optimal early-exercise point.

Put-call parity also weakens. For European contracts it holds as an equation; for American contracts on a non-dividend-paying asset it becomes a band, with the call minus put difference bounded below by spot minus strike and above by spot minus the present value of the strike.

Style is a contract specification, not a market-wide rule. Individual equity options in the United States are conventionally American, broad index options are often European and cash-settled, and options on futures and many currency contracts are frequently American. Bermudan contracts sit between the two, permitting exercise only on stated dates. Writers of American options carry assignment risk: they can be called upon at any time, most commonly on calls immediately before an ex-dividend date, which turns a hedged book into an unintended stock position overnight.

Candidates should be able to state that the American price is at least the European price, identify the cases where the two coincide, run early-exercise checks through a two-period binomial tree, and apply the parity inequality rather than the equality when the style is American.