European Option

The holder makes no exercise decision during the life of the contract. On the expiration date, value is determined mechanically: a call pays the underlying price minus the exercise price when that difference is positive and zero otherwise; a put pays the exercise price minus the underlying price under the same condition. Because no decision is required before that single date, the terminal payoff is a fixed function of the underlying's price at one point in time.

Settlement follows two conventions. Physically settled contracts deliver the underlying against payment of the exercise price; cash-settled contracts transfer the payoff amount only, which is why index options, where delivering a basket of hundreds of constituents would be impractical, are almost universally cash-settled and European-style. The exercise restriction also removes assignment risk for the writer before expiry, a practical reason institutions prefer European terms in over-the-counter documentation.

Put-call parity states that a European call plus the exercise price discounted at the risk-free rate to expiration equals the corresponding European put plus the underlying's spot price, for options sharing the same underlying, strike and maturity. The relation is an arbitrage identity only because neither side can be terminated early: both portfolios are known to produce identical values on one specific date. Introduce early exercise and the equality collapses into a pair of inequalities.

The same logic drives Black-Scholes-Merton, which prices a claim on a single terminal distribution and therefore requires no optimal-stopping analysis. In a binomial tree, European style means the model rolls back discounted expected values under risk-neutral probabilities without ever comparing a node value to immediate intrinsic value; that comparison at each node is precisely what distinguishes American pricing and what makes closed-form American solutions unavailable in general.

Exchange-traded index options, currency options traded over the counter, and most interest rate options embedded in swap and loan structures use European exercise. Each caplet within an interest rate cap is a European call on a reference rate fixing on a single date; a swaption grants a one-date right to enter a swap. Structured notes, convertible features priced by dealers, and employee-style contracts on indices typically specify European terms so the issuer can hedge a fixed-maturity exposure rather than an uncertain one.

The restriction is on exercise, not on liquidity. Positions are routinely closed by an offsetting trade in the secondary market or, in the OTC world, by negotiated unwind or novation, capturing the option's full market value including its remaining time value. Time value is in fact what early exercise destroys, which is why exercising an American call on a non-dividend-paying stock is economically inferior to selling it.

A candidate should be able to compute the lower bound of a European call as the greater of zero and the spot price minus the present value of the exercise price, and the lower bound of a European put as the greater of zero and the present value of the exercise price minus the spot price. The put bound explains a frequently tested result: a deep in-the-money European put can trade below intrinsic value, because the exercise proceeds arrive only at expiration and must be discounted.

Parity should be usable in every direction: solving for any one of the four instruments, constructing a synthetic underlying from a long call, short put and a risk-free bond, or identifying arbitrage when quoted prices violate the identity. Extending parity to a dividend-paying underlying means subtracting the present value of expected dividends from the spot price. Candidates should also recognise that dividends raise the value of early exercise for American calls, widening the gap between the two exercise styles.