European Option

A European option is an option that gives its holder the right to buy or sell the underlying asset at the strike price on one date only, the expiry date. The holder cannot exercise early. This distinguishes it from an American option, which may be exercised on any day up to expiry. The terms refer to the exercise style and not to geography: both kinds trade in markets worldwide.

A European option can still be bought and sold at any time before expiry, so a holder who wants to realize a gain early can sell the contract in the market. The single exercise date makes European options simpler to value and to hedge, and most of option pricing theory was first worked out for them.

Most options on broad stock indices are European. Options on the S&P 500 index listed by Cboe, and index options on Eurex and on the National Stock Exchange of India, are of this style and are settled in cash, since delivering every share in an index would be impractical. In over-the-counter markets, the standard FX option is European, as is the standard swaption and most interest rate caps and floors.

Options on individual shares differ by country. In the United States they are American. Several other markets, India among them, use European exercise for single-stock options as well. Exotic contracts such as the Asian option are usually European in exercise, with the payoff worked out at the end.

At expiry a European call option is worth the amount by which the underlying price exceeds the strike price, or nothing if it does not. A put option is worth the amount by which the strike exceeds the underlying price, or nothing. Exchanges exercise in-the-money contracts automatically at option expiry. With cash settlement, the holder receives the difference in money, calculated from an official settlement price.

The prices of European calls and puts with the same strike and expiry are tied together by a no-arbitrage relationship called put-call parity. For an underlying that pays no income before expiry:

• $C$ and $P$ are the prices of the call and the put.
• $S$ is the current price of the underlying.
• $K$ is the strike price.
• $r$ is the risk-free rate, continuously compounded, and $T$ is the time to expiry in years.

The relationship holds because a portfolio of a call plus cash equal to the present value of the strike pays the same at expiry as a portfolio of a put plus the underlying. If the prices differed, a trader could buy the cheaper portfolio and sell the dearer one for a riskless profit through arbitrage.

Suppose a share trades at $100, and one-year options have a strike of $100 with the interest rate at 5 percent. The present value of the strike is 100 × $e^{-0.05}$ = 100 × 0.9512 = $95.12. If the call costs $10.45, the put must cost 10.45 − 100 + 95.12 = $5.57. Parity is exact only for European options. With American options, the right to exercise early turns the equality into a pair of bounds.

The Black-Scholes model gives a closed formula for the price of a European call or put from five inputs: the underlying price, strike, time to expiry, interest rate and volatility. The formula applies directly because the model assumes exercise only at expiry. Variants cover options on assets that pay a dividend, on currencies and on futures. The figures in the parity example are consistent with the model at a volatility of 20 percent.

Where the model's assumptions are too restrictive, European options are valued numerically. The binomial option pricing model converges to the Black-Scholes value as the number of steps grows, and simulation methods handle more complex payoffs. In practice, traders quote European options by implied volatility, the volatility that makes the formula match the market price.

An American option carries every right of the European one and the right of early exercise besides, so it can never be worth less. For a call on a share that pays no dividend, early exercise is never worthwhile, and the two styles have the same value. For puts, and for calls on shares with large dividends, the American version is worth more.

One consequence is that a European put that is deep in the money can trade below its intrinsic value. The holder must wait until expiry to collect the strike, and the delay has a cost when interest rates are positive. An American put would be exercised instead.

For sellers, the European style removes the risk of early assignment. A trader who has sold a European option, or holds a spread built from European options, knows that nothing can be exercised until the final day. This predictability is one reason index option strategies are popular with institutional traders. A middle style, the Bermudan option, allows exercise on a set of agreed dates and is common in interest rate markets.

• Cboe Global Markets
• The Options Clearing Corporation