European Option

A European option can only be exercised at the end of its life, at its maturity. European options tend to trade at a discount when compared to similar American options. This is because American options allow the option buyer to exercise the option at any time up to and including the expiration date. European options normally trade over the counter, while American options usually trade on standardized exchanges. 

European options are contracts that give the owner the right, but not the obligation, to buy or sell the underlying security at a specific price, known as the strike price, on the option's expiration date. A European call option gives the owner the right to purchase the underlying security, while a European put option gives the owner the right to sell the underlying security.

A buyer of a European option who does not want to wait for maturity to exercise it can sell the option to close the position at any time. Exercising is utilizing the rights of the option in regards to the underlying security. Closing the option position, prior to expiration, means the trader is realizing the loss or profit on the option itself since option prices change based on the movement and volatility of the underlying security and the time until expiration.

The payoffs for a European call option and European put options are as follows:

Call option payoff = max((S - K), 0)

Put option payoff = max((K - S), 0)

S is equal to the spot price of the underlying security and K equals the strike price of the option.

For example, an investor purchases a July call option on stock XYZ with a $50 strike price. At expiration, the spot price of stock XYZ is $75. In this case, the owner of the call option has the right to purchase the stock at $50 and exercises the option, making $25, or ($75 - $50), per share. However, in this scenario, if the spot price of stock XYZ is $30 at expiration, it does not make sense to exercise the option to purchase the stock at $50 when the same stock could be purchased in the spot market for $30. In this case, the payoff is $0.

Note the payoff and profit are different. To calculate the profit from the option, the cost of the contract must be subtracted from the payoff. In this sense, the most an investor in the option can lose is the premium paid for the option. If the investor paid $5 for the call option, and the underlying stock goes to $75, that option is worth $25, and the trader's profit is $20 per share ($25 - $5). An option typically represents 100 shares. If the stock drops to $40, or even $1, the call option is worthless at expiration and the trader loses the $5 they paid for the option.

Owners of European options can only exercise the option on the expiration date. American options allow the owner to exercise the option any time between purchase and expiration dates. Bermuda options are custom contracts that give the owner the right to exercise the option on a variety of specified dates in the contract.