Option Style: American vs European
The "style" of an option contract refers to whether the contract can be exercised prior to the expiration date.
The buyer of an American option can choose to exercise the option any time up until and including the expiration date.
However, the buyer of a European option can only exercise the option at the expiration date.
Note: the option to exercise always resides with the buyer; the seller of the option has the obligation to deliver if exercised by the option buyer.
Almost all listed equity options (options listed on stocks) are American options; as a buyer you can exercise at any time during the life of the contract.
Say AMZN stock is trading at 1,468 and you buy a $1,500 call option that expires in 3 weeks and you paid $45 for the option.
As the stock price is below the strike price for the call, the exercise value of the option is 0 (more commonly known as intrinsic value). However, if the stock price rallies to $1,580 then the option now has some exercise value, which is $80.
Because the option is American styled you can exercise the call option and you will be assigned AMZN stock at the strike price of $1,500. Now that you own the stock at a purchase price of $1,500 you can sell it back on the market for $1,580 and make an immediate gain of $80.
Subtract the $45 paid for the option and your net profit is $35.
The advantage here is that if the stock rises above the strike price prior to the expiration date then you can exercise, take the stock and either continue to hold it or sell it for profit; you're not stuck with the option until it expires before deciding.
By taking advantage of an early favourable price move means you don't the risk of the stock falling back below the strike and having the option expire worthless.
Having this extra flexibility means the American options generally trade at a premium to European options.
American vs. European Style Options
This means that, all other things being equal, an American call/put option will have a slightly higher purchase price than the same option that has a European exercise style. For this reason, traders who value options theoretically will need to ensure that they use the correct pricing model for American options.
The most common type of model used for options that have early exercise is the Binomial Model.
Options that have a European exercise style mean that the buyer can only exercise the option at the expiration date of the contract.
While almost all stock options are American style, all index option contracts are European styled.
This is due to the complexities that would arise as a result of early exercising a cash index product.
Take the S&P 500 for example, which is an index comprised of the top 500 stocks, weighted by market capitalization, in the US market.
If you were long an index call option and the S&P 500 traded above this strike and you decided to exercise the option for delivery, this would be a major problem for the brokers and clearing houses to facilitate this transaction.
At the time of "exercise", every component stock price as well as the weight of each compared to the number of options, would need to be recorded.
These values would then be used to determine the split of stocks at each strike price, which would be the component price at the time of exercised. Administratively this is just too difficult for brokers to consider, which is why these products do not allow for early exercise.
European options, however, are easier to price theoretically and are able to be price using Analytical models.
The most common for European options is the Black and Scholes.