Options that can be exercised anytime up to their expiration date, are American Style Options (American Options). In the United States, most options are of this type. The American Style option provides an investor with a greater degree of flexibility than a European-style option.
Options which can only be exercised at their expiration, are European Style options.
A typically used formula to evaluate European options, is the Black-Scholes or Black model formula. There are no general formulas for American Style options.
In the investment world, the options style is determined in relation to the date the options could be exercised
The majority of options are either European Style or American Style options. The most of the U.S. listed options are American style options that could be exercised on any trading day from the day the options were purchased and until the expiration date. In opposite the European style options are less flexible and could be exercised only on the expiration date. The majority of the over-the counter options (OTC options) are European style options.
The different options styles could be exercise on the different dates and the payoff value (value when the options are exercised) of the different options styles could be calculated by different methods. The American and European (style) options payoff is calculated in similar way and these options are called "vanilla options". The options that payoff are calculated by different a method called "exotic options".
For both American and European options payoff is calculated by the following formula:
Where Strike is a strike price of the exercised options and S is a spot price of the underlying assets at the moment when the option is exercised.
From the formula above you may see that if on the options expiration date the price of the underlying assets is higher than the striker price of the put options then the put options expire worthless and if the price of the underlying assets is lower than the strike price then call options expire worthless.
Even the American options could be exercised before the expiration date; it's rarely done on the practice. It simply because unexercised options are worth more. A trader who is willing to receive the full value of the options before the expiration date would rather sell it then exercise it. Due to the high liquidity it is not a big problem for listed options. One of the reasons why a trader may be wishing to exercise the call options before the expiration date is the dividend that could be paid on the underlying stock.
Other then the expiration rights American and European options are identical.
Naked options trading is very risky - many people lose money trading them. It is recommended contacting your broker or investment professional to find out about trading risk and margin requirements before getting involved into trading uncovered options.