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In finance, the style or family of an option is the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American style options. These options—as well as others where the payoff is calculated similarly—are referred to as " vanilla options ".

Options where the payoff is calculated differently are categorized as " exotic options ". Exotic options can pose challenging problems in valuation and hedging. The key difference between American and European options relates to when the options can be exercised:. Where K is the strike price and S is the spot price of the underlying asset. Option contracts traded on futures exchanges are mainly American-style, whereas those traded over-the-counter are mainly European.

Nearly all stock and equity options are American options, while indexes are generally represented by European options. Commodity options can be either style. Traditional monthly American options expire the third Saturday of every month. They are closed for trading the Friday prior. European options expire the Friday prior to the third Saturday of every month. Therefore, they are closed for trading the Thursday prior to the third Saturday of every month.

Assuming an arbitrage-free market, a partial differential equation known as the Black-Scholes equation can be derived to describe the prices of derivative securities as a function of few parameters. Under simplifying assumptions of the widely adopted Black model , the Black-Scholes equation for European options has a closed-form solution known as the Black-Scholes formula.

In general, no corresponding formula exist for American options, but a choice of methods to approximate the price are available for example Roll-Geske-Whaley, Barone-Adesi and Whaley, Bjerksund and Stensland, binomial options model by Cox-Ross-Rubinstein, Black's approximation and others; there is no consensus on which is preferable.

An investor holding an American-style option and seeking optimal value will only exercise it before maturity under certain circumstances. Owners who wish to realise the full value of their option will mostly prefer to sell it on, rather than exercise it immediately, sacrificing the time value. Where an American and a European option are otherwise identical having the same strike price , etc.

If it is worth more, then the difference is a guide to the likelihood of early exercise. In practice, one can calculate the Black—Scholes price of a European option that is equivalent to the American option except for the exercise dates of course. The difference between the two prices can then be used to calibrate the more complex American option model. To account for the American's higher value there must be some situations in which it is optimal to exercise the American option before the expiration date.

This can arise in several ways, such as:. There are other, more unusual exercise styles in which the payoff value remains the same as a standard option as in the classic American and European options above but where early exercise occurs differently:. These options can be exercised either European style or American style; they differ from the plain vanilla option only in the calculation of their payoff value:.

The following " exotic options " are still options, but have payoffs calculated quite differently from those above. Although these instruments are far more unusual they can also vary in exercise style at least theoretically between European and American:.

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