In finance, a general term that is used to denote the class into which the option falls is the style or family of an option. It is 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 known as "vanilla options". Options where the payoff is calculated differently are referred to as "exotic options". Challenging problems in valuation and hedging can be posed by exotic options.

American and European Options

The key difference between American and European options is related to when the options can be exercised:

• It is possible that a European option might be exercised only at the expiry date of the option, i.e. at a single pre-defined point in time.
• On the other hand, an American option might be exercised at any time before the expiry date.

For both, the pay-off when it occurs is:

For a call option it is via Max [ (S – K), 0 ],

For a put option it is via Max [ (K – S), 0 ],:

(Here K is the Strike price and S is the spot price of the underlying asset)

Option contracts that are traded on futures exchanges are mainly American-style, whereas those that are traded over-the-counter are mainly European-style.

Difference in Value

Typically European options are valued by using the Black-Scholes or Black model formula. This is a simple equation having a closed-form solution that has become standard in the financial community. For American options there are no general formulae, but a choice of models to approximate the price are available (for example Whaley, binomial options model, and others but there is no idea on which one is preferable). Song-Ping Zhu has published, a semi-closed exact formula for the price of American puts . There had been some debate over whether this formula should be considered as a tractable analytic solution or whether it defines the basis for a genre of numerical methods of solving the problem.

An investor that is holding an American-style option and looking for an optimal value will only exercise it under certain circumstances before maturity. Any option posses a non-negative time value and is usually worth more unexercised. Owners of these options who are willing to realize the full value of their option will mostly prefer to sell it on, rather than to exercise it immediately, sacrificing the time value.

A point where a European option and an American option are otherwise identical (having the same strike price, etc.), then the American option will be worth at least as much as the European (which it entails). If it is worth more, then the difference between the two will guide to the likelihood of early exercise. In practice, the Black-Scholes price of a European option can be calculated  that is equivalent to the American option (except for the exercise dates of course). Then the difference between the two prices can be used in order to calibrate the more complex American option model.

In order to account for the American’s higher value there must be some circumstances in which it is optimal to exercise the American option before that the expiration date is reached. There are several ways in which it can arise, such as:

• An in the money (ITM) call option on a stock is often exercised just before a dividend is paid by the stock pays due to which its value would be lowered by more than the option’s remaining time value.

• A deep ITM currency option (FX option) where the strike currency has a lower interest rate than the currency to be received will often be exercised early it is due to the reason that the time value sacrificed is less valuable than the expected depreciation of the received currency against the strike.

• If ITM and a coupon is due then as American bond option on the dirty price of a bond (such as some convertible bonds) may be exercised immediately.

• A put option on gold will be exercised earlier than the deep ITM, it is due to the reason that gold tends to hold its value whereas the currency that is used as the strike is usually expected to lose value through inflation if the holder waits until final maturity to exercise the option.

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