In finance, an option is a type of derivative which can be classified in several ways. In general terms they are divided into two groups: the vanilla which comprise of the basic contracts of options – call, put and exotic which incorporate more complex variants.
European option is typically exercised on a given date (exercise date), while American option is carried throughout its tenure, that is, until the date of exercise. The method mostly used to evaluate European options is the Black-Scholes.
For American options, which are rarely exercised before the date of exercise, you can use the same method, assuming that their behavior is similar.
American options expire on the third Saturday of each month, while European options expire on a Friday before the third Saturday of each month. Hence, trading is finalized on a Thursday before the third Saturday of each month.
Exotic options are more complex than commonly traded options (plain vanilla). These products are usually traded over-the-counter (OTC). They incorporate different variants which may complicate calculating the valuation of the option by far. In many cases, value is used to generate random numbers based on the statistical method of Monte Carlo.
Asian options rely on the average of the underlying value in a given period, while the lookback option is calculated on the basis of the maximum (or minimum) attained by the underlying over a given period. Russian option is a lookback option operating in perpetuity.
With a digital/binary option payment can be a certain amount, asset or no payment at all. The swing option allows the buyer to swing the underlying price, and is mainly used for energy. And the parisian option depends on how far the asset is above (or below) the strike.
Bermuda options are exercised at various points, for example, quarterly. Canary options combine the characteristics of a classic European option and a bermuda and can be exercised at various times but never before a fixed period. With a barrier option, the option ceases to exist (knock-out) or begins to exist (knock in) in the event that the underlying touches a given value (otherwise known as barrier level).
Basket option is dependent on a weighted average of various factors. The rainbow option is based on a basket whose weighting of the components relies on their final behavior. On the other hand, an exchange option involves assets being exchanged for another.
Cross/composite option: the underlying currency is traded in a strike and is designated in another. While a quanto option has a fixed exchange rate from the beginning.