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Any option contract that is not an American or European option. That is, an exotic option contains some provision that makes it different from a straightforward option contract with a strike price, underlying asset, and expiration date. For example, a chooser option allows the holder to decide whether the contract is a call or a put at some point over the contract's life. Also, an Asian option has no set strike price and is calculated as the average of some price listed in the contract and the market value of the underlying asset at the time of exercise. Most exotic options trade over-the-counter because their provisions make them too complex for the regulations of exchanges.
An option with a nonstandard feature. A lookback option that allows the owner flexibility in selecting an exercise price is an example of an exotic option. Exotic options are traded in the over-the-counter market.