An option that confers on its holder the right to choose between two risky assets (the two underlying assets), whichever is associated with the maximum gain. For example, a call option on the maximum (with a European-style exercise) gives its holder the right to buy the asset with the higher market price for the set exercise price at the option’s expiration date (in the case where the prices of the two assets have traded above the exercise price), while the holder of a put option on the maximum (with a European-style exercise) has the right to exercise on (sell) the asset with the higher market price (of course, in the case where the prices of the two assets have dropped below the exercise price, otherwise the option expires worthless).

An option on the maximum has many other names including a better-of-two-assets option or an outperformance option.