American Rainbow Option Pricing Formulae in Uncertain Environment

American rainbow option is a compound option on multiple underlying assets and its gain depends upon the variations of these assets' prices. This option can entitle investors a right that this option be traded and waived at the discretion of investors. Considering the flexibility of exercising time of American option and the investment preferences of purchasing multiple underlying assets, this paper predominantly discusses American rainbow option pricing under the framework of uncertainty theory. Assuming that independent stock prices obey the corresponding uncertain differential equations, we derive the option pricing formulae. They are American rainbow call on max option, American rainbow call on min option, American rainbow put on max option, American rainbow put on min option, American rainbow put 2 and call 1 option, and American rainbow better of assets option, respectively. Moreover, we verify that the option pricing formulae is valid through some numerical examples. And the effect of relevant parameters on option price is also analyzed.