American Barrier Option Pricing Formulas for Stock Model in Uncertain Environment

In the foundation of uncertainty theory, uncertain stock model has been put forward to portray the price fluctuation of stocks in a market with uncertain information. In this paper, the model is depicted by uncertain differential equations involved by a Liu process that is a sequence of uncertain variables varying with time. According to this model, we mainly investigate the formulas to price the American barrier option for rights of buying or selling the stock with a set price in the uncertain financial market. Then, four new types of concepts are introduced that are, respectively, American call options, including both up-and-in and down-and-out, and American put options, including both down-and-in and up-and-out. Moreover, several formulas are derived for giving the price of the corresponding four types of options. At the same time, some examples are given.