Uncertain Currency Option Pricing Based on the Fractional Differential Equation in the Caputo Sense

Abstract

The foreign exchange market comprises the largest global volume, so the pricing of foreign exchange options has always been a hot issue in the foreign exchange market. This paper treats the exchange rate as an uncertain process that is described by an uncertain fractional differential equation, and establishes a new uncertain fractional currency model. The uncertain process is driven by Liu process, and, with the application of the Mittag-Leffler function, the solution of the fractional differential equation in a Caputo sense is presented. Then, according to the uncertain fractional currency model, the pricing formulas of European and American currency options are given. Lastly, the two numerical examples of European and American currency options are given; the price of the currency option increased when p changed from 1.0 to 1.1, and prices with different p were all decreasing functions of exercise price K.