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Uncertain fractional differential equations (UFDEs) are excellent tools for describing complicated dynamic systems. This study analyzes the valuation problems of currency options based on UFDE under the optimistic value criterion. Firstly, a new uncertain fractional currency model is formulated to describe the dynamics of the foreign exchange rate. Then, the pricing formulae of European, American, and Asian currency options are obtained under the optimistic value criterion. Numerical simulations are performed to discuss the properties of the option prices with respect to some parameters. Finally, a real-world example is provided to show that the uncertain fractional currency model is superior to the classical stochastic model. [ABSTRACT FROM AUTHOR]

In this scientific research note, certain necessary-basic information in relation to some types of the operators specified by fractional calculus of arbitrary order will be firstly introduced, various argument properties of certain analytic functions specified by those fractional-order type operators will be then determined, and a number of special consequences of those extensive properties will be also pointed out. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper studies the smallest fraction of money, or currency divisor, the "Meaja", in the territories of the Castilian crown during the lower Middle Ages, which is initially related to the “obolo”. This coin was extremely important in order to give exactness to transactions and facilitate operations: physically coined (occasionally) or handmadecut of money (more often). The meaja allowed the smallest payments (such as alms, portazgos, sisas, or dues of fellowships). The other divisive fraction (which would become a multiple of dinero in the period), the cinquén, is also contemplated. The meaja evolved in its relative value, being smaller and smaller (within the progressive devaluation of the money that fractioned), arriving at the XV century without just liberating value and its disuse, ending just used for accounting as “imagined money”.

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