Reliability Index And Option Pricing Formulas Of The First-Hitting Time Model Based On The Uncertain Fractional-Order Differential Equation With Caputo Type

Since the ability to control the investor's income or loss within a certain range, barrier option has been among the most popular path-dependent options where its payoff depends on whether or not the underlying asset's price reaches a given "barrier". First, assuming the underlying asset as an uncertain variable for the case that the Caputo fractional-order derivative is adopted instead of the ordinary derivative, the real financial market is better modeled by the uncertain fractional-order differential equation with Caputo type. Then, a first-hitting time model which can measure the exercise ability is innovatively presented. Second, based on the first-hitting time theorem of the uncertain fractional-order differential equation, the reliability index (including validity and survival index) for the proposed model is obtained, and four types of European barrier option (including up-and-in call, down-and-in put, up-and-out put, and down-and-out call options) pricing formulas are obtained accordingly. Lastly, applying the predictor-corrector method, numerical algorithms are provided for calculating European barrier and the reliability index, numerical experiments and corresponding sensitivity analysis are also illustrated concerning various conditions.