Reliability index and Asian barrier option pricing formulas of the uncertain fractional first-hitting time model with Caputo type

Abstract

Abstract This paper mainly studies the pricing problem of arithmetic average Asian barrier options in the continuous case and analyzes the corresponding reliability index. Owing to the fact that the return function of the option is closely related to the average price of the underlying asset in a certain period, which can effectively alleviate the market speculation, the Asian barrier option is widely active in the financial market as a derivative product. Firstly, considering that there exists difficult to predict and measure reliability based on historical data, traditional stock models relied on stochastic theory are abandoned, and further assumes that the underlying assets follow an uncertain process. Then, we introduce the uncertain fractional differential equations with Caputo-type to describe the dynamic changes of risky asset prices. Secondly, in order to analyze the impact on the first hitting time of the barrier being triggered and the option execution result theoretically, a novel first-hitting time model is established to measure the relationship between the option reliability index and the value of risky assets. Meanwhile, the pricing formulas of four Asian barrier options and the corresponding reliability index under the first-hitting time model are derived, respectively. Finally, numerical algorithms and corresponding methods are designed to verify the results of our model.