Pricing of Asian knock-out options in uncertain environment

Asian knock-out options combine the insensitivity of Asian options to price fluctuations and the risk control function of knock-out options. This paper explores two types of options, namely Asian down-and-out call options and Asian up-and-out put options. Initially, we employ an uncertain mean-reverting model and the corresponding uncertain differential equation for floating interest rates to construct a refined model to capture the dynamics of stock price volatility. Building on this foundation, we derive detailed pricing formulas for these options. Subsequently, based on real financial data, we utilize uncertain maximum likelihood estimation and conduct rigorous hypothesis testing to validate the suitability of the uncertain differential equation. Lastly, through the analysis of two examples, we confirm the effectiveness and usability of the derived pricing formulas.