an uncertain renewal stock model for barrier options pricing with floating interest rate

Option pricing is a main topic in contemporary financial theories, captivating the attention of numerous financial analysts and economists. barrier option, classified as an exotic option, derives its value from the behavior of an underlying asset. the outcome of this option is based on whether or not the price of the underlying asset has reached a predetermined barrier level. over the years, the stock price has been represented through continuous stochastic processes, with the prominent one being the brownian motion process. correspondingly, the widely used black-scholes model has been employed. nevertheless, it has become evident that utilizing stochastic differential equations to characterize the stock price process is unsuitable and leads to a perplexing paradox. as a result, many researchers have turned to incorporating fuzzy or uncertain environments in such situations. this study presents a methodology for pricing barrier options on stocks in an uncertain environment, in which the interarrival times are uncertain variables. the approach employs the liu process and renewal uncertain process, considering the interest rate as dynamic and floating. the pricing formulas for knock-in barrier options are derived using α-paths of uncertain differential equations with jumps.