This paper presents a novel numerical method, named extended fractional neural stochastic differential equation (fNSDE)-Net, which combines the generative adversarial network (GAN) and fNSDE with a self-attention module. The method is designed to generate and forecast the stock price of Contemporary Amperex Technology Co., Ltd. (CATL) in China. The primary challenge of this study lies in the fact that the input consists of a single, irregular time-series dataset with long-range dependencies (i.e., Hurst index $H>\frac{1}{2}$), and its inherent noise cannot be directly modeled using pure Brownian motion. The proposed method not only generates multiple sample paths based on the initial data in a probabilistic sense but also preserves the long-term memory characteristics of the generated samples. Moreover, the pricing of a Bermuda call option, induced by stock prices, is explored. Through a series of numerical error comparisons and estimator reliability tests, the proposed method outperforms both the pure fNSDE-GAN method and the NSDE-GAN method in terms of fitting and generalization performance, thereby demonstrating its effectiveness.