The empirical observations show that the constant elasticity of variance (CEV) model is a practical approach to capture the implied volatility smile phenomenon. Also, due to the appearance of long-range dependence (or long memory effect) in stock returns or volatility, the fractional Black–Scholes models became particularly important in finance. We will generalize the CEV model to a fractional-CEV model with Jumarie’s fractional model (Jumarie, 2008) to capture the long memory effect in addition to show the negative relationship between stock price and its return volatility. The asset price dynamics of this model follows from a fractional stochastic differential equation, and its volatility is a function of the underlying asset price. We derive the fractional Black–Scholes equation by using the It$\hat{\mathrm{o}}$ Lemma and fractional Taylor’s series. Previously, Jumarie’s model was used to determine the value of European and American options. In this study, given the importance of the barrier option, we use the proposed model for pricing a European double barrier option. Since most fractional PDEs do not have closed-form analytical solutions, we solve the option pricing problem numerically. Then, we investigate the stability and convergence of the proposed scheme by applying the Fourier analysis. Finally, some numerical results are given in the last section by computing the European double barrier option.

实验结果表明，恒定弹性方差（CEV）模型是一种捕获隐含波动率微笑现象的实用方法。而且，由于股票收益或波动率中存在长期依赖关系（或长期记忆效应），分数布莱克-斯科尔斯模型在金融中变得尤为重要。我们将利用Jumarie的分数模型（Jumarie，2008年）将CEV模型推广到分数CEV模型（Jumarie，2008年），以捕捉长记忆效应，并证明股票价格与其收益波动率之间存在负相关关系。该模型的资产价格动态来自分数随机微分方程，其波动率是基础资产价格的函数。我们使用It推导分数式Black-Scholes方程$\widehat{\mathrm{Ø}}$引理和分数泰勒级数。以前，Jumarie的模型用于确定欧洲和美国期权的价值。在这项研究中，鉴于障碍期权的重要性，我们使用提议的模型对欧洲双重障碍期权定价。由于大多数分数PDE都没有封闭形式的分析解决方案，因此我们以数字方式解决了期权定价问题。然后，我们通过应用傅立叶分析来研究所提出方案的稳定性和收敛性。最后，通过计算欧洲双重障碍选项，在最后一节中给出了一些数值结果。