The value of an option plays an important role in finance. In this paper, we use the Black–Scholes equation, which is described by the nonsingular fractional-order derivative, to determine the value of an option. We propose both a numerical scheme and an analytical solution. Recent studies in fractional calculus have included new fractional derivatives with exponential kernels and Mittag-Leffler kernels. These derivatives have been found to be applicable in many real-world problems. As fractional derivatives without nonsingular kernels, we use a Caputo–Fabrizio fractional derivative and a Mittag-Leffler fractional derivative. Furthermore, we use the Adams–Bashforth numerical scheme and fractional integration to obtain the numerical scheme and the analytical solution, and we provide graphical representations to illustrate these methods. The graphical representations prove that the Adams–Bashforth approach is helpful in getting the approximate solution for the fractional Black–Scholes equation. Finally, we investigate the volatility of the proposed model and discuss the use of the model in finance. We mainly notice in our results that the fractional-order derivative plays a regulator role in the diffusion process of the Black–Scholes equation.

中文翻译：

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Mittag-Leffler分数阶导数描述的分数阶Black-Scholes方程解的新颖方法

期权的价值在金融中起着重要的作用。在本文中，我们使用由非奇异分数阶导数描述的Black-Scholes方程来确定期权的价值。我们提出了数值方案和解析解。分数微积分的最新研究包括具有指数核和Mittag-Leffler核的新分数导数。已经发现这些导数可应用于许多实际问题。作为没有非奇异内核的分数导数，我们使用Caputo–Fabrizio分数导数和Mittag-Leffler分数导数。此外，我们使用Adams–Bashforth数值方案和分数积分来获得数值方案和解析解，并提供图形表示形式来说明这些方法。图形表示法证明了Adams–Bashforth方法有助于获得分数Black–Scholes方程的近似解。最后，我们调查了该模型的波动性，并讨论了该模型在金融中的使用。我们主要在结果中注意到，分数阶导数在Black-Scholes方程的扩散过程中起调节作用。