This paper aims to solve the Black–Scholes (B–S) model for the European options pricing problem using a hybrid method called fractional generalized homotopy analysis method (FGHAM). The convergence region of the B–S model solutions are clearly identified using h-curve and the closed form series solutions are produced using FGHAM. To verify the convergence of the proposed series solutions, sequence of errors are obtained by estimating the deviation between the exact solution and the series solution, which is increased in number of terms in the series. The convergence of sequence of errors is verified using the convergence criteria and the results are graphically illustrated. Moreover, the FGHAM approach has overcome the difficulties of applying multiple integration and differentiation procedures while obtaining the solution using well-established methods such as homotopy analysis method and homotopy perturbation method. The computational efficiency of the proposed method is analyzed using a comparative study. The advantage of the proposed method is shown with a numerical example using the comparative study between FGHAM and Monte Carlo simulation. Using the numerical example, analytical expression for the implied volatility is derived and the non-local behavior is studied for the various values of the fractional parameter. The results of FGHAM are statistically validated with the exact solution and the other existing computational methods.

中文翻译：

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使用分数广义同伦分析方法求解Black-Scholes方程

本文旨在使用一种称为分数广义同伦分析方法（FGHAM）的混合方法来解决欧洲期权定价问题的Black-Scholes（BS）模型。使用h可以清楚地识别出BS模型解的收敛区域-曲线和封闭形式系列解决方案使用FGHAM生产。为了验证所提出的级数解的收敛性，通过估计精确解和级数解之间的偏差来获得误差序列，该偏差在级数上增加了。使用收敛准则验证错误序列的收敛性，并以图形方式说明结果。此外，FGHAM方法克服了应用多重积分和微分程序的困难，同时使用了完善的方法（例如同伦分析方法和同伦摄动方法）获得了解决方案。通过比较研究分析了所提出方法的计算效率。通过FGHAM和Monte Carlo仿真的比较研究，通过数值例子说明了该方法的优点。使用数值示例，推导了隐含波动率的解析表达式，并研究了分数参数的各种值的非局部行为。FGHAM的结果已通过精确解和其他现有的计算方法进行了统计验证。