Following a financial loss in trades due to lack of risk management in previous models from market practitioners, Fisher Black and Myron Scholes visited the academic setting and were able to mathematically develop an option pricing equation named the Black–Scholes model. In this study, we address the solution of a Caputo fractional-order Black–Scholes model using an analytic method named the modified initial guess homotopy perturbation method. Foremost, the classical Black Scholes model relaxed for European option style is generalized to be of Caputo derivative. The introduced method is established by coupling a power series function of arbitrary order with the renown He’s homotopy perturbation method. The convergence of the method is demonstrated using the fixed point theorem, and its methodology is illustrated by solving a generalized theoretical form of the fractional order Black Scholes model. The applicability of the method is proven by solving three different fractional order Black–Scholes equations derived from different market scenarios and its effectiveness is confirmed as feasible series of arbitrary orders that accelerate fast to the exact solution at an integer order were obtained. The computation of these results was carried out using Mathematica 12 software. Subsequently, the obtained outcomes were utilized in Maple 18 software to conduct a series of numerical simulations. These simulations aimed to analyze the influence of the fractional order on the dynamics of payoff functions regarding the share value as the option approached its expiration date under varying market constraints. In all three scenarios, the results showed that option values decrease as the expiration date approaches the integer order. Furthermore, the comparative outcomes reveal that Caputo fractional order derivatives control the flexibility of the classical Black–Scholes model because its payoff curve exhibits more sensitivity to changes associated with market characteristic parameters, such as volatility and interest rates. We propose that the results of this work should be examined and implemented by mathematicians and economists to better comprehend the influence of Caputo-fractional order derivatives in understanding the dynamics of option price evolution of financial assets.

中文翻译：

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改进的同伦摄动法及其在Caputo分数阶Black Scholes模型价格演化动力学中的应用

由于市场从业者之前的模型缺乏风险管理而导致交易出现财务损失，费舍尔·布莱克（Fisher Black）和迈伦·斯科尔斯（Myron Scholes）访问了学术界，并能够在数学上开发出一个名为布莱克-斯科尔斯模型的期权定价方程。在本研究中，我们使用一种称为修正初始猜测同伦摄动法的分析方法来解决 Caputo 分数阶 Black-Scholes 模型的求解。首先，针对欧式期权风格放松的经典Black Scholes模型被推广为Caputo导数。所提出的方法是通过将任意阶的幂级数函数与著名的He同伦摄动方法耦合而建立的。使用不动点定理证明了该方法的收敛性，并通过求解分数阶 Black Scholes 模型的广义理论形式来说明其方法。通过求解源自不同市场场景的三个不同分数阶 Black-Scholes 方程证明了该方法的适用性，并通过获得可快速加速到整数阶精确解的可行任意阶序列来证实其有效性。这些结果的计算是使用 Mathematica 12 软件进行的。随后，获得的结果在Maple 18软件中进行了一系列数值模拟。这些模拟旨在分析当期权在不同的市场约束下接近到期日时，分数阶对关于股票价值的支付函数动态的影响。在所有三种情况下，结果表明，随着到期日接近整数阶，期权价值会下降。此外，比较结果表明，Caputo 分数阶导数控制了经典 Black-Scholes 模型的灵活性，因为其收益曲线对与市场特征参数（例如波动性和利率）相关的变化表现出更高的敏感性。我们建议数学家和经济学家应检查和实施这项工作的结果，以更好地理解卡普托分数阶导数对理解金融资产期权价格演化动态的影响。