In the finance market, it is well known that the price change of the underlying fractal transmission system can be modeled with the Black-Scholes equation. This article deals with finding the approximate analytic solutions for the time-fractional Black-Scholes equation with the fractional integral boundary condition for a European option pricing problem in the Katugampola fractional derivative sense. It is well known that the Katugampola fractional derivative generalizes both the Riemann–Liouville fractional derivative and the Hadamard fractional derivative. The technique used to find the approximate analytic solutions of the time-fractional Black-Scholes equation is the generalized Laplace homotopy perturbation method, the combination of the generalized Laplace transform and homotopy perturbation method. The approximate analytic solution for the problem is in the form of the generalized Mittag-Leffler function. This shows that the generalized Laplace homotopy perturbation method is one of the most effective methods to construct approximate analytic solutions of the fractional differential equations. Finally, the approximate analytic solutions of the Riemann–Liouville and Hadamard fractional Black-Scholes equation with the European option are also shown.

中文翻译：

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基于Katugampola分数阶导数的带欧式期权的时间分形Black-Scholes方程的近似解析解

在金融市场上，众所周知，基本的分形传输系统的价格变化可以用Black-Scholes方程建模。本文讨论的是具有分数阶积分边界条件的时间分数阶Black-Scholes方程在Katugampola分数导数意义上的欧式期权定价问题的近似解析解。众所周知，Katugampola分数导数同时推广了Riemann-Liouville分数导数和Hadamard分数导数。用于找到时间分数阶Black-Scholes方程的近似解析解的技术是广义Laplace同伦摄动方法，广义Laplace变换和同伦摄动方法的组合。该问题的近似解析解采用广义Mittag-Leffler函数的形式。这表明广义拉普拉斯同伦摄动方法是构造分数阶微分方程的近似解析解的最有效方法之一。最后，还显示了带有欧洲期权的Riemann-Liouville和Hadamard分数Black-Scholes方程的近似解析解。