A study on the fractional Black–Scholes option pricing model of the financial market via the Yang-Abdel-Aty-Cattani operator

IF 1.5 4区 工程技术 Q3 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS
Surath Ghosh
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引用次数: 0

Abstract

Purpose

Financial mathematics is one of the most rapidly evolving fields in today’s banking and cooperative industries. In the current study, a new fractional differentiation operator with a nonsingular kernel based on the Robotnov fractional exponential function (RFEF) is considered for the Black–Scholes model, which is the most important model in finance. For simulations, homotopy perturbation and the Laplace transform are used and the obtained solutions are expressed in terms of the generalized Mittag-Leffler function (MLF).

Design/methodology/approach

The homotopy perturbation method (HPM) with the help of the Laplace transform is presented here to check the behaviours of the solutions of the Black–Scholes model. HPM is well known for its accuracy and simplicity.

Findings

In this attempt, the exact solutions to a famous financial market problem, namely, the BS option pricing model, are obtained using homotopy perturbation and the LT method, where the fractional derivative is taken in a new YAC sense. We obtained solutions for each financial market problem in terms of the generalized Mittag-Leffler function.

Originality/value

The Black–Scholes model is presented using a new kind of operator, the Yang-Abdel-Aty-Cattani (YAC) operator. That is a new concept. The revised model is solved using a well-known semi-analytic technique, the homotopy perturbation method (HPM), with the help of the Laplace transform. Also, the obtained solutions are compared with the exact solutions to prove the effectiveness of the proposed work. The different characteristics of the solutions are investigated for different values of fractional-order derivatives.

通过 Yang-Abdel-Aty-Cattani 算子研究金融市场的分数 Black-Scholes 期权定价模型
目的金融数学是当今银行业和合作行业发展最迅速的领域之一。在当前的研究中,针对金融学中最重要的布莱克-斯科尔斯模型(Black-Scholes model),考虑了一种基于罗伯托诺夫分数指数函数(Robotnov fractional exponential function,RFEF)的具有非星核的新分数微分算子。在模拟过程中,使用了同调扰动和拉普拉斯变换,并用广义米塔格-勒弗勒函数(MLF)来表示所得到的解。设计/方法/方法本文介绍了借助拉普拉斯变换的同调扰动法(HPM),以检验布莱克-斯科尔斯模型解的行为。研究结果在这一尝试中,利用同调扰动和 LT 方法获得了著名金融市场问题(即布莱克-斯科尔斯期权定价模型)的精确解,其中分数导数是在新的 YAC 意义上取的。我们用广义 Mittag-Leffler 函数得到了每个金融市场问题的解。原创性/价值布莱克-斯科尔斯模型是用一种新的算子--Yang-Abdel-Aty-Cattani(YAC)算子提出的。这是一个新概念。在拉普拉斯变换的帮助下,使用著名的半解析技术同调扰动法(HPM)求解了修订后的模型。此外,还将获得的解与精确解进行了比较,以证明拟议工作的有效性。针对不同的分数阶导数值,研究了求解的不同特征。
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来源期刊
Engineering Computations
Engineering Computations 工程技术-工程:综合
CiteScore
3.40
自引率
6.20%
发文量
61
审稿时长
5 months
期刊介绍: The journal presents its readers with broad coverage across all branches of engineering and science of the latest development and application of new solution algorithms, innovative numerical methods and/or solution techniques directed at the utilization of computational methods in engineering analysis, engineering design and practice. For more information visit: http://www.emeraldgrouppublishing.com/ec.htm
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