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# Application of the Generalized Laplace Homotopy Perturbation Method to the Time-Fractional Black–Scholes Equations Based on the Katugampola Fractional Derivative in Caputo Type

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Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
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Center of Excellence in Mathematics, Commission on Higher Education, Bangkok 10400, Thailand
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Department of Mathematics, Faculty of Applyied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
*
Author to whom correspondence should be addressed.
Academic Editors: Yongwimon Lenbury, Ravi P. Agarwal, Philip Broadbridge and Dongwoo Sheen
Received: 28 January 2021 / Revised: 22 February 2021 / Accepted: 4 March 2021 / Published: 12 March 2021
In the finance market, the Black–Scholes equation is used to model the price change of the underlying fractal transmission system. Moreover, the fractional differential equations recently are accepted by researchers that fractional differential equations are a powerful tool in studying fractal geometry and fractal dynamics. Fractional differential equations are used in modeling the various important situations or phenomena in the real world such as fluid flow, acoustics, electromagnetic, electrochemistry and material science. There is an important question in finance: “Can the fractional differential equation be applied in the financial market?”. The answer is “Yes”. Due to the self-similar property of the fractional derivative, it can reply to the long-range dependence better than the integer-order derivative. Thus, these advantages are beneficial to manage the fractal structure in the financial market. In this article, the classical Black–Scholes equation with two assets for the European call option is modified by replacing the order of ordinary derivative with the fractional derivative order in the Caputo type Katugampola fractional derivative sense. The analytic solution of time-fractional Black–Scholes European call option pricing equation with two assets is derived by using the generalized Laplace homotopy perturbation method. The used method is the combination of the homotopy perturbation method and generalized Laplace transform. The analytic solution of the time-fractional Black–Scholes equation is carried out in the form of a Mittag–Leffler function. Finally, the effects of the fractional-order in the Caputo type Katugampola fractional derivative to change of a European call option price are shown. View Full-Text
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MDPI and ACS Style

Thanompolkrang, S.; Sawangtong, W.; Sawangtong, P. Application of the Generalized Laplace Homotopy Perturbation Method to the Time-Fractional Black–Scholes Equations Based on the Katugampola Fractional Derivative in Caputo Type. Computation 2021, 9, 33. https://0-doi-org.brum.beds.ac.uk/10.3390/computation9030033

AMA Style

Thanompolkrang S, Sawangtong W, Sawangtong P. Application of the Generalized Laplace Homotopy Perturbation Method to the Time-Fractional Black–Scholes Equations Based on the Katugampola Fractional Derivative in Caputo Type. Computation. 2021; 9(3):33. https://0-doi-org.brum.beds.ac.uk/10.3390/computation9030033

Chicago/Turabian Style

Thanompolkrang, Sirunya, Wannika Sawangtong, and Panumart Sawangtong. 2021. "Application of the Generalized Laplace Homotopy Perturbation Method to the Time-Fractional Black–Scholes Equations Based on the Katugampola Fractional Derivative in Caputo Type" Computation 9, no. 3: 33. https://0-doi-org.brum.beds.ac.uk/10.3390/computation9030033

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