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The Analytical Solution for the Black-Scholes Equation with Two Assets in the Liouville-Caputo Fractional Derivative Sense

Author

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  • Panumart Sawangtong

    (Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand)

  • Kamonchat Trachoo

    (Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand)

  • Wannika Sawangtong

    (Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
    Centre of Excellence in Mathematics, Commission on Higher Education, Ministry of Education, 328 Sri Ayuthaya Road, Bangkok 10400, Thailand)

  • Benchawan Wiwattanapataphee

    (School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Perth, WA 6845, Australia)

Abstract

It is well known that the Black-Scholes model is used to establish the behavior of the option pricing in the financial market. In this paper, we propose the modified version of Black-Scholes model with two assets based on the Liouville-Caputo fractional derivative. The analytical solution of the proposed model is investigated by the Laplace transform homotopy perturbation method.

Suggested Citation

  • Panumart Sawangtong & Kamonchat Trachoo & Wannika Sawangtong & Benchawan Wiwattanapataphee, 2018. "The Analytical Solution for the Black-Scholes Equation with Two Assets in the Liouville-Caputo Fractional Derivative Sense," Mathematics, MDPI, vol. 6(8), pages 1-14, July.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:8:p:129-:d:159861
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    References listed on IDEAS

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    1. Kim, Junseok & Kim, Taekkeun & Jo, Jaehyun & Choi, Yongho & Lee, Seunggyu & Hwang, Hyeongseok & Yoo, Minhyun & Jeong, Darae, 2016. "A practical finite difference method for the three-dimensional Black–Scholes equation," European Journal of Operational Research, Elsevier, vol. 252(1), pages 183-190.
    2. M. F. M. Osborne, 1959. "Brownian Motion in the Stock Market," Operations Research, INFORMS, vol. 7(2), pages 145-173, April.
    3. Tomas Björk & Henrik Hult, 2005. "A note on Wick products and the fractional Black-Scholes model," Finance and Stochastics, Springer, vol. 9(2), pages 197-209, April.
    4. Lina Song & Weiguo Wang, 2013. "Solution of the Fractional Black-Scholes Option Pricing Model by Finite Difference Method," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-10, June.
    5. Ji-Hun Yoon, 2014. "Mellin Transform Method for European Option Pricing with Hull-White Stochastic Interest Rate," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-7, October.
    6. Kleinert, H. & Korbel, J., 2016. "Option pricing beyond Black–Scholes based on double-fractional diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 449(C), pages 200-214.
    7. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    8. Mauricio Contreras & Alejandro Llanquihu'en & Marcelo Villena, 2015. "On the Solution of the Multi-asset Black-Scholes model: Correlations, Eigenvalues and Geometry," Papers 1510.02768, arXiv.org.
    9. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
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    Cited by:

    1. Sarit Maitra & Vivek Mishra & Goutam Kr. Kundu & Kapil Arora, 2023. "Integration of Fractional Order Black-Scholes Merton with Neural Network," Papers 2310.04464, arXiv.org, revised Oct 2023.

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