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Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations

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  • Jumarie, Guy

Abstract

Stock exchange dynamics of fractional order are usually modeled as a non-random exponential growth process driven by a fractional Brownian motion. Here we propose to use rather a non-random fractional growth driven by a (standard) Brownian motion. The key is the Taylor's series of fractional order where E[alpha](.) denotes the Mittag-Leffler function, and is the so-called modified Riemann-Liouville fractional derivative which we introduced recently to remove the effects of the non-zero initial value of the function under consideration. Various models of fractional dynamics for stock exchange are proposed, and their solutions are obtained. Mainly, the Itô's lemma of fractional order is illustrated in the special case of a fractional growth with white noise. Prospects for the Merton's optimal portfolio are outlined, the path probability density of fractional stock exchange dynamics is obtained, and two fractional Black-Scholes equations are derived. This approach avoids using fractional Brownian motion and thus is of some help to circumvent the mathematical difficulties so involved.

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  • Jumarie, Guy, 2008. "Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 271-287, February.
    • Handle: RePEc:eee:insuma:v:42:y:2008:i:1:p:271-287
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    References listed on IDEAS

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    1. 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.
    2. Cioczek-Georges, R. & Mandelbrot, B. B., 1996. "Alternative micropulses and fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 64(2), pages 143-152, November.
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    14. Nuugulu, Samuel M & Gideon, Frednard & Patidar, Kailash C, 2021. "A robust numerical scheme for a time-fractional Black-Scholes partial differential equation describing stock exchange dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    15. Xin Cai & Yihong Wang, 2024. "A Novel Fourth-Order Finite Difference Scheme for European Option Pricing in the Time-Fractional Black–Scholes Model," Mathematics, MDPI, vol. 12(21), pages 1-23, October.
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