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Solving Black-Schole Equation Using Standard Fractional Brownian Motion

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  • Didier Alain Njamen Njomen
  • Eric Djeutcha

Abstract

In this paper, we emphasize the Black-Scholes equation using standard fractional Brownian motion BHwith the hurst index H ∈ [0,1]. N. Ciprian (Necula, C. (2002)) and Bright and Angela (Bright, O., Angela, I., & Chukwunezu (2014)) get the same formula for the evaluation of a Call and Put of a fractional European with the different approaches. We propose a formula by adapting the non-fractional Black-Scholes model using a λHfactor to evaluate the european option. The price of the option at time t ∈]0,T[ depends on λH(T − t), and the cost of the action St, but not only from t − T as in the classical model. At the end, we propose the formula giving the implied volatility of sensitivities of the option and indicators of the financial market.

Suggested Citation

  • Didier Alain Njamen Njomen & Eric Djeutcha, 2019. "Solving Black-Schole Equation Using Standard Fractional Brownian Motion," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 11(2), pages 142-157, April.
    • Handle: RePEc:ibn:jmrjnl:v:11:y:2019:i:2:p:142
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    References listed on IDEAS

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    1. Robert J. Elliott & John Van Der Hoek, 2003. "A General Fractional White Noise Theory And Applications To Finance," Mathematical Finance, Wiley Blackwell, vol. 13(2), pages 301-330, April.
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    3. Mandelbrot, Benoit B, 1971. "When Can Price Be Arbitraged Efficiently? A Limit to the Validity of the Random Walk and Martingale Models," The Review of Economics and Statistics, MIT Press, vol. 53(3), pages 225-236, August.
    4. Rosanna Coviello & Cristina Di Girolami & Francesco Russo, 2011. "On stochastic calculus related to financial assets without semimartingales," Papers 1102.2050, arXiv.org.
    5. Cipian Necula, 2008. "Option Pricing in a Fractional Brownian Motion Environment," Advances in Economic and Financial Research - DOFIN Working Paper Series 2, Bucharest University of Economics, Center for Advanced Research in Finance and Banking - CARFIB.
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    Cited by:

    1. Paula Morales-Bañuelos & Nelson Muriel & Guillermo Fernández-Anaya, 2022. "A Modified Black-Scholes-Merton Model for Option Pricing," Mathematics, MDPI, vol. 10(9), pages 1-16, April.

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    More about this item

    Keywords

    stock price; black-Scholes model; fractional Brownian motion; options; volatility;
    All these keywords.

    JEL classification:

    • R00 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - General - - - General
    • Z0 - Other Special Topics - - General

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