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Solving Arbitrage Problem on the Financial Market Under the Mixed Fractional Brownian Motion With Hurst Parameter H ∈]1/2,3/4[

Author

Listed:
  • Eric Djeutcha
  • Didier Alain Njamen Njomen
  • Louis-Aimé Fono

Abstract

This study deals with the arbitrage problem on the financial market when the underlying asset follows a mixed fractional Brownian motion. We prove the existence and uniqueness theorem for the mixed geometric fractional Brownian motion equation. The semi-martingale approximation approach to mixed fractional Brownian motion is used to eliminate the arbitrage opportunities.

Suggested Citation

  • Eric Djeutcha & Didier Alain Njamen Njomen & Louis-Aimé Fono, 2019. "Solving Arbitrage Problem on the Financial Market Under the Mixed Fractional Brownian Motion With Hurst Parameter H ∈]1/2,3/4[," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 11(1), pages 76-92, February.
    • Handle: RePEc:ibn:jmrjnl:v:11:y:2019:i:1:p:76
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    References listed on IDEAS

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    1. Nualart, David & Saussereau, Bruno, 2009. "Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 391-409, February.
    2. 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.
    3. 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.
    4. Nguyen Tien, Dung, 2013. "The existence of a positive solution for a generalized delay logistic equation with multifractional noise," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1240-1246.
    5. Nourdin, Ivan & Simon, Thomas, 2006. "On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 76(9), pages 907-912, May.
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    Citations

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    Cited by:

    1. Djeutcha, Eric & Kamdem, Jules Sadefo, 2021. "Local and implied volatilities with the mixed-modified-fractional-Dupire model," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    2. Eric Djeutcha & Jules Sadefo Kamdem, 2024. "Pricing for a vulnerable bull spread options using a mixed modified fractional Hull–White–Vasicek model," Annals of Operations Research, Springer, vol. 334(1), pages 101-131, March.
    3. Eric Djeutcha & Jules Sadefo-Kamdem & Louis Aimé Fono, 2021. "Mixed Modified Fractional Merton model of the bear spread Basket put option using the multidimensional Mellin transform," Working Papers hal-03330043, HAL.
    4. Amit K. Sinha, 2021. "The reliability of geometric Brownian motion forecasts of S&P500 index values," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 40(8), pages 1444-1462, December.

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    More about this item

    Keywords

    mixed fractional process; asset pricing; Gaussian process;
    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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