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Local and implied volatilities with the Mixed-Modified-Fractional-Dupire Model

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Listed:
  • Eric Djeutcha

    (UMa - University of Maroua)

  • Jules Sadefo-Kamdem

    (MRE - Montpellier Recherche en Economie - UM - Université de Montpellier)

Abstract

In this paper, we use the Mellin transform to obtain the analytical formulas of European option (call or put) values, when the evolution of the underlying asset return is governed by a mixed modified fractional stochastic process. As an extension of the Dupire Model [12], we also introduce the so-called 'Mixed-Modified-Fractional-Dupire model', by giving the expression of it's local volatility and it's sensitivity in relation to the Hurst coefficient H. Finally, in the same vein, we highlight an analytical relationship between local volatility and implied volatility.

Suggested Citation

  • Eric Djeutcha & Jules Sadefo-Kamdem, 2021. "Local and implied volatilities with the Mixed-Modified-Fractional-Dupire Model," Post-Print hal-03324320, HAL.
    • Handle: RePEc:hal:journl:hal-03324320
    DOI: 10.1016/j.chaos.2021.111328
    Note: View the original document on HAL open archive server: https://hal.science/hal-03324320
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    References listed on IDEAS

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    1. Andrew W. Lo, A. Craig MacKinlay, 1988. "Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test," The Review of Financial Studies, Society for Financial Studies, vol. 1(1), pages 41-66.
    2. Fama, Eugene F & French, Kenneth R, 1988. "Permanent and Temporary Components of Stock Prices," Journal of Political Economy, University of Chicago Press, vol. 96(2), pages 246-273, April.
    3. Lo, Andrew W, 1991. "Long-Term Memory in Stock Market Prices," Econometrica, Econometric Society, vol. 59(5), pages 1279-1313, September.
    4. Bwo-Nung Huang & Chin Yang, 1995. "The fractal structure in multinational stock returns," Applied Economics Letters, Taylor & Francis Journals, vol. 2(3), pages 67-71.
    5. Frennberg, Per & Hansson, Bjorn, 1993. "Testing the random walk hypothesis on Swedish stock prices: 1919-1990," Journal of Banking & Finance, Elsevier, vol. 17(1), pages 175-191, February.
    6. 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.
    7. repec:adr:anecst:y:1999:i:54:p:03 is not listed on IDEAS
    8. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    9. Akgül, Esra Karatas & Akgül, Ali & Yavuz, Mehmet, 2021. "New Illustrative Applications of Integral Transforms to Financial Models with Different Fractional Derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    10. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
    11. Lennart Berg & Johan Lyhagen, 1998. "Short and long-run dependence in Swedish stock returns," Applied Financial Economics, Taylor & Francis Journals, vol. 8(4), pages 435-443.
    12. 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.
    13. Mounir Zili, 2006. "On the mixed fractional Brownian motion," International Journal of Stochastic Analysis, Hindawi, vol. 2006, pages 1-9, August.
    14. Sandrine Lardic & Valérie Mignon, 1999. "Prévision ARFIMA des taux de change : les modélisateurs doivent-ils encore exhorter à la naïveté des prévisions ?," Annals of Economics and Statistics, GENES, issue 54, pages 47-68.
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