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A multifractional option pricing formula

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  • Axel A. Araneda

Abstract

Fractional Brownian motion has become a standard tool to address long-range dependence in financial time series. However, a constant memory parameter is too restrictive to address different market conditions. Here we model the price fluctuations using a multifractional Brownian motion assuming that the Hurst exponent is a time-deterministic function. Through the multifractional Ito calculus, both the related transition density function and the analytical European Call option pricing formula are obtained. The empirical performance of the multifractional Black-Scholes model is tested by calibration of option market quotes for the SPX index and offers best fit than its counterparts based on standard and fractional Brownian motions.

Suggested Citation

  • Axel A. Araneda, 2023. "A multifractional option pricing formula," Papers 2303.16314, arXiv.org, revised Jun 2024.
    • Handle: RePEc:arx:papers:2303.16314
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    References listed on IDEAS

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    1. S. Bianchi & A. Pantanella & A. Pianese, 2013. "Modeling stock prices by multifractional Brownian motion: an improved estimation of the pointwise regularity," Quantitative Finance, Taylor & Francis Journals, vol. 13(8), pages 1317-1330, July.
    2. Sergio Bianchi & Augusto Pianese, 2015. "Asset Price Modeling: From Fractional to Multifractional Processes," International Series in Operations Research & Management Science, in: Alain Bensoussan & Dominique Guegan & Charles S. Tapiero (ed.), Future Perspectives in Risk Models and Finance, edition 127, pages 247-285, Springer.
    3. Wang, Xiao-Tian, 2010. "Scaling and long range dependence in option pricing, IV: Pricing European options with transaction costs under the multifractional Black–Scholes model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(4), pages 789-796.
    4. Wang, Xiao-Tian, 2010. "Scaling and long-range dependence in option pricing I: Pricing European option with transaction costs under the fractional Black–Scholes model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(3), pages 438-444.
    5. Sergio Bianchi, 2005. "Pathwise Identification Of The Memory Function Of Multifractional Brownian Motion With Application To Finance," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 8(02), pages 255-281.
    6. Bladt, Mogens & Rydberg, Tina Hviid, 1998. "An actuarial approach to option pricing under the physical measure and without market assumptions," Insurance: Mathematics and Economics, Elsevier, vol. 22(1), pages 65-73, May.
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