IDEAS home Printed from https://ideas.repec.org/p/hal/wpaper/hal-03230167.html
   My bibliography  Save this paper

Forecasting with fractional Brownian motion: a financial perspective

Author

Listed:
  • Matthieu Garcin

    (Research Center - Léonard de Vinci Pôle Universitaire - De Vinci Research Center)

Abstract

The fractional Brownian motion (fBm) extends the standard Brownian motion by introducing some dependence between non-overlapping increments. Consequently, if one considers for example that log-prices follow an fBm, one can exploit the non-Markovian nature of the fBm to forecast future states of the process and make statistical arbitrages. We provide new insights into forecasting an fBm, by proposing theoretical formulas for accuracy metrics relevant to a systematic trader, from the hit ratio to the expected gain and risk of a simple strategy. In addition, we answer some key questions about optimizing trading strategies in the fBm framework: Which lagged increments of the fBm, observed in discrete time, are to be considered? If the predicted increment is close to zero, up to which threshold is it more profitable not to invest? We also propose empirical applications on high-frequency FX rates, as well as on realized volatility series, exploring the rough volatility concept in a forecasting perspective.

Suggested Citation

  • Matthieu Garcin, 2021. "Forecasting with fractional Brownian motion: a financial perspective," Working Papers hal-03230167, HAL.
    • Handle: RePEc:hal:wpaper:hal-03230167
    Note: View the original document on HAL open archive server: https://hal.science/hal-03230167v3
    as

    Download full text from publisher

    File URL: https://hal.science/hal-03230167v3/document
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Matthieu Garcin & Martino Grasselli, 2020. "Long vs Short Time Scales: the Rough Dilemma and Beyond," Papers 2008.07822, arXiv.org, revised Nov 2021.
    2. Eom, Cheoljun & Choi, Sunghoon & Oh, Gabjin & Jung, Woo-Sung, 2008. "Hurst exponent and prediction based on weak-form efficient market hypothesis of stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(18), pages 4630-4636.
    3. Cajueiro, Daniel O & Tabak, Benjamin M, 2004. "The Hurst exponent over time: testing the assertion that emerging markets are becoming more efficient," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 336(3), pages 521-537.
    4. Kristoufek, Ladislav & Vosvrda, Miloslav, 2016. "Gold, currencies and market efficiency," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 449(C), pages 27-34.
    5. Alvarez-Ramirez, Jose & Alvarez, Jesus & Rodriguez, Eduardo & Fernandez-Anaya, Guillermo, 2008. "Time-varying Hurst exponent for US stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(24), pages 6159-6169.
    6. Di Matteo, T. & Aste, T. & Dacorogna, M.M., 2003. "Scaling behaviors in differently developed markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(1), pages 183-188.
    7. Matteo, T. Di & Aste, T. & Dacorogna, Michel M., 2005. "Long-term memories of developed and emerging markets: Using the scaling analysis to characterize their stage of development," Journal of Banking & Finance, Elsevier, vol. 29(4), pages 827-851, April.
    8. Alexandra Chronopoulou & Frederi Viens, 2012. "Estimation and pricing under long-memory stochastic volatility," Annals of Finance, Springer, vol. 8(2), pages 379-403, May.
    9. Garcin, Matthieu, 2017. "Estimation of time-dependent Hurst exponents with variational smoothing and application to forecasting foreign exchange rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 483(C), pages 462-479.
    10. Kristoufek, Ladislav & Vosvrda, Miloslav, 2013. "Measuring capital market efficiency: Global and local correlations structure," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(1), pages 184-193.
    11. Eduardo Abi Jaber, 2018. "Lifting the Heston model," Papers 1810.04868, arXiv.org, revised Nov 2019.
    12. Oomen, Roel C.A., 2006. "Properties of Realized Variance Under Alternative Sampling Schemes," Journal of Business & Economic Statistics, American Statistical Association, vol. 24, pages 219-237, April.
    13. Matthieu Garcin, 2019. "Hurst Exponents And Delampertized Fractional Brownian Motions," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(05), pages 1-26, August.
    14. T. Di Matteo, 2007. "Multi-scaling in finance," Quantitative Finance, Taylor & Francis Journals, vol. 7(1), pages 21-36.
    15. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
    16. Eduardo Abi Jaber, 2019. "Lifting the Heston model," Quantitative Finance, Taylor & Francis Journals, vol. 19(12), pages 1995-2013, December.
    17. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    18. Eduardo Abi Jaber, 2019. "Lifting the Heston model," Post-Print hal-01890751, HAL.
    19. Sánchez Granero, M.A. & Trinidad Segovia, J.E. & García Pérez, J., 2008. "Some comments on Hurst exponent and the long memory processes on capital markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(22), pages 5543-5551.
    20. Bianchi, Sergio & Pianese, Augusto, 2018. "Time-varying Hurst–Hölder exponents and the dynamics of (in)efficiency in stock markets," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 64-75.
    21. Paolo Guasoni & Yuliya Mishura & Miklós Rásonyi, 2021. "High-frequency trading with fractional Brownian motion," Finance and Stochastics, Springer, vol. 25(2), pages 277-310, April.
    22. Surgailis, Donatas & Teyssière, Gilles & Vaiciulis, Marijus, 2008. "The increment ratio statistic," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 510-541, March.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Matthieu Garcin, 2021. "Forecasting with fractional Brownian motion: a financial perspective," Papers 2105.09140, arXiv.org, revised Sep 2021.
    2. Matthieu Garcin, 2019. "Fractal analysis of the multifractality of foreign exchange rates [Analyse fractale de la multifractalité des taux de change]," Working Papers hal-02283915, HAL.
    3. Brouty, Xavier & Garcin, Matthieu, 2024. "Fractal properties, information theory, and market efficiency," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).
    4. Anagnostidis, P. & Varsakelis, C. & Emmanouilides, C.J., 2016. "Has the 2008 financial crisis affected stock market efficiency? The case of Eurozone," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 447(C), pages 116-128.
    5. Sukpitak, Jessada & Hengpunya, Varagorn, 2016. "The influence of trading volume on market efficiency: The DCCA approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 458(C), pages 259-265.
    6. Vogl, Markus, 2023. "Hurst exponent dynamics of S&P 500 returns: Implications for market efficiency, long memory, multifractality and financial crises predictability by application of a nonlinear dynamics analysis framewo," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    7. Li, Daye & Nishimura, Yusaku & Men, Ming, 2016. "The long memory and the transaction cost in financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 442(C), pages 312-320.
    8. Ayoub Ammy-Driss & Matthieu Garcin, 2021. "Efficiency of the financial markets during the COVID-19 crisis: time-varying parameters of fractional stable dynamics," Working Papers hal-02903655, HAL.
    9. Ayoub Ammy-Driss & Matthieu Garcin, 2020. "Efficiency of the financial markets during the COVID-19 crisis: time-varying parameters of fractional stable dynamics," Papers 2007.10727, arXiv.org, revised Nov 2021.
    10. Sensoy, Ahmet & Tabak, Benjamin M., 2016. "Dynamic efficiency of stock markets and exchange rates," International Review of Financial Analysis, Elsevier, vol. 47(C), pages 353-371.
    11. Ma, Pengcheng & Li, Daye & Li, Shuo, 2016. "Efficiency and cross-correlation in equity market during global financial crisis: Evidence from China," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 444(C), pages 163-176.
    12. Kristoufek, Ladislav & Vosvrda, Miloslav, 2014. "Commodity futures and market efficiency," Energy Economics, Elsevier, vol. 42(C), pages 50-57.
    13. Ladislav Kristoufek & Miloslav Vosvrda, 2014. "Measuring capital market efficiency: long-term memory, fractal dimension and approximate entropy," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 87(7), pages 1-9, July.
    14. Ammy-Driss, Ayoub & Garcin, Matthieu, 2023. "Efficiency of the financial markets during the COVID-19 crisis: Time-varying parameters of fractional stable dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).
    15. Miguel Ángel Sánchez & Juan E Trinidad & José García & Manuel Fernández, 2015. "The Effect of the Underlying Distribution in Hurst Exponent Estimation," PLOS ONE, Public Library of Science, vol. 10(5), pages 1-17, May.
    16. Siew Ann Cheong, 2013. "Econophysics: An Experimental Course for Advanced Undergraduates in the Nanyang Technological University," IIM Kozhikode Society & Management Review, , vol. 2(2), pages 79-99, July.
    17. Sensoy, Ahmet & Tabak, Benjamin M., 2015. "Time-varying long term memory in the European Union stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 436(C), pages 147-158.
    18. Kristoufek, Ladislav, 2018. "On Bitcoin markets (in)efficiency and its evolution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 503(C), pages 257-262.
    19. Eduardo Abi Jaber, 2020. "The Laplace transform of the integrated Volterra Wishart process," Working Papers hal-02367200, HAL.
    20. Ashok Chanabasangouda Patil & Shailesh Rastogi, 2020. "Multifractal Analysis of Market Efficiency across Structural Breaks: Implications for the Adaptive Market Hypothesis," JRFM, MDPI, vol. 13(10), pages 1-18, October.

    More about this item

    Keywords

    rough volatility; foreign-exchange rates; fractional Brownian motion; Hurst exponent; systematic trading;
    All these keywords.

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hal:wpaper:hal-03230167. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.