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Fractal properties, information theory, and market efficiency

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  • Xavier Brouty
  • Matthieu Garcin

Abstract

Considering that both the entropy-based market information and the Hurst exponent are useful tools for determining whether the efficient market hypothesis holds for a given asset, we study the link between the two approaches. We thus provide a theoretical expression for the market information when log-prices follow either a fractional Brownian motion or its stationary extension using the Lamperti transform. In the latter model, we show that a Hurst exponent close to 1/2 can lead to a very high informativeness of the time series, because of the stationarity mechanism. In addition, we introduce a multiscale method to get a deeper interpretation of the entropy and of the market information, depending on the size of the information set. Applications to Bitcoin, CAC 40 index, Nikkei 225 index, and EUR/USD FX rate, using daily or intraday data, illustrate the methodological content.

Suggested Citation

  • Xavier Brouty & Matthieu Garcin, 2023. "Fractal properties, information theory, and market efficiency," Papers 2306.13371, arXiv.org.
    • Handle: RePEc:arx:papers:2306.13371
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    References listed on IDEAS

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    1. Bacry, E. & Delour, J. & Muzy, J.F., 2001. "Modelling financial time series using multifractal random walks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 299(1), pages 84-92.
    2. Alvarez-Ramirez, Jose & Rodriguez, Eduardo & Alvarez, Jesus, 2012. "A multiscale entropy approach for market efficiency," International Review of Financial Analysis, Elsevier, vol. 21(C), pages 64-69.
    3. Alvarez-Ramirez, Jose & Rodriguez, Eduardo, 2021. "A singular value decomposition entropy approach for testing stock market efficiency," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 583(C).
    4. 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).
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    Cited by:

    1. Matthieu Garcin, 2023. "Complexity measure, kernel density estimation, bandwidth selection, and the efficient market hypothesis," Papers 2305.13123, arXiv.org.
    2. Matthieu Garcin, 2023. "Complexity measure, kernel density estimation, bandwidth selection, and the efficient market hypothesis," Working Papers hal-04102815, HAL.

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