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A note on geometric method-based procedures to calculate the Hurst exponent

Author

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  • Trinidad Segovia, J.E.
  • Fernández-Martínez, M.
  • Sánchez-Granero, M.A.

Abstract

Geometric method-based procedures, which we will call GM algorithms hereafter, were introduced in M.A. Sánchez-Granero, J.E. Trinidad Segovia, J. García Pérez, Some comments on Hurst exponent and the long memory processes on capital markets, Phys. A 387 (2008) 5543–5551, to calculate the Hurst exponent of a time series. The authors proved that GM algorithms, based on a geometrical approach, are more accurate than classical algorithms, especially with short length time series. The main contribution of this paper is to provide a mathematical background for the validity of these two algorithms to calculate the Hurst exponent H of random processes with stationary and self-affine increments. In particular, we show that these procedures are valid not only for exploring long memory in classical processes such as (fractional) Brownian motions, but also for estimating the Hurst exponent of (fractional) Lévy stable motions.

Suggested Citation

  • Trinidad Segovia, J.E. & Fernández-Martínez, M. & Sánchez-Granero, M.A., 2012. "A note on geometric method-based procedures to calculate the Hurst exponent," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(6), pages 2209-2214.
  • Handle: RePEc:eee:phsmap:v:391:y:2012:i:6:p:2209-2214
    DOI: 10.1016/j.physa.2011.11.044
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    References listed on IDEAS

    as
    1. 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.
    2. Marco Corazza & A. G. Malliaris, 2005. "Multi-Fractality in Foreign Currency Markets," World Scientific Book Chapters,in: Economic Uncertainty, Instabilities And Asset Bubbles Selected Essays, chapter 11, pages 151-184 World Scientific Publishing Co. Pte. Ltd..
    3. Laskin, Nick, 2000. "Fractional market dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 482-492.
    4. MacBeth, James D & Merville, Larry J, 1979. "An Empirical Examination of the Black-Scholes Call Option Pricing Model," Journal of Finance, American Finance Association, vol. 34(5), pages 1173-1186, December.
    5. Sun, Wei & Rachev, Svetlozar & Fabozzi, Frank J., 2007. "Fractals or I.I.D.: Evidence of long-range dependence and heavy tailedness from modeling German equity market returns," Journal of Economics and Business, Elsevier, vol. 59(6), pages 575-595.
    6. Jean-Philippe Bouchaud & Didier Sornette, 1994. "The Black-Scholes option pricing problem in mathematical finance: generalization and extensions for a large class of stochastic processes," Science & Finance (CFM) working paper archive 500040, Science & Finance, Capital Fund Management.
    7. Rafal Weron, 2001. "Measuring long-range dependence in electricity prices," Papers cond-mat/0103621, arXiv.org.
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    Cited by:

    1. Fernández-Martínez, M. & Sánchez-Granero, M.A. & Trinidad Segovia, J.E., 2013. "Measuring the self-similarity exponent in Lévy stable processes of financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(21), pages 5330-5345.
    2. Martín-Montoya, L.A. & Aranda-Camacho, N.M. & Quimbay, C.J., 2015. "Long-range correlations and trends in Colombian seismic time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 421(C), pages 124-133.
    3. M. Fern'andez-Mart'inez & M. A S'anchez-Granero & Mar'ia Jos'e Mu~noz Torrecillas & Bill McKelvey, 2016. "A comparison among some Hurst exponent approaches to predict nascent bubbles in $500$ company stocks," Papers 1601.04188, arXiv.org.

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