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Martingales, nonstationary increments, and the efficient market hypothesis

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  • McCauley, Joseph L.
  • Bassler, Kevin E.
  • Gunaratne, Gemunu H.

Abstract

We discuss the deep connection between nonstationary increments, martingales, and the efficient market hypothesis for stochastic processes x(t) with arbitrary diffusion coefficients D(x,t). We explain why a test for a martingale is generally a test for uncorrelated increments. We explain why martingales look Markovian at the level of both simple averages and 2-point correlations. But while a Markovian market has no memory to exploit and cannot be beaten systematically, a martingale admits memory that might be exploitable in higher order correlations. We also use the analysis of this paper to correct a misstatement of the ‘fair game’ condition in terms of serial correlations in Fama’s paper on the EMH. We emphasize that the use of the log increment as a variable in data analysis generates spurious fat tails and spurious Hurst exponents.

Suggested Citation

  • McCauley, Joseph L. & Bassler, Kevin E. & Gunaratne, Gemunu H., 2008. "Martingales, nonstationary increments, and the efficient market hypothesis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(15), pages 3916-3920.
    • Handle: RePEc:eee:phsmap:v:387:y:2008:i:15:p:3916-3920
    DOI: 10.1016/j.physa.2008.01.049
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    References listed on IDEAS

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    1. McCauley, J.L. & Gunaratne, G.H. & Bassler, K.E., 2007. "Martingale option pricing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 380(C), pages 351-356.
    2. McCauley, Joseph L. & Gunaratne, Gemunu H. & Bassler, Kevin E., 2006. "Hurst exponents, Markov processes, and fractional Brownian motion," MPRA Paper 2154, University Library of Munich, Germany.
    3. J. L. McCauley & G. H. Gunaratne & K. E. Bassler, 2006. "Martingale Option Pricing," Papers physics/0606011, arXiv.org, revised Feb 2007.
    4. McCauley, Joseph L. & Bassler, Kevin E. & Gunaratne, Gemunu H., 2008. "Martingales, detrending data, and the efficient market hypothesis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(1), pages 202-216.
    5. Bassler, Kevin E. & McCauley, Joseph L. & Gunaratne, Gemunu H., 2006. "Nonstationary increments, scaling distributions, and variable diffusion processes in financial markets," MPRA Paper 2126, University Library of Munich, Germany.
    6. McCauley, Joseph L. & Gunaratne, Gemunu H. & Bassler, Kevin E., 2007. "Martingale option pricing," MPRA Paper 2151, University Library of Munich, Germany.
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    Cited by:

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    2. Politi, Mauro & Millot, Nicolas & Chakraborti, Anirban, 2012. "The near-extreme density of intraday log-returns," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(1), pages 147-155.
    3. Miśkiewicz, Janusz & Ausloos, Marcel, 2008. "Correlation measure to detect time series distances, whence economy globalization," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(26), pages 6584-6594.
    4. Mauro Politi & Nicolas Millot & Anirban Chakraborti, 2011. "The near-extreme density of intraday log-returns," Papers 1106.0039, arXiv.org.
    5. Tomoyuki Ichiba & Guodong Pang & Murad S. Taqqu, 2022. "Path Properties of a Generalized Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 35(1), pages 550-574, March.
    6. Jonathan A. Batten & Cetin Ciner & Brian M. Lucey & Peter G. Szilagyi, 2013. "The structure of gold and silver spread returns," Quantitative Finance, Taylor & Francis Journals, vol. 13(4), pages 561-570, March.

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