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Martingales, detrending data, and the efficient market hypothesis

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

Abstract

We discuss martingales, detrending data, and the efficient market hypothesis (EMH) for stochastic processes x(t) with arbitrary diffusion coefficients D(x,t). Beginning with x-independent drift coefficients R(t) we show that martingale stochastic processes generate uncorrelated, generally non-stationary increments. Generally, a test for a martingale is therefore a test for uncorrelated increments. A detrended process with an x-dependent drift coefficient is generally not a martingale, and so we extend our analysis to include the class of (x,t)-dependent drift coefficients of interest in finance. We explain why martingales look Markovian at the level of both simple averages and 2-point correlations. And while a Markovian market has no memory to exploit and presumably cannot be beaten systematically, it has never been shown that martingale memory cannot be exploited in 3-point or higher correlations to beat the market. We generalize our Markov scaling solutions presented earlier, and also generalize the martingale formulation of the EMH to include (x,t)-dependent drift in log returns. 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 end with a discussion of Levy's characterization of Brownian motion and prove that an arbitrary martingale is topologically inequivalent to a Wiener process.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:phsmap:v:387:y:2008:i:1:p:202-216
    DOI: 10.1016/j.physa.2007.08.019
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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. J. L. McCauley & G. H. Gunaratne & K. E. Bassler, 2006. "Martingale Option Pricing," Papers physics/0606011, arXiv.org, revised Feb 2007.
    3. Skjeltorp, Johannes A, 2000. "Scaling in the Norwegian stock market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 283(3), pages 486-528.
    4. McCauley, Joseph L. & Gunaratne, Gemunu H. & Bassler, Kevin E., 2007. "Hurst exponents, Markov processes, and fractional Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 379(1), pages 1-9.
    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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    Citations

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    Cited by:

    1. McCauley, Joseph L., 2008. "Time vs. ensemble averages for nonstationary time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(22), pages 5518-5522.
    2. 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.
    3. Hua, Jia-Chen & Chen, Lijian & Falcon, Liberty & McCauley, Joseph L. & Gunaratne, Gemunu H., 2015. "Variable diffusion in stock market fluctuations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 419(C), pages 221-233.
    4. McCauley, Joseph L., 2008. "Nonstationarity of efficient finance markets: FX market evolution from stability to instability," International Review of Financial Analysis, Elsevier, vol. 17(5), pages 820-837, December.
    5. McCauley, Joseph L., 2009. "ARCH and GARCH models vs. martingale volatility of finance market returns," International Review of Financial Analysis, Elsevier, vol. 18(4), pages 151-153, September.
    6. McCauley, Joseph L. & Bassler, Kevin E. & Gunaratne, Gemunu H., 2009. "Is integration I(d) applicable to observed economics and finance time series?," International Review of Financial Analysis, Elsevier, vol. 18(3), pages 101-108, June.

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