McCauley, Joseph L. and Bassler, Kevin E. and Gunaratne, Gemunu H. (2007): Martingales, Detrending Data, and the Efficient Market Hypothesis.
Download (1382Kb) | Preview
We discuss martingales, detrending data, and the efficient market hypothesis 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 nonstationary 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 efficient market hypothesis (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’scharacterization of Brownian motion and prove that an arbitrary martingale is topologically inequivalent to a Wiener process.
|Item Type:||MPRA Paper|
|Institution:||University of Houston, NUI Galway|
|Original Title:||Martingales, Detrending Data, and the Efficient Market Hypothesis|
|Keywords:||Martingales; Markov processes; detrending; memory; stationary and nonstationary increments; correlations; efficient market hypothesis|
|Subjects:||C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C53 - Forecasting and Prediction Methods; Simulation Methods
G - Financial Economics > G0 - General
C - Mathematical and Quantitative Methods > C2 - Single Equation Models; Single Variables
|Depositing User:||Joseph L. McCauley|
|Date Deposited:||15. Mar 2007|
|Last Modified:||18. Feb 2013 21:42|
1. J. L. McCauley , G.H. Gunaratne, & K.E. Bassler, Hurst Exponents, Markov Processes, and Fractional Brownian Motion, Physica A (2007), in press. 2. P. Hänggi, H. Thomas, H. Grabert, and P. Talkner, J. Stat. Phys. 18, 155, 1978. 1 If the transition density has fat tails, then higher moments will diverge. There, one must be more careful with the boundary terms. 3. J. L. McCauley, K.E. Bassler, & G.H. Gunaratne, On the Analysis of Time Series with Nonstationary Increments in Handbook of Complexity Research, ed. B. Rosser, 2007. 4. K.E. Bassler, J. L. McCauley, & G.H. Gunaratne, Nonstationary Increments, Scaling Distributions, and Variable Diffusion Processes in Financial Markets, 2006. 5. R.L. Stratonovich. Topics in the Theory of Random Noise, Gordon & Breach: N.Y., tr. R. A. Silverman, 1963. 6. M.C. Wang & G.E. Uhlenbeck in Selected Papers on Noise and Stochastic Processes, ed. N. Wax, Dover: N.Y., 1954. 7. W. Feller, The Annals of Math. Statistics 30, No. 4, 1252, 1959. 8. M. Courbage & D. Hamdan, The Annals of Probability 22, No. 3, 1662, 1994. 9. K.E. Bassler, G.H. Gunaratne, & J. L. McCauley , Physica A 369, 343 (2006). 10. A. Friedman, Stochastic Differential Equations and Applications, pg. 23, Academic, N.Y., 1975. 11. P. Hänggi and H. Thomas, Zeitschr. Für Physik B26, 85, 1977. 12. R. Durrett, Brownian Motion and Martingales in Analysis, Wadsworth, Belmont, 1984. 13. M. Baxter and A. Rennie,. Financial Calculus, Cambridge, Cambridge, 1995. 14. J.M. Steele, Stochastic Calculus and Financial Applications. Springer-Verlag, N.Y., 2000. 15. Joseph L. McCauley, Stochastic processes for physics and finance, book manuscript to be submitted (2007). 16. B. Mandelbrot & J. W. van Ness, SIAM Rev. 10, 2, 422,1968. 17. B. Mandelbrot, J. Business 39, 242, 1966. 18. J. L. McCauley, Markov vs. nonMarkovian processes: A comment on the paper ‘Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations’ by T.D. Frank, submitted (2007). 19. J. L. McCauley, Fokker-Planck and Chapman-Kolmogorov equations for Ito processes with finite memory, submitted (2007). 20. P. Embrechts and M. Maejima, Self-similar Processes, Princeton University Press, Princeton, 2002. 21. L. Arnold, Stochastic Differential Equations, Krieger, Malabar, 1992. 22. E. Scalas, R. Gorenflo and F. Mainardi, Physica A284, 376, 2000. 23. F. Mainardi, M. Raberto, R. Gorenflo and E. Scalas Physica A 287, 468, 2000. 24. J.L. McCauley, Dynamics of Markets: Econophysics and Finance, Cambridge, Cambridge, 2004. 25. A. W. Lo, H. Mamaysky, and J. Wang, J. Finance LV, Nr. 4, 1705, 2000. 26. E. Fama, J. Finance 25, 383-417, 1970. 27. Johannes A. Skjeltorp, Scaling in the Norwegian stock market, Physica A 283, 486-528, 2000. 28. F. Black, J. of Finance 3, 529, 1986. 29. C.C. Heyde & N.N. Leonenko, Adv. Appl. Prob. 37, 342, 2005. 30. K.E. Bassler, G.H. Gunaratne, & J. L. McCauley, work in progress (2007). 31. S. Gallucio, G. Caldarelli, M. Marsili, and Y.-C. Zhang, Physica A245, 423, 1997. 32. J. L. McCauley, G.H. Gunaratne, & K.E. Bassler, Martingale option pricing, submitted, 2006. 33. J. L. Snell, A Conversation with Joe Doob, http://www.dartmouth.edu/~chance/Doob/conversation. html; Statistical Science 12, No. 4, 301, 1997. 34. H. P. McKean, Stochastic Integrals, AMS Chelsea, Providence, 2000. 35. V.I. Arnol’d, Ordinary Differential Equations, transl. by R. Cooke, springer, N.Y., 1992. 36. J.L. McCauley, Classical Mechanics: flows, transformations, integrability and chaos, Cambridge Univ. Pr.: Cambridge, 1997. 37. R. Durrett, Stochastic Calculus, CRC, Boca Raton, 1996. 38. G. Bottazzi, private communication, March, 2006