Abstract
We investigate the random walk of prices by developing a simple model relating the properties of the signs and absolute values of individual price changes to the diffusion rate (volatility) of prices at longer time scales. We show that this benchmark model is unable to reproduce the diffusion properties of real prices. Specifically, we find that for one hour intervals this model consistently over-predicts the volatility of real price series by about 70%, and that this effect becomes stronger as the length of the intervals increases. By selectively shuffling some components of the data while preserving others we are able to show that this discrepancy is caused by a subtle but long-range non-contemporaneous correlation between the signs and sizes of individual returns. We conjecture that this is related to the long-memory of transaction signs and the need to enforce market efficiency.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Bachelier, in The Random Character of Stock Prices, edited by H.P. Cooper (Cambridge, 1964)
P. Cootner, The random character of stock prices (MIT Press, Cambridge, MA, 1964)
R.F. Engle, Econometrica 50, 987 (1982)
Z. Ding, C.W.J. Granger, R.F. Engle, J. Empir. Finance 1, 83 (1993)
F.J. Breidt, N. Crato, P.J.F. de Lima, Working paper (Johns Hopkins University, 1993)
A.C. Harvey, Working paper (London School of Economics, 1993)
M. Montero, J. Perello, J. Masoliver, F. Lillo, S. Micciche, R. Mantegna, Phys. Rev. E 72, 056101 (2005)
Y.J. Campbell, A.W. Lo, A.C. Mackinlay, The Econometrics of Financial Markets (Princeton University Press, Princeton, 1997)
R. Engle, J. Rangel, Tech. rep., NYU and UCSD (2005)
R. Engle, E. Ghysels, B. Sohn, Tech. rep., New York University and University of North Carolina at Chapel Hill (2006)
R. Roll, Am. Econ. Rev. 861 (1984)
K. French, R. Roll, J. Financ. Econ. 17, 5 (1986)
D.M. Cutler, J.M. Poterba, L.H. Summers, The Journal of Portfolio Management 15, 4 (1989)
P.K. Clark, Econometrica 41, 135 (1973)
V. Plerou, P. Gopikrishnan, L.A.N. Amaral, X. Gabaix, H.E. Stanley, Phys. Rev. E 62, R3023 (2000)
T. Ane, H. Geman, J. Financ. 55, 2259 (2000)
T. Andersen, T. Bollerslev, F. Diebold, P. Labys, Econometrica 71, 579 (2003)
L. Gillemot, J.D. Farmer, F. Lillo, Quant. Finance 6, 371 (2006)
J.P. Bouchaud, Y. Gefen, M. Potters, M. Wyart, Quantit. Finance 4, 176 (2004)
F. Lillo, J.D. Farmer, Studies in Nonlinear Dynamics & Econometrics 8, 1 (2004)
J.P. Bouchaud, J. Kockelkoren, M. Potters, Quant. Finance 6, 115 (2006)
J. Farmer, A. Gerig, F. Lillo, S. Mike, Quantit. Finance 6, 107 (2006)
J. Hamilton, Time series analysis (Princeton University Press, Princeton, 1994)
E.F. Fama, The Journal of Business 38, 34 (1965)
B. Mandelbrot, The Journal of Business 36, 394 (1963)
F.M. Longin, The Journal of Business 69, 383 (1996)
T. Lux, Appl. Fin. Econ. 6, 463 (1996)
P. Gopikrishnan, M. Meyer, L. Amaral, H. Stanley, Eur. Phys. J. B. 3, 139 (1998)
B.M. Hill, Ann. Stat. 3, 1163 (1975)
J. Beran, Statistics for Long-Memory Processes (Chapman & Hall, New York, 1994)
C.K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, A.L. Goldberger, Phys. Rev. E 49, 1685 (1994)
P. Weber, Phys. Rev. E. 75, 016105 (2006)
J.P. Bouchaud, A. Matacz, M. Potters, Phys. Rev. Lett. 87, 228701 (2001)
A. Gerig, Ph.D. thesis, University of Illinois (2007), http://www.santafe.edu/~gerig/Gerig07_TheoryMarketImpact.pdf
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
La Spada, G., Farmer, J.D. & Lillo, F. The non-random walk of stock prices: the long-term correlation between signs and sizes. Eur. Phys. J. B 64, 607–614 (2008). https://doi.org/10.1140/epjb/e2008-00244-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjb/e2008-00244-4