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A theory of power-law distributions in financial market fluctuations

Nature volume 423, pages 267–270 (2003)Cite this article

Abstract

Insights into the dynamics of a complex system are often gained by focusing on large fluctuations. For the financial system, huge databases now exist that facilitate the analysis of large fluctuations and the characterization of their statistical behaviour1,2. Power laws appear to describe histograms of relevant financial fluctuations, such as fluctuations in stock price, trading volume and the number of trades3,4,5,6,7,8,9,10. Surprisingly, the exponents that characterize these power laws are similar for different types and sizes of markets, for different market trends and even for different countries—suggesting that a generic theoretical basis may underlie these phenomena. Here we propose a model, based on a plausible set of assumptions, which provides an explanation for these empirical power laws. Our model is based on the hypothesis that large movements in stock market activity arise from the trades of large participants. Starting from an empirical characterization of the size distribution of those large market participants (mutual funds), we show that the power laws observed in financial data arise when the trading behaviour is performed in an optimal way. Our model additionally explains certain striking empirical regularities that describe the relationship between large fluctuations in prices, trading volume and the number of trades.

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Figure 1: Cumulative distributions of the normalized 15-min absolute returns of the 1,000 largest companies in the ‘Trades and Quotes’ database for the 2-yr period 1994–1995.
Figure 2: Conditional expectation of the squared return r2 given the volume V.
Figure 3: Conditional expectations for E(r∣V′), E(V′∣r), E(N∣V′), E(N∣N′), and E(N′/N∣V′).

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Acknowledgements

We thank the National Science Foundation's economics program and the Russell Sage Foundation for support, K. Doran for research assistance, and M. Avellaneda, R. Barro, O. Blanchard, J. Campbell, A. Dixit, J. Hasbrouck, C. Hopman, D. Laibson, L. Pedersen, T. Philippon, R. Ramalho, J. Reuter, G. Saar, J. Scheinkman, A. Shleifer, D. Vayanos and J. Wang for discussions.

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Author notes

  1. Parameswaran Gopikrishnan

    Present address: Goldman Sachs and Co., 10 Hanover Square, New York, New York, 10005, USA

Authors and Affiliations

  1. Department of Economics, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02142, USA

    Xavier Gabaix

  2. Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts, 02215, USA

    Parameswaran Gopikrishnan, Vasiliki Plerou & H. Eugene Stanley

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  1. Xavier Gabaix
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Correspondence to Xavier Gabaix.

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Gabaix, X., Gopikrishnan, P., Plerou, V. et al. A theory of power-law distributions in financial market fluctuations. Nature 423, 267–270 (2003). https://doi.org/10.1038/nature01624

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