Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

A theory of power-law distributions in financial market fluctuations

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.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

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(rV′), E(Vr), E(NV′), E(NN′), and E(N′/NV′).

References

  1. 1

    Takayasu, H. (ed.) Empirical Science of Financial Fluctuations: The Advent of Econophysics (Springer, Berlin, 2002)

  2. 2

    Bunde, A., Schellnhuber, H. J. & Kropp, J. (eds) The Science of Disasters: Climate Disruptions, Heart Attacks, and Market Crashes (Springer, Berlin, 2002)

  3. 3

    Mandelbrot, B. B. The variation of certain speculative prices. J. Business 36, 394–419 (1963)

    Article  Google Scholar 

  4. 4

    Lux, T. The stable Paretian hypothesis and the frequency of large returns: an examination of major German stocks. Appl. Fin. Econ. 6, 463–475 (1996)

    ADS  Article  Google Scholar 

  5. 5

    Liu, Y. et al. The statistical properties of the volatility of price fluctuations. Phys. Rev. E 60, 1390–1400 (1999)

    ADS  CAS  Article  Google Scholar 

  6. 6

    Guillaume, D. M. et al. From the bird's eye to the microscope: a survey of new stylized facts of the intra-daily foreign exchange markets. Fin. Stochastics 1, 95–129 (1997)

    Article  Google Scholar 

  7. 7

    Plerou, V., Gopikrishnan, P., Amaral, L. A. N., Meyer, M. & Stanley, H. E. Scaling of the distribution of price fluctuations of individual companies. Phys. Rev. E 60, 6519–6529 (1999)

    ADS  CAS  Article  Google Scholar 

  8. 8

    Gopikrishnan, P., Plerou, V., Amaral, L. A. N., Meyer, M. & Stanley, H. E. Scaling of the distributions of fluctuations of financial market indices. Phys. Rev. E 60, 5305–5316 (1999)

    ADS  CAS  Article  Google Scholar 

  9. 9

    Gopikrishnan, P., Plerou, V., Gabaix, X. & Stanley, H. E. Statistical properties of share volume traded in financial markets. Phys. Rev. E 62, R4493–R4496 (2000)

    ADS  CAS  Article  Google Scholar 

  10. 10

    Plerou, V., Gopikrishnan, P., Amaral, L. A. N., Gabaix, X. & Stanley, H. E. Economic fluctuations and anomalous diffusion. Phys. Rev. E 62, R3023–R3026 (2000)

    ADS  CAS  Article  Google Scholar 

  11. 11

    Keim, D. & Madhavan, A. Transactions costs and investment style: An inter-exchange analysis of institutional equity trades. J. Fin. Econ. 46, 265–292 (1997)

    Article  Google Scholar 

  12. 12

    Chan, L. & Lakonishok, J. The behavior of stock prices around institutional trades. J. Fin. 50, 1147–1174 (1995)

    Article  Google Scholar 

  13. 13

    Wurgler, J. & Zhuravskaya, K. Does arbitrage flatten demand curves for stocks? J. Business 75, 583–608 (2002)

    Article  Google Scholar 

  14. 14

    Bagwell, L. Dutch auction repurchases: An analysis of shareholder heterogeneity. J. Fin. 47, 71–105 (1992)

    Article  Google Scholar 

  15. 15

    Kyle, A. S. Continuous auctions and insider trading. Econometrica 53, 1315–1335 (1985)

    Article  Google Scholar 

  16. 16

    Grossman, S. & Miller, M. Liquidity and market structure. J. Fin. 43, 617–633 (1988)

    Article  Google Scholar 

  17. 17

    O'Hara, M. Market Microstructure Theory (Blackwell, Oxford, 1997)

    Google Scholar 

  18. 18

    Cutler, D., Poterba, J. M. & Summers, L. H. What moves stock prices? J. Portfolio Management 15, 4–12 (1989)

    Article  Google Scholar 

  19. 19

    Zipf, G. Human Behavior and the Principle of Least Effort (Addiston-Wesley, Cambridge, 1949)

    Google Scholar 

  20. 20

    Okuyama, K., Takayasu, M. & Takayasu, H. Zipf's law in income distribution of companies. Physica A 269, 125–131 (1999)

    ADS  Article  Google Scholar 

  21. 21

    Axtell, R. Zipf distribution of U.S. firm sizes. Science 293, 1818–1820 (2001)

    ADS  CAS  Article  Google Scholar 

  22. 22

    Gabaix, X. Zipf's law for cities: An explanation. Q. J. Econ. 114, 739–767 (1999)

    Article  Google Scholar 

  23. 23

    Hasbrouck, J. Measuring the information content of stock trades. J. Fin. 46, 179–207 (1991)

    Article  Google Scholar 

  24. 24

    Plerou, V., Gopikrishnan, P., Gabaix, X. & Stanley, H. E. Quantifying stock price response to demand fluctuations. Phys. Rev. E 66, 027104 (2002)

    ADS  Article  Google Scholar 

  25. 25

    Daniel, K., Hirshleifer, D. & Subrahmanyam, A. Investor psychology and security market under- and over-reactions. J. Fin. 53, 1839–1885 (1998)

    Article  Google Scholar 

  26. 26

    Shleifer, A. Inefficient Markets: An Introduction to Behavioral Finance (Oxford Univ Press, Oxford, 2000)

    Book  Google Scholar 

  27. 27

    Gabaix, X., Ramalho, R. & Reuter, J. Power laws and mutual fund dynamics (MIT Mimeo, Massachusetts Institute of Technology, Cambridge, 2003)

  28. 28

    Lee, C. M. C. & Ready, M. J. Inferring trade direction from intraday data. J. Fin. 46, 733–746 (1991)

    Article  Google Scholar 

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xavier Gabaix.

Ethics declarations

Competing interests

The authors declare that they have no competing financial interests.

Supplementary information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing