Long range dependence in financial markets

  • Rama Cont


The notions of self-similarity, scaling, fractional processes and long range dependence have been repeatedly used to describe properties of financial time series: stock prices, foreign exchange rates, market indices and commodity prices. We discuss the relevance of these properties in the context of financial modelling, their relation with the basic principles of financial theory and possible economic explanations for their presence in financial time series.


Stock Market Long Range Stock Return Fractional Brownian Motion Asset Return 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    V. Akgiray and G. Booth, The stable law model of stock returns, J. Business Econom. Statis., 6 (1988), pp. 51–57.CrossRefGoogle Scholar
  2. 2.
    T. Andersen and T. Bollerslev, Heterogeneous information arrivals and returns volatility dynamics, Journal of finance, 52 (1997), pp. 975–1005.Google Scholar
  3. 3.
    V. V. Anh, C. C. Heyde, and N. N. Leonenko, Dynamic models of long-memory processes driven by Lévy noise, J. Appl. Probab., 39 (2002), pp. 730–747.MathSciNetCrossRefGoogle Scholar
  4. 4.
    W. Arthur, J. Holland, B. LeBaron, J. Palmer, and P. Tayler, Asset pricing under endogeneous expectations in an artificial stock market, in The economy as an Evolving Complex System, W. Arthur, S. Durlauf, and D. Lane, eds., vol. II, Reading MA, 1997, Perseus Books, pp. 15–44.Google Scholar
  5. 5.
    R. T. Baillie, Long memory processes and fractional integration in econometrics, J. Econometrics, 73 (1996), pp. 5–59.zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    R. T. Baillie, T. Bollerslev, and H. O. Mikkelsen, Fractionally integrated generalized autoregressive conditional heteroskedasticity, J. Econometrics, 74 (1996), pp. 3–30.MathSciNetCrossRefGoogle Scholar
  7. 7.
    O. E. Barndorff-Nielsen and N. Shephard, Modelling by Lévy processes for financial econometrics, in Lévy processes—Theory and Applications, O. Barndorff-Nielsen, T. Mikosch, and S. Resnick, eds., Birkhäuser, Boston, 2001, pp. 283–318.Google Scholar
  8. 8.
    E. Bayraktar, U. Horst, and K. R. Sircar, A limit theorem for financial markets with inert investors, working paper, Princeton University, 2003.Google Scholar
  9. 9.
    J. Beran, Statistics for long-memory processes, vol. 61 of Monographs on Statistics and Applied Probability, Chapman and Hall, New York, 1994.Google Scholar
  10. 10.
    R. Blattberg and N. Gonedes, A comparison of the stable Paretian and Student t distributions as statistial models for prices, Journal of Business, 47 (1974), pp. 244–280.CrossRefGoogle Scholar
  11. 11.
    T. Bollerslev, R. Chou, and K. Kroner, ARCH modeling in finance, J. Econometrics, 52 (1992), pp. 5–59.CrossRefGoogle Scholar
  12. 12.
    P. Cheridito, Arbitrage in fractional Brownian motion models, Finance Stoch., 7 (2003), pp. 533–553.zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    —, Gaussian moving averages, semimartingales and option pricing, Stochastic Process. Appl., 109 (2004), pp. 47–68.zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    F. Comte and E. Renault, Long memory continuous time models, J. Econometrics, 73 (1996), pp. 101–149.MathSciNetCrossRefGoogle Scholar
  15. 15.
    R. Cont, Empirical properties of asset returns: Stylized facts and statistical issues, Quant. Finance, 1 (2001), pp. 1–14.CrossRefGoogle Scholar
  16. 16.
    R. Cont, J.-P. Bouchaud, and M. Potters, Scaling in financial data: Stable laws and beyond, in Scale Invariance and Beyond, B. Dubrulle, F. Graner, and D. Sornette, eds., Springer, Berlin, 1997.Google Scholar
  17. 17.
    R. Cont, F. Ghoulmië, and J.-P. Nadal, Heterogeneity and feedback in an agent-based market model, Journal of Physics: Condensed Matter, 17 (2005), pp. S1259–S1268.CrossRefGoogle Scholar
  18. 18.
    D. Cutler, J. Poterba, and L. Summers, What moves stock prices?, Journal of Portfolio Management, (1989), pp. 4–12.Google Scholar
  19. 19.
    F. Delbaen and W. Schachermayer, The fundamental theorem of asset pricing for unbounded stochastic processes, Math. Ann., 312 (1998), pp. 215–250.MathSciNetCrossRefGoogle Scholar
  20. 20.
    C. Dellacherie and P. Meyer, Probabilités et Potentiel. Chapitres I à IV, Hermann, Paris, 1975.Google Scholar
  21. 21.
    Z. Ding, C. Granger, and R. Engle, A long memory property of stock market returns and a new model, Journal of empirical finance, 1 (1983), pp. 83–106.CrossRefGoogle Scholar
  22. 22.
    Z. Ding and C. W. J. Granger, Modeling volatility persistence of speculative returns: a new approach, J. Econometrics, 73 (1996), pp. 185–215.MathSciNetCrossRefGoogle Scholar
  23. 23.
    P. Doukhan, G. Oppenheim, and M. S. Taqqu, eds., Theory and applications of long-range dependence, Birkhäuser Boston Inc., Boston, MA, 2003.Google Scholar
  24. 24.
    R. Engle, ARCH models, Oxford University Press, Oxford, 1995.Google Scholar
  25. 25.
    I. Giardina and J.-P. Bouchaud, Bubbles, crashes and intermittency in agent based market models, Eur. Phys. J. B Condens. Matter Phys., 31 (2003), pp. 421–437.MathSciNetCrossRefGoogle Scholar
  26. 26.
    C. Gourieroux, A. Monfort, and E. Renault, Indirect inference, Journal of Applied Econometrics, (1993).Google Scholar
  27. 27.
    C. Granger and N. Hyng, Occasional structural breaks and long memory with an application to the SP500 absolute stock returns, Journal of Empirical Finance, 11 (2004), pp. 399–421.CrossRefGoogle Scholar
  28. 28.
    C. W. J. Granger, Long memory relationships and the aggregation of dynamic models, J. Econometrics, 14 (1980), pp. 227–238.zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    C. W. J. Granger, Essays in econometrics: collected papers of Clive W. J. Granger. Vol. II, vol. 33 of Econometric Society Monographs, Cambridge University Press, Cambridge, 2001. Causality, integration and cointegration, and long memory, Edited by Eric Ghysels, Norman R. Swanson and Mark W. Watson.Google Scholar
  30. 30.
    D. Guillaume, M. Dacorogna, R. Davé, U. Müller, R. Olsen, and O. Pictet, From the birds eye view to the microscope: A survey of new stylized facts of the intraday foreign exchange markets, Finance Stoch., 1 (1997), pp. 95–131.CrossRefGoogle Scholar
  31. 31.
    J. M. Harrison and S. R. Pliska, Martingales and stochastic integrals in the theory of continuous trading, Stochastic Process. Appl., 11 (1981), pp. 215–260.MathSciNetCrossRefGoogle Scholar
  32. 32.
    C. C. Heyde, On modes of long-range dependence, J. Appl. Probab., 39 (2002), pp. 882–888.zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    M. Hols and C. De Vries, The limiting distribution of extremal exchange rate returns, Journal of Applied Econometrics, 6 (1991), pp. 287–302.Google Scholar
  34. 34.
    C. H. Hommes, A. Gaunersdorfer, and F. O. Wagener, Bifurcation routes to volatility clustering under evolutionary learning, working paper, CenDEF, 2003.Google Scholar
  35. 35.
    D. Jansen and C. De Vries, On the frequency of large stock returns, Rev. Econ. Stat., 73 (1991), pp. 18–24.CrossRefGoogle Scholar
  36. 36.
    A. Kirman, Ants, rationality, and recruitment, Quarterly Journal of Economics, 108 (1993), pp. 137–156.CrossRefGoogle Scholar
  37. 37.
    A. Kirman and G. Teyssiere, Microeconomic models for long-memory in the volatility of financial time series, Studies in nonlinear dynamics and econometrics, 5 (2002), pp. 281–302.CrossRefGoogle Scholar
  38. 38.
    B. LeBaron, Evolution and time horizons in an agent-based stock market, Macroeconomic Dynamics, 5 (2001), pp. 225–254.zbMATHCrossRefGoogle Scholar
  39. 39.
    —, Stochastic volatility as a simple generator of apparent financial power laws and long memory, Quant. Finance, 1 (2001), pp. 621–631.MathSciNetCrossRefGoogle Scholar
  40. 40.
    B. LeBaron, B. Arthur, and R. Palmer, Time series properties of an artificial stock market, Journal of Economic Dynamics and Control, 23 (1999), pp. 1487–1516.CrossRefGoogle Scholar
  41. 41.
    M. Liu, Modeling long memory in stock market volatility, Journal of Econometrics, 99 (2000), pp. 139–171.zbMATHCrossRefGoogle Scholar
  42. 42.
    A. Lo, Long-term memory in stock market prices, Econometrica, 59 (1991), pp. 1279–1313.zbMATHMathSciNetGoogle Scholar
  43. 43.
    I. N. Lobato and Velasco, Long memory in stock market trading volume, J. Bus. Econom. Statist., 18 (2000), pp. 410–427.MathSciNetCrossRefGoogle Scholar
  44. 44.
    F. Longin, The asymptotic distribution of extreme stock market returns, Journal of Business, 69 (1996), pp. 383–408.CrossRefGoogle Scholar
  45. 45.
    M. Loretan and P. Phillips, Testing the covariance stationarity of heavy-tailed time series, Journal of empirical finance, 1 (1994), pp. 211–248.CrossRefGoogle Scholar
  46. 46.
    T. Lux, On moment condition failure in German stock returns, Empirical economics, 25 (2000), pp. 641–652.CrossRefGoogle Scholar
  47. 47.
    T. Lux and M. Marchesi, Volatility clustering in financial markets: a microsimulation of interacting agents, Int. J. Theor. Appl. Finance, 3 (2000), pp. 675–702.MathSciNetCrossRefGoogle Scholar
  48. 48.
    B. B. Mandelbrot, The variation of certain speculative prices, Journal of Business, XXXVI (1963), pp. 392–417.Google Scholar
  49. 49.
    —, When can price be arbitraged efficiently? A limit to the validity of the random walk and martingale models, Rev. Econom. Statist., 53 (1971), pp. 225–236.MathSciNetCrossRefGoogle Scholar
  50. 50.
    —, Fractals and Scaling in Finance: Discontinuity, Concentration, Risk., Springer, New York, 1997.Google Scholar
  51. 51.
    —, Stochastic volatility, power laws and long memory, Quant. Finance, 1 (2001), pp. 558–559.MathSciNetCrossRefGoogle Scholar
  52. 52.
    B. B. Mandelbrot and M. S. Taqqu, Robust R/S analysis of long-run serial correlation, in Proceedings of the 42nd session of the International Statistical Institute, Vol. 2 (Manila, 1979), vol. 48, 1979, pp. 69–99.MathSciNetGoogle Scholar
  53. 53.
    B. B. Mandelbrot and J. Van Ness, Fractional Brownian motion, fractional noises and applications, SIAM Review, 10 (1968), pp. 422–437.MathSciNetCrossRefGoogle Scholar
  54. 54.
    R. Mantegna and H. Stanley, Scaling behavior of an economic index, Nature, 376 (1995), pp. 46–49.CrossRefGoogle Scholar
  55. 55.
    M. Métivier and J. Pellaumail, Stochastic integration, Academic Press, New York, 1980. Probability and Mathematical Statistics.Google Scholar
  56. 56.
    T. Mikosch and C. StĂricĂ, Limit theory for the sample autocorrelations and extremes of a GARCH (1; 1) process, Ann. Statist., 28 (2000), pp. 1427–1451.MathSciNetCrossRefGoogle Scholar
  57. 57.
    T. Mikosch and C. Stărică, Long-range dependence effects and ARCH modeling, in Theory and applications of long-range dependence, Birkhäuser Boston, Boston, MA, 2003, pp. 439–459.Google Scholar
  58. 58.
    J. Muzy, J. Delour, and E. Bacry, Modeling fluctuations of financial time series: From cascade processes to stochastic volatility models, Eur. J. Phys. B, 17 (2000), pp. 537–548.CrossRefGoogle Scholar
  59. 59.
    A. Pagan, The econometrics of financial markets, Journal of Empirical Finance, 3 (1996), pp. 15–102.CrossRefGoogle Scholar
  60. 60.
    M. Pourahmadi, Stationarity of the solution of Xt = AtXt−1 + εt and analysis of non-Gaussian dependent random variables, J. Time Ser. Anal., 9 (1988), pp. 225–239.zbMATHMathSciNetGoogle Scholar
  61. 61.
    P. Protter, Stochastic integration and differential equations, Springer, Berlin, 1990.Google Scholar
  62. 62.
    S. Resnick, G. Samorodnitsky, and F. Xue, How misleading can sample ACFs of stable MAs be? (Very!), Ann. Appl. Probab., 9 (1999), pp. 797–817.MathSciNetCrossRefGoogle Scholar
  63. 63.
    S. Resnick and E. van den Berg, Sample correlation behavior for the heavy tailed general bilinear process, Comm. Statist. Stochastic Models, 16 (2000), pp. 233–258.MathSciNetGoogle Scholar
  64. 64.
    L. C. G. Rogers, Arbitrage with fractional Brownian motion, Math. Finance, 7 (1997), pp. 95–105.zbMATHMathSciNetCrossRefGoogle Scholar
  65. 65.
    G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian Random Processes, Chapman & Hall, New York, 1994.Google Scholar
  66. 66.
    M. S. Taqqu and J. B. Levy, Using renewal processes to generate long-range dependence and high variability, in Dependence in probability and statistics (Oberwolfach, 1985), vol. 11 of Progr. Probab. Statist., Birkhäuser Boston, Boston, MA, 1986, pp. 73–89.Google Scholar
  67. 67.
    V. Teverovsky, M. S. Taqqu, and W. Willinger, A critical look at Lo's modified R/S statistic, J. Statist. Plann. Inference, 80 (1999), pp. 211–227.MathSciNetCrossRefGoogle Scholar
  68. 68.
    W. Willinger, M. Taqqu, and V. Teverovsky, Long range dependence and stock returns, Finance and Stochastics, 3 (1999), pp. 1–13.CrossRefGoogle Scholar

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© Springer-Verlag London Limited 2005

Authors and Affiliations

  • Rama Cont
    • 1
  1. 1.Centre de Mathématiques appliquéesEcole PolytechniqueFrance

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