The Leverage Effect and Other Stylized Facts Displayed by Bitcoin Returns


In this paper, we explore some stylized facts of the Bitcoin market using the BTC-USD exchange rate time series of historical intraday data from 2013 to 2020. Despite Bitcoin presenting some very peculiar idiosyncrasies, like the absence of macroeconomic fundamentals or connections with underlying assets or benchmarks, an asymmetry between demand and supply and the presence of inefficiency in the form of strong arbitrage opportunity, all these elements seem to be marginal in the definition of the structural statistical properties of this virtual financial asset, which result to be analogous to general individual stocks or indices. In contrast, we find some clear differences, compared to fiat money exchange rates time series, in the values of the linear autocorrelation and, more surprisingly, in the presence of the leverage effect. We also explore the dynamics of correlations, monitoring the shifts in the evolution of the Bitcoin market. This analysis is able to distinguish between two different regimes: a stochastic process with weaker memory signatures and closer to Gaussianity between the Mt. Gox incident and the late 2015, and a dynamics with relevant correlations and strong deviations from Gaussianity before and after this interval.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. 1.

    Total Market Capitalization, Available from: (2020)

  2. 2.

    S. Nakamoto, Bitcoin: A Peer-to-Peer Electronic Cash System. Available from:

  3. 3.

    D.G. Baur, K. Hong, A.D. Lee, Bitcoin: Medium of exchange or speculative assets?. J. Int. Finance. Markets Institut. Money. 54, 177 (2018)

    Article  Google Scholar 

  4. 4.

    S. Ammous, Can cryptocurrencies fulfil the functions of money?. Q. Rev. Econ. Finance. 70, 38–51 (2018)

    Article  Google Scholar 

  5. 5.

    Y. Liu, A. Tsyvinski. Risks and returns of cryptocurrency. Technical Report (National Bureau of Economic Research, Cambridge, 2018)

    Google Scholar 

  6. 6.

    A.F. Bariviera, M.J. Basgall, W. Hasperué, M. Naiouf, Some stylized facts of the Bitcoin market. Physica A. 484, 82 (2017)

    ADS  Article  Google Scholar 

  7. 7.

    A. Urquhart, H. Zhang, Is Bitcoin a hedge or safe haven for currencies? An intraday analysis. Int. Rev. Financial Anal. 63, 49 (2019)

    Article  Google Scholar 

  8. 8.

    E.F. Fama, Efficient capital markets: II. J. Finance. 46, 575 (1991)

    Article  Google Scholar 

  9. 9.

    M.T. Greene, B.D. Fielitz, Long-term dependence in common stock returns. J. Financ. Econ. 4, 339 (1977)

    Article  Google Scholar 

  10. 10.

    F. Lillo, J.D. Farmer, The long memory of the efficient market. Stud. Nonlinear Dyn. Econom. 8, 1 (2004)

    ADS  MATH  Google Scholar 

  11. 11.

    J.T. Barkoulas, C.F. Baum, Long-term dependence in stock returns. Econ. Lett. 53, 253 (1996)

    MATH  Article  Google Scholar 

  12. 12.

    J. Tolvi, Long memory and outliers in stock market returns. Appl. Financial Econ. 13, 495 (2003)

    Article  Google Scholar 

  13. 13.

    S. Kasman, E. Turgutlu, A.D. Ayhan, Long memory in stock returns: Evidence from the major emerging central European stock markets. Appl. Econ. Lett. 16, 1763 (2009)

    Article  Google Scholar 

  14. 14.

    C. Cheong, Estimating the hurst parameter in financial time series via heuristic approaches. J. Appl. Stat. 37, 201 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    A.W. Lo, Long-term memory in stock market prices. econometrica. 59, 1279 (1991)

    MATH  Article  Google Scholar 

  16. 16.

    A.W. Lo, A.C. MacKinlay. Long-Term Memory in Stock Market Prices. A Non-Random Walk Down Wall Street (Princeton University Press, Princeton, 1999)

    Google Scholar 

  17. 17.

    J. Bartos, Does Bitcoin follow the hypothesis of efficient market?. Int. J. Econ. Sci. IV, 10 (2015)

    Google Scholar 

  18. 18.

    A. Urquhart, The inefficiency of Bitcoin. Econ. Lett. 148, 80 (2016)

    Article  Google Scholar 

  19. 19.

    S. Nadarajah, J. Chu, On the inefficiency of Bitcoin. Econ. Lett. 150, 6 (2017)

    Article  Google Scholar 

  20. 20.

    V. Dimitrova, M. Fernández-Martínez, M.A. Sánchez-Granero, J.E. Trinidad Segovia, Some comments on Bitcoin market (in)efficiency. Plos One. 14, e0219243 (2019)

    Article  Google Scholar 

  21. 21.

    Z. Nan, T. Kaizoji, Market efficiency of the Bitcoin exchange rate: Weak and semi-strong form tests with the spot, futures and forward foreign exchange rates. Int. Rev. Financial Anal. 64, 273 (2019)

    Article  Google Scholar 

  22. 22.

    R. Cont, Empirical properties of asset returns: stylized facts and statistical issues. Quant. Finance. 1, 223 (2001)

    MATH  Article  Google Scholar 

  23. 23.

    A. Chakraborti, I.M. Toke, M. Patriarca, F. Abergel, Econophysics review: I. Emp. Facts Quant. Finance. 11, 991 (2011)

    MathSciNet  Article  Google Scholar 

  24. 24.

    BitcoinCharts, Available from: (2020)

  25. 25.

    N. Kaldor, in Capital accumulation and economic growth. The Theory of Capital, ed. by F.A. Lutz, D.C. Hague, (Macmillan, 1961), pp. 177–222

  26. 26.

    D.M. Guillaume, M.M. Dacorogna, R.R. Davé, U.A. Muller, R.B. Olsen, O.V. Pictet, From the bird’s eye to the microscope: A survey of new stylized facts of the Intra-Daily foreign exchange markets. Finance Stochast. 1, 95 (1997)

    MATH  Article  Google Scholar 

  27. 27.

    X. Gabaix, et al., Institutional investors and stock market volatility. Quart. J. Econ. 461 (2006)

  28. 28.

    M.E.J. Newman, Power laws, Pareto distributions and Zipf’s, law. Contemp. Phys. 46, 323 (2005)

    ADS  Article  Google Scholar 

  29. 29.

    Clauset A., C. Shalizi, M. Newman, Power-Law distributions in empirical data. SIAM Rev. 54, 661 (2009)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    A. Clauset, M. Young, K.S. Gleditsch, On the frequency of severe terrorist events. J. Confl. Resolut. 51, 58 (2007)

    Article  Google Scholar 

  31. 31.

    A. Pagan, The econometrics of financial markets. J. Empirical Finance. 3, 15 (1996)

    Article  Google Scholar 

  32. 32.

    E.F. Fama, Efficient capital markets: a review of theory and empirical work. J. Finance. 25, 383 (1971)

    MathSciNet  Article  Google Scholar 

  33. 33.

    C.E. Spearman, The proof and measurement of association between two things. Amer. J. Psychol. 15, 72 (1904)

    Article  Google Scholar 

  34. 34.

    C.E. Spearman, General intelligence, objectively determined and measured. Amer. J. Psychol. 15, 201 (1904)

    Article  Google Scholar 

  35. 35.

    C.E. Spearman, Correlation calculated from faulty data. British J. Psychol. 3, 271 (1910)

    Google Scholar 

  36. 36.

    E. Lehmann, H. D’Abrera. Nonparametrics: Statistical Methods Based on Ranks (Springer, New York, 2006)

    Google Scholar 

  37. 37.

    Y. Malevergne, D. Sornette. Extreme Financial Risks: from Dependence to Risk Management (Springer, Heidelberg, 2006)

    Google Scholar 

  38. 38.

    F. Comte, E. Renault, Long memory continuous time models. J. Econ. 73, 101 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    C.W.J. Granger, Z. Ding, Varieties of long memory models. J. Econ. 73, 61 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    O.V. Pictet, M. Dacorogna, U.A. Muller, R.B. Olsen, J.R. Ward, Statistical study of foreign exchange rates. J. Banking Finance. 14, 189 (1997)

    Google Scholar 

  41. 41.

    R. Cont, M. Potters, J.P. Bouchaud, in Scaling in stock market data: stable laws and beyond. Scale Invariance and Beyond. (Proceedings CNRS Workshop on Scale Invariance, Les Houches), ed. by D Graner, Sornette (Springer, Berlin, 1997)

  42. 42.

    Y. Liu, P. Cizeau, M. Meyer, C. K. Peng, H.E. Stanley, Correlations in economic time series. Physica A. 245, 437 (1997)

    ADS  MathSciNet  Article  Google Scholar 

  43. 43.

    J.P. Bouchaud, A. Matacz, M. Potters, Leverage effect in financial markets: The retarded volatility model. Phys. Rev. Lett. 87, 228701 (2001)

    ADS  Article  Google Scholar 

  44. 44.

    A.M. Calvão, E. Brigatti, Collective movement in alarmed animals groups: A simple model with positional forces and a limited attention field. Physica A. 520, 450 (2019)

    ADS  MathSciNet  Article  Google Scholar 

  45. 45.

    T. Bollerslev, J. Litvinova, G. Tauchen, Leverage and volatility feedback, effects in High-Frequency data. J. Financial Econ. 4(3), 353 (2006)

    Google Scholar 

  46. 46.

    J.-P. Bouchaud, M. Potters, More stylized facts of financial markets: Leverage effect and downside correlations. Physica A. 299, 60 (2001)

    ADS  MATH  Article  Google Scholar 

  47. 47.

    A. Pagan, The econometrics of financial markets. J. Empirical Finance. 3, 15 (1996)

    Article  Google Scholar 

  48. 48.

    J.C. Hull. Options, Futures and Other Derivatives, 6th edn. (Pearson Prentice Hall, New Jersey, 2006)

    Google Scholar 

  49. 49.

    P.R. Hansen, A. Lunde, A forecast comparison of volatility models: Does anything beat a garch (1, 1)?. J. Appl. Econ. 20, 873 (2005)

    MathSciNet  Article  Google Scholar 

  50. 50.

    T. Bollerslev, R.Y. Chou, K.F. Kroner, ARCH modeling in finance: A review of theory and empirical evidence. J. Econ. 52, 5 (1992)

    MATH  Article  Google Scholar 

  51. 51.

    D.K. Zuegel, What happened at MtGox? The collapse of the world’s largest Bitcoin exchange. The Stanford Review. Available from:

  52. 52.

    S. Begušic, Z. Kostanjčar, E.H. Stanley, B. Podobnik, Scaling properties of extreme price fluctuations in Bitcoin markets. Physica A. 510, 400 (2018)

    ADS  Article  Google Scholar 

  53. 53.

    C.W. Senarathne, The Leverage Effect and Information Flow Interpretation for Speculative Bitcoin Prices: Bitcoin Volume vs ARCH Effect. European J. Econ. Stud. 8, 77 (2019)

    Google Scholar 

  54. 54.

    A.K. Tiwari, S. Kumar, R. Pathak, Modelling the dynamics of Bitcoin and Litecoin: GARCH versus stochastic volatility models. Appl. Econ. 51, 4073 (2019)

    Article  Google Scholar 

  55. 55.

    D. Ardia, K. Bluteau, M. Rüede, Regime changes in Bitcoin GARCH volatility dynamics. Fin. Res. Lett. 29, 266 (2019)

    Article  Google Scholar 

  56. 56.

    J. Liu, A. Serletis, Volatility in the cryptocurrency market, open. Econ. Rev. 30, 779 (2019)

    MATH  Google Scholar 

  57. 57.

    J.C. Gerlach, G. Demos, D. Sornette, Dissection of Bitcoin’s multiscale bubble history from January 2012 to February 2018. R. Soc. open sci. 6, 180643 (2019)

    ADS  Article  Google Scholar 

  58. 58.

    S. Drożdż, et al, Bitcoin market route to maturity? Evidence from return fluctuations, temporal correlations and multiscaling effects. Chaos. 28, 071101 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  59. 59.

    P. Katsiampa, S. Corbet, B. Lucey, Volatility spillover effects in leading cryptocurrencies: A BEKK-MGARCH analysis. Finance Res Lett. 29, 68 (2019)

    Article  Google Scholar 

Download references


F.N.M.S.F. received partial financial support from the PIBIC program of Universidade Federal do Rio de Janeiro. M.A.B. received support from Fapesp (Grant 2018/22562-4) and CNPQ (Grant 303986/2017-4 and 428433/2018-9).

Author information



Corresponding author

Correspondence to E. Brigatti.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

de Sousa Filho, F.N.M., Silva, J.N., Bertella, M.A. et al. The Leverage Effect and Other Stylized Facts Displayed by Bitcoin Returns. Braz J Phys (2021).

Download citation


  • Fluctuation phenomena
  • Random processes
  • Noise
  • Brownian motion