Volatility, Long Memory, and Chaos: A Discussion on some “Stylized Facts” in Financial Markets with a Focus on High Frequency Data

  • Amitava Sarkar
  • Gagari Chakrabarti
  • Chitrakalpa Sen


Starting from the “Tulip Mania’ in the seventeenth century, financial sector crises have come in waves and in many different guises. While some of these remained confined to the regional boundaries, some acquired global dimension with the ultimate devastating impact on the real economy. The fact that the global economy has collapsed many a time following a financial panic has instigated the researchers, particularly after the recent global financial melt down, to explore the dynamics of global financial markets: the old issue of ‘finance-growth nexus’ is resurrected once again. The traditional school of the literature, however, is bifurcated on the issue. While one school perceives finance to be a ‘side show’ of growth, others believe in the considerable control that financial markets exercise on the real sector. The true nature and direction of the causality, however, is yet to be exposed. More recently, a parallel school of thought has developed that concentrate on the endogenous factors that generate dynamics within the stock market independent of the real sector. This growing body of the literature conjectures the global financial markets to be mostly deterministic, and in some cases, chaotic in nature. The markets are characterized by nonperiodic cycles and trends where volatility and fluctuations generate endogenously. The global markets, thus, are supposed to be inherently instable, or at best, stable on knife edge. Cycles and crashes are manifestations of this inherent instability. There is no determinate equilibrium in the market and no external shocks would be required to gear financial crises at regular interval. The implications of these are tremendous. A chaotic financial market puts efficient market hypothesis on trial, renders traditional asset pricing models useless, and makes long-term forecasting less reliable. The most devastating implication of the fact that the financial markets implode from within is perhaps for the developing markets as it makes government intervention ineffective. Hence, an introspection of financial market dynamics following this rather offbeat line of thought could be a researcher’s delight. The results obtained might lead to a reframing of old ideas and beliefs regarding financial market dynamics, boom and bust cycle, and predictability of financial crashes.


Stock Market Financial Market GARCH Model Foreign Exchange Market Maximum Lyapunov Exponent 
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.


  1. Alexander C (2001) Market Models: a guide to financial model analysis, Chichester, J. WileyGoogle Scholar
  2. Annis AA, Lloyd EH (1976) The expected value of the adjusted rescaled Hurst range of independent normal summands. Biometrika 63:111–116CrossRefGoogle Scholar
  3. Barkoulas J, Baum T, Christopher F, Travlos N (2000) Long memory in the Greek stock market. Appl Financ Econ 10(2):177–184CrossRefGoogle Scholar
  4. Bask M (2002) A positive Lyapunov exponent in Swedish exchange rates? Chaos, Solitons Fractals 14(8):1295–1304CrossRefGoogle Scholar
  5. Black, F (1976) Studies in stock price volatility changes. In: American Statistical Association, Proceedings of the Business and Economic Statistics Section, 177–181Google Scholar
  6. Bollerslev T (1986) Generalized autoregressive conditional heteroscedasticity. J Econometrics 31:307–327CrossRefGoogle Scholar
  7. Brock W, Hommes C (1998) Heterogeneous beliefs and routes to chaos in a simple asset pricing model. J Econ Dyn Control 22:1235–1274CrossRefGoogle Scholar
  8. Brock WA, Dechert W, Scheinkman J (1996) A test for independence based on the correlation dimension. Econometric Rev 15(3):197–235CrossRefGoogle Scholar
  9. Brooks C (1995) A measure of persistence in daily pound exchange rates. Appl Econ Lett 2:428–431CrossRefGoogle Scholar
  10. Cavalcante J, Assaf A (2004) Long range dependence in the returns and volatility of the Brazilian stock market. Eur Rev Econ Finance 3:5–22Google Scholar
  11. Chakrabarti G (2010) Dynamics of global stock market: crisis and beyond. (Vdm-Verlag Dr. Müller), GermanyGoogle Scholar
  12. Chakrabarti G, Sen C (2012) Anatomy of global stock market crashes: an empirical analysis. Springer, NewdelhiGoogle Scholar
  13. Cheung Y (1993) Long memory in foreign-exchange rates. J Bus Econ Stat 11(1):93–101Google Scholar
  14. Cheung Y, Lai K (1995) A search for long memory in international stock market returns. J Int Money Financ 14:597–615CrossRefGoogle Scholar
  15. Chou RY (1988) Volatility persistence and stock valuations: some empirical evidence using Garch. J Appl Econometrics 3(4):279–294CrossRefGoogle Scholar
  16. Cuaresma JC (1998) Deterministic chaos versus stochastic process: an empirical study on the Austrian Schilling-US Dollar Exchange Rate. Economic Series No. 60Google Scholar
  17. De Grauwe P, Grimaldi M (2006) Exchange rate puzzles: a tale of switching attractors. Eur Econ Rev 50:1–33CrossRefGoogle Scholar
  18. De Grauwe P, Dewachter H, Embrechts M (1993) Exchange rate theory: chaotic models of foreign exchange markets. Blackwell, OxfordGoogle Scholar
  19. Ding Z, Granger CWJ (1996) Modeling volatility persistence of speculative returns: a new approach. J Econometrics 73:185–215CrossRefGoogle Scholar
  20. Engle R (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation. Econometrica 50:987–1008CrossRefGoogle Scholar
  21. Engle RF, Lee GGJ (1999) A permanent and transitory component model of stock return volatility. In: Engle R, White H (eds.) Cointegration, Causality, and Forecasting: A Festschrift in Honor of Clive W. J. Granger. Oxford University Press, New YorkGoogle Scholar
  22. Engle RF, Victor K Ng (1993) Measuring and testing the impact of news on volatility. J Financ 48(5): 1749–1778Google Scholar
  23. Engle RF, Bollerslev T (1986) Modeling the persistence of conditional variances. Econometric Rev 5:1–50CrossRefGoogle Scholar
  24. Engle RF (1990) Discussion: stock market volatility and the crash of ‘87. Rev Financ Stud 3:103–106CrossRefGoogle Scholar
  25. Engle RF, Lilien DM, Robbins RP (1987) Estimating time varying risk premia in the term structure: The ARCH-M model. Econometrica 55:391–407CrossRefGoogle Scholar
  26. Fraser AM, Swinney HL (1986) Independent coordinates for strange attractors from mutual information. Phys Rev A 33:1134–1140CrossRefGoogle Scholar
  27. Geweke J, Porter-Hudak S (1983) The estimation and application of long memory time series models. J Time Ser Anal 4:221–238CrossRefGoogle Scholar
  28. Glosten LR, Jagannathan P, Runkle DE (1993) On the relation between expected value and the volatility of excess returns on stocks. J Financ 48:1779–1801CrossRefGoogle Scholar
  29. Granger CWJ, Joyeux R (1980) An introduction to long-memory time series models and fractional differencing. J Time Ser Anal 1:15–30CrossRefGoogle Scholar
  30. Greene MT, Fielitz BD (1977) Long-term dependence in common stock returns. J Financ Econ 4(3):339–349CrossRefGoogle Scholar
  31. Guillaume DM (1995) A low-dimensional fractal attractor in the foreign exchange markets? In: Robert Trippi (ed) Chaos & nonlinear dynamics in the financial markets, IRWIN Professional Publishing, pp. 269–294Google Scholar
  32. Guillaume DM (2000) Chaos in the foreign exchange market. In: Intra-daily exchange rate movements, Chapter 3. Springer, Boston, pp 59–79Google Scholar
  33. Hsieh DA (1989) Testing for nonlinear dependence in daily foreign exchange rates. J Bus 62(3):339–368CrossRefGoogle Scholar
  34. Hurst H (1951) Long Term Storage Capacity of Reservoirs, Transactions of the American Society of Civil Engineers, 116: 770–799Google Scholar
  35. Jacobsen B (1996) Long-term dependence in stock returns. J Empirical Financ 3:393–417CrossRefGoogle Scholar
  36. Kang SH, Yoon S (2007) Long memory properties in return and volatility: evidence from the Korean stock market. Physica A 385(2):591–600CrossRefGoogle Scholar
  37. Kantz H, Schreiber T (1997) Nonlinear time series analysis. Cambridge University Press, CambridgeGoogle Scholar
  38. Kaplan DT, Glass L (1992) Direct test for determinism in a time series. Phys Rev Lett 68:427–430CrossRefGoogle Scholar
  39. Kasman A, Torun E (2007) Long memory in the Turkish stock market return and volatility. Central Bank Rev 20(2):13–27Google Scholar
  40. Kennel MB, Brown R, Abarbanel HDI (1992) Determining embedding dimension for phase space reconstruction using a geometrical construction. Phys Rev A45:3403–3411Google Scholar
  41. Kočenda E (1996) Volatility of a seemingly fixed exchange rate. East Eur Econ 34(6):37–67Google Scholar
  42. Kodba S, Perc M, Marhl M (2004) Detecting chaos from a time series. Eur J Phys 26(2005):205–215Google Scholar
  43. Lee J, Kim TS, Lee HK (2000) Long memory in volatility of Korean stock market returns. Proceedings of Informs & KormsGoogle Scholar
  44. Lo AW (1991) Long-term memory in stock market prices. Econometrica 59:1279–1313CrossRefGoogle Scholar
  45. Lo AW, MacKinlay C (1988) Stock market prices do not follow random walks: evidence from a simple specification test. Rev Financ Stud 1:41–66CrossRefGoogle Scholar
  46. Lo AW, McKinlay AC (1990) Data snooping biases in tests of financial asset pricing models, Review of Financial Studies, Society for Financial Studies, 3(3): 431–67Google Scholar
  47. Lux T (1996) Long-term stochastic dependence in financial prices: evidence from the german stock market. Appl Econ Lett 3(11):701–706CrossRefGoogle Scholar
  48. Mandelbrot BB (1965) Forecasts of Future Prices, Unbiased Markets, and "Martingale" Models, The Journal of Business, University of Chicago Press, 39: 242–255Google Scholar
  49. Mandelbrot B, Wallis J (1969) Noah, joseph and operational hydrology. Water Resour Res 4:909–918CrossRefGoogle Scholar
  50. Mandelbrot BB (1971) When Can Price Be Arbitraged Efficiently? A Limit to the Validity of the Random Walk and Martingale Models, The Review of Economics and Statistics, MIT Press, 53(3): 225–236Google Scholar
  51. Mandelbrot BB (1972) Statistical methodology for non-periodic cycles; From the covariance to R/S analysis. Ann Econ Soc Measur 1:259–260Google Scholar
  52. Mandelbrot BB (1975) A fast fractional Gaussian noise generator. Water Resour Res 7:543–553CrossRefGoogle Scholar
  53. Mantegna R, Stanley E (2000) Introduction to econophysics. Cambridge University Press, CambridgeGoogle Scholar
  54. Mukherjee I, Sen C, Sarkar A (2011) Long memory in stock returns: insights from the Indian market. Int J Appl Econ Financ 5(1):62–74CrossRefGoogle Scholar
  55. Nath G. (2001) Long memory and Indian stock market-an empirical evidence, UTIICM Conference Paper. Accessed 3rd Jan 2009
  56. Nelson DB (1991) Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59:347–370CrossRefGoogle Scholar
  57. Newey WK, West KD (1994) Automatic Lag Selection in Covariance Matrix Estimation. Rev Econ Stud 61(4):631–653CrossRefGoogle Scholar
  58. Perc M (2005a) Nonlinear time series analysis of the human electrocardiogram. Eur J Phys 26:757–768CrossRefGoogle Scholar
  59. Perc M (2005b) The dynamics of human gait. Eur J Phys 26:525–534CrossRefGoogle Scholar
  60. Peters EE (1991) Chaos and order in the capital markets. John Wiley & Sons, New York Google Scholar
  61. Peters EE (1989) Fractal structure in the capital markets. Financ Anal J 32–37 (July–August)Google Scholar
  62. Rabemananjara R, Zakoian JM (1993) Threshold arch models and asymmetries in volatility. J Appl Econometrics 8(1):31–49CrossRefGoogle Scholar
  63. Rhode C, Morari M (1997) False-nearest-neighbors algorithm and noise-corrupted time series. Phys Rev E 55(5):6162–6170CrossRefGoogle Scholar
  64. Rickles D (2008) Econophysics and the complexity of financial markets. In Collier J, Hooker C (eds.) Handbook of the philosophy of science, Vol 10. Philosophy of complex systems. Elsevier, North HollandGoogle Scholar
  65. Rodrìguez SA, Kharitonov V, Dion JM, Dugard L (2004) Robust stability of neutral systems: A Lyapunov-Krasovskii constructive approach. Int J Robust Nonlinear Control 14(16):1345–1358CrossRefGoogle Scholar
  66. Rubio BO, Rodriguez F, Rivero S (1992) Chaotic behavior in exchange rate series: first results for the Peseta-U.S. dollar case. Econ Lett 39(2):207–211CrossRefGoogle Scholar
  67. Samorodnitsky G (2006) Long range dependence. Found Trends Stochast Syst 1(3):163–257CrossRefGoogle Scholar
  68. Sarkar A, Chakrabarti G, Sen C (2009) Indian stock market volatility in recent years: transmission from global and regional contagion and traditional domestic sectors. J Asset Manage 10(4):63–71CrossRefGoogle Scholar
  69. Scheinkman J, LeBaron B (1989) Nonlinear dynamics and stock returns. J Bus 62(3):311–337CrossRefGoogle Scholar
  70. Schwert GW (1989) Business cycles, financial crises, and stock volatility. Carnegie-Rochester Conf Ser Public Policy 31:83–126CrossRefGoogle Scholar
  71. Sen C, Mukherjee I, Chakrabarti G, Sarkar A (2009) Long memory in some selected stock markets: an autocorrelation based approach. Indian J Capital Market 3(1):11–16Google Scholar
  72. Sen C (2012) Indian foreign exchange market: a study into volatility and regime switch. LAP Lambert Academic Publishing, GermanyGoogle Scholar
  73. Sen C, Chakrabarti G, Sarkar A (2011) Evidence of chaos: a tale of two exchange rates. Empirical Econ Lett 10(8):777–784Google Scholar
  74. Taylor SJ (1986) Modeling financial time series. John Wiley and Sons, ChichesterGoogle Scholar
  75. Tolvi J (2003) Long memory and outliers in stock market returns. Appl Financ Econ 13:495–502CrossRefGoogle Scholar
  76. Tsay RS (2001) Analysis of financial time series. Wiley-Interscience, New YorkGoogle Scholar
  77. Tschernig R (1995) Long memory in foreign exchange rates revisited. J Int Financ Markets Inst Money 5(2/3):53–78Google Scholar
  78. Vandrovych V (2007) Nonlinearities in exchange-rate dynamics: chaos? SSRN: Accessed 24 July 2008
  79. Wolf A, Swift JB, Swinney HL, Vastano JA (1985) Determining Lyapunov exponents from a time series. Physica D 16:285–317CrossRefGoogle Scholar
  80. Zakoian JM (1994) Threshold heteroscedastic models. J Econ Dynam Control 18:931–955CrossRefGoogle Scholar

Copyright information

© Springer India 2013

Authors and Affiliations

  • Amitava Sarkar
    • 1
  • Gagari Chakrabarti
    • 2
  • Chitrakalpa Sen
    • 3
  1. 1.Indian Statistical InstituteKolkataIndia
  2. 2.Department of EconomicsPresidency UniversityKolkataIndia
  3. 3.Auro UniversitySuratIndia

Personalised recommendations