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Autocorrelation as a source of truncated Lévy flights in foreign exchange rates

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  • Figueiredo, Annibal
  • Gleria, Iram
  • Matsushita, Raul
  • Da Silva, Sergio

Abstract

We suggest that the ultraslow speed of convergence associated with truncated Lévy flights (Phys. Rev. Lett. 73 (1994) 2946) may well be explained by autocorrelations in data. We show how a particular type of autocorrelation generates power laws consistent with a truncated Lévy flight. Stock exchanges have been suggested to be modeled by a truncated Lévy flight (Nature 376 (1995) 46; Physica A 297 (2001) 509; Econom. Bull. 7 (2002) 1). Here foreign exchange rate data are taken instead. Scaling power laws in the “probability of return to the origin” are shown to emerge for most currencies. A novel approach to measure how distant a process is from a Gaussian regime is presented.

Suggested Citation

  • Figueiredo, Annibal & Gleria, Iram & Matsushita, Raul & Da Silva, Sergio, 2003. "Autocorrelation as a source of truncated Lévy flights in foreign exchange rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 323(C), pages 601-625.
    • Handle: RePEc:eee:phsmap:v:323:y:2003:i:c:p:601-625
    DOI: 10.1016/S0378-4371(03)00029-3
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    1. Andrew W. Lo, A. Craig MacKinlay, 1988. "Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test," The Review of Financial Studies, Society for Financial Studies, vol. 1(1), pages 41-66.
    2. repec:ebl:ecbull:v:7:y:2002:i:3:p:1-12 is not listed on IDEAS
    3. Sergio Da Silva & Raul Matsushita & Iram Gleria, 2002. "Scaling power laws in the Sao Paulo Stock Exchange," Economics Bulletin, AccessEcon, vol. 7(3), pages 1-12.
    4. Blattberg, Robert C & Gonedes, Nicholas J, 1974. "A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices," The Journal of Business, University of Chicago Press, vol. 47(2), pages 244-280, April.
    5. Miranda, L.Couto & Riera, R., 2001. "Truncated Lévy walks and an emerging market economic index," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 297(3), pages 509-520.
    6. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
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    Cited by:

    1. Annibal Figueiredo & Iram Gleria & Raul Matsushita & Sergio Da Silva, 2004. "On the origins of truncated Lévy flights," Finance 0404013, University Library of Munich, Germany.
    2. Matsushita, Raul & Gleria, Iram & Figueiredo, Annibal & Rathie, Pushpa & Da Silva, Sergio, 2004. "Exponentially damped Lévy flights, multiscaling, and exchange rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 333(C), pages 353-369.
    3. Liu, Lei & Hu, Fei, 2013. "Cascade-like and scaling behavior of wind velocity increments in the atmospheric surface layer," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(23), pages 5808-5816.
    4. da Silva, Roberto & Zembrzuski, Marcelo & Correa, Fabio C. & Lamb, Luis C., 2010. "Stock markets and criticality in the current economic crisis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(23), pages 5460-5467.
    5. Medino, Ary V. & Lopes, Sílvia R.C. & Morgado, Rafael & Dorea, Chang C.Y., 2012. "Generalized Langevin equation driven by Lévy processes: A probabilistic, numerical and time series based approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(3), pages 572-581.
    6. Sergio Da Silva, 2004. "Autocorrelation and the Sum of Stochastic Variables," Finance 0405020, University Library of Munich, Germany.
    7. Gleria, Iram & Figueiredo, Annibal & Matsushita, Raul & Rathie, Pushpa & Da Silva, Sergio, 2004. "Exponentially damped Lévy flights, multiscaling and slow convergence in stockmarkets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 342(1), pages 200-206.
    8. Figueiredo, Annibal & Gleria, Iram & Matsushita, Raul & Da Silva, Sergio, 2004. "Lévy flights, autocorrelation, and slow convergence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 337(3), pages 369-383.
    9. Matsushita, Raul & Rathie, Pushpa & Da Silva, Sergio, 2003. "Exponentially damped Lévy flights," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 326(3), pages 544-555.
    10. De Domenico, Federica & Livan, Giacomo & Montagna, Guido & Nicrosini, Oreste, 2023. "Modeling and simulation of financial returns under non-Gaussian distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 622(C).
    11. Sergio Da Silva, 2004. "International Finance, Levy Distributions, and the Econophysics of Exchange Rates," International Finance 0405018, University Library of Munich, Germany.

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