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Exponentially damped Lévy flights

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Author Info

  • Matsushita, Raul
  • Rathie, Pushpa
  • Da Silva, Sergio

Abstract

Since real processes seem to departure from standard Lévy distributions, modifications to the latter have been suggested in literature. These include (abruptly) truncated (Phys. Rev. Lett. 73 (1994) 2946), smoothly truncated (Phys. Rev. E 52 (1995) 1197; Phys. Lett. A 266 (2000) 282) and gradually truncated Lévy flights (Physica A 268 (1999) 231; Physica A 275 (2000) 531). We put forward what we call an exponentially damped Lévy flight which encompasses the previous cases. In the presence of increasing and positive feedbacks, our distribution is assumed to deviate from the Lévy in both a smooth and gradual fashion. We estimate the truncation parameters by nonlinear least squares to optimally fit the distribution tails. That is a novel approach for estimating parameters α and γ of the Lévy. The method is illustrated with daily data on exchange rates for 15 countries against the US dollar. Our results show that the exponentially damped Lévy flight fits the data well when increasing and positive deviations are present.

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Bibliographic Info

Article provided by Elsevier in its journal Physica A: Statistical Mechanics and its Applications.

Volume (Year): 326 (2003)
Issue (Month): 3 ()
Pages: 544-555

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Handle: RePEc:eee:phsmap:v:326:y:2003:i:3:p:544-555

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Web page: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/

Related research

Keywords: Lévy flights; Foreign exchange rates;

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References

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  1. Benoit Mandelbrot, 1963. "The Variation of Certain Speculative Prices," The Journal of Business, University of Chicago Press, vol. 36, pages 394.
  2. Andrew W. Lo & A. Craig MacKinlay, 1989. "Stock Market Prices Do Not Follow Random Walks: Evidence From a Simple Specification Test," NBER Working Papers 2168, National Bureau of Economic Research, Inc.
  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. 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.
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Citations

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Cited by:
  1. Vinogradov, Dmitry V., 2010. "Cumulant approach of arbitrary truncated Levy flight," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(24), pages 5794-5800.
  2. Sergio Da Silva, 2004. "International Finance, Levy Distributions, and the Econophysics of Exchange Rates," International Finance 0405018, EconWPA.
  3. Sergio Da Silva, 2004. "Exponentially Damped Levy Flights, Multiscaling, and Slow Convergence in Stockmarkets," Finance 0405028, EconWPA.
  4. Imai, Junichi & Kawai, Reiichiro, 2011. "On finite truncation of infinite shot noise series representation of tempered stable laws," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(23), pages 4411-4425.
  5. Schinckus, C., 2013. "Between complexity of modelling and modelling of complexity: An essay on econophysics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(17), pages 3654-3665.
  6. Sergio Da Silva, 2004. "Exponentially Damped Levy Flights, Multiscaling, and Exchange Rates," Finance 0405027, EconWPA.
  7. 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.
  8. Lemmens, D. & Liang, L.Z.J. & Tempere, J. & De Schepper, A., 2010. "Pricing bounds for discrete arithmetic Asian options under Lévy models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(22), pages 5193-5207.

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