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Lévy processes on the cryptocurrency market


  • Damian Zięba

    () (Faculty of Economic Sciences, University of Warsaw)


Lévy processes are very often used in financial modelling since they address various characteristics of financial data. One of those characteristics is the heavy-tailedness of probability density functions - a very common empirical stylized fact on the cryptocurrency market. The aim of this study was to determine which type of Lévy motion fits the data of cryptocurrencies better, namely Alpha-Stable distribution or one of distributions from the family of generalized hyperbolic motions. The log-returns of 227 cryptocurrencies, standardized by the realized volatility estimated with the GARCH (1,1), were fitted to 11 types of distributions. The results show that the generalized hyperbolic motions fit the cryptocurrency data much more accurately than the Alpha-Stable distribution, similarly as in the case of TOP100 NASDAQ stocks. In the further stage of the analysis, it is shown how the distribution of cryptocurrency data varies over time, i.e. before, during, and after the ‘boom-period’ of 2017/2018.

Suggested Citation

  • Damian Zięba, 2019. "Lévy processes on the cryptocurrency market," Working Papers 2019-15, Faculty of Economic Sciences, University of Warsaw.
  • Handle: RePEc:war:wpaper:2019-15

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    References listed on IDEAS

    1. Gkillas, Konstantinos & Katsiampa, Paraskevi, 2018. "An application of extreme value theory to cryptocurrencies," Economics Letters, Elsevier, vol. 164(C), pages 109-111.
    2. Stephen Chan & Jeffrey Chu & Saralees Nadarajah & Joerg Osterrieder, 2017. "A Statistical Analysis of Cryptocurrencies," Journal of Risk and Financial Management, MDPI, Open Access Journal, vol. 10(2), pages 1-23, May.
    3. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold & Paul Labys, 2000. "Exchange Rate Returns Standardized by Realized Volatility are (Nearly) Gaussian," Multinational Finance Journal, Multinational Finance Journal, vol. 4(3-4), pages 159-179, September.
    4. Corbet, Shaen & Lucey, Brian & Urquhart, Andrew & Yarovaya, Larisa, 2019. "Cryptocurrencies as a financial asset: A systematic analysis," International Review of Financial Analysis, Elsevier, vol. 62(C), pages 182-199.
    5. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-524, October.
    6. Joerg Osterrieder & Julian Lorenz, 2017. "A Statistical Risk Assessment Of Bitcoin And Its Extreme Tail Behavior," Annals of Financial Economics (AFE), World Scientific Publishing Co. Pte. Ltd., vol. 12(01), pages 1-19, March.
    7. Phillip, Andrew & Chan, Jennifer S.K. & Peiris, Shelton, 2018. "A new look at Cryptocurrencies," Economics Letters, Elsevier, vol. 163(C), pages 6-9.
    8. Borri, Nicola, 2019. "Conditional tail-risk in cryptocurrency markets," Journal of Empirical Finance, Elsevier, vol. 50(C), pages 1-19.
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    More about this item


    cryptocurrency market; distribution fitting; Generalized Hyperbolic distribution; Alpha-Stable distribution; Lévy process;

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C30 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - General
    • G15 - Financial Economics - - General Financial Markets - - - International Financial Markets

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