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Microscopic origin of non-Gaussian distributions of financial returns

Author

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  • Biró, T.S.
  • Rosenfeld, R.
Abstract
In this paper we study the possible microscopic origin of heavy-tailed probability density distributions for the price variation of financial instruments. We extend the standard log-normal process to include another random component in the so-called stochastic volatility models. We study these models under an assumption, akin to the Born–Oppenheimer approximation, in which the volatility has already relaxed to its equilibrium distribution and acts as a background to the evolution of the price process. In this approximation, we show that all models of stochastic volatility should exhibit a scaling relation in the time lag of zero-drift modified log-returns. We verify that the Dow–Jones Industrial Average index indeed follows this scaling. We then focus on two popular stochastic volatility models, the Heston and Hull–White models. In particular, we show that in the Hull–White model the resulting probability distribution of log-returns in this approximation corresponds to the Tsallis (t-Student) distribution. The Tsallis parameters are given in terms of the microscopic stochastic volatility model. Finally, we show that the log-returns for 30 years Dow Jones index data is well fitted by a Tsallis distribution, obtaining the relevant parameters.

Suggested Citation

  • Biró, T.S. & Rosenfeld, R., 2008. "Microscopic origin of non-Gaussian distributions of financial returns," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(7), pages 1603-1612.
    • Handle: RePEc:eee:phsmap:v:387:y:2008:i:7:p:1603-1612
    DOI: 10.1016/j.physa.2007.10.067
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    Citations

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    Cited by:

    1. Xu, Dan & Beck, Christian, 2016. "Transition from lognormal to χ2-superstatistics for financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 453(C), pages 173-183.
    2. Geoffrey Ducournau, 2021. "Bayesian inference and superstatistics to describe long memory processes of financial time series," Papers 2105.04171, arXiv.org.
    3. Ramos, Antônio M.T. & Carvalho, J.A. & Vasconcelos, G.L., 2016. "Exponential model for option prices: Application to the Brazilian market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 445(C), pages 161-168.
    4. López Martín, María del Mar & García, Catalina García & García Pérez, José, 2012. "Treatment of kurtosis in financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(5), pages 2032-2045.

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