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A model for stock returns and volatility

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  • Ma, Tao
  • Serota, R.A.

Abstract

We prove that Student’s t-distribution provides one of the better fits to returns of S&P component stocks and the generalized inverse gamma distribution best fits VIX and VXO volatility data. We further prove that stock returns are best fit by the product distribution of the generalized inverse gamma and normal distributions. We find Brown noise in VIX and VXO time series and explain the mean and the variance of the relaxation times on approach to the steady-state distribution.

Suggested Citation

  • Ma, Tao & Serota, R.A., 2014. "A model for stock returns and volatility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 398(C), pages 89-115.
    • Handle: RePEc:eee:phsmap:v:398:y:2014:i:c:p:89-115
    DOI: 10.1016/j.physa.2013.11.032
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    References listed on IDEAS

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    1. Eckhard Platen & Renata Rendek, 2007. "Empirical Evidence on Student-t Log-Returns of Diversified World Stock Indices," Research Paper Series 194, Quantitative Finance Research Centre, University of Technology, Sydney.
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    5. Jin‐Chuan Duan, 1995. "The Garch Option Pricing Model," Mathematical Finance, Wiley Blackwell, vol. 5(1), pages 13-32, January.
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    Citations

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    Cited by:

    1. Nicolas Langren'e & Geoffrey Lee & Zili Zhu, 2015. "Switching to non-affine stochastic volatility: A closed-form expansion for the Inverse Gamma model," Papers 1507.02847, arXiv.org, revised Mar 2016.
    2. Zhiyuan Liu & M. Dashti Moghaddam & R. A. Serota, 2017. "Distributions of Historic Market Data - Stock Returns," Papers 1711.11003, arXiv.org, revised Dec 2017.
    3. Wang, Xiao-Tian & Li, Zhe & Zhuang, Le, 2017. "European option pricing under the Student’s t noise with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 469(C), pages 848-858.
    4. M. Dashti Moghaddam & Zhiyuan Liu & R. A. Serota, 2018. "Distributions of Historic Market Data -- Implied and Realized Volatility," Papers 1804.05279, arXiv.org.
    5. Nicolas Langrené & Geoffrey Lee & Zili Zhu, 2016. "Switching To Nonaffine Stochastic Volatility: A Closed-Form Expansion For The Inverse Gamma Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(05), pages 1-37, August.
    6. Zhiyuan Liu & R. A. Serota, 2017. "On absence of steady state in the Bouchaud-M\'ezard network model," Papers 1704.02377, arXiv.org.
    7. M. Dashti Moghaddam & Zhiyuan Liu & R. A. Serota, 2019. "Distributions of Historic Market Data -- Relaxation and Correlations," Papers 1907.05348, arXiv.org, revised Feb 2020.
    8. Jiong Liu & R. A. Serota, 2023. "Rethinking Generalized Beta family of distributions," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 96(2), pages 1-14, February.
    9. Nicolas Langrené & Geoffrey Lee & Zili Zhu, 2016. "Switching to nonaffine stochastic volatility: a closed-form expansion for the Inverse Gamma model," Post-Print hal-02909113, HAL.
    10. Andrés Felipe Galeano Zurbaran, 2018. "Distribuciones no normales para la selección de activos en el mercado Colombiano," Documentos de Trabajo 17208, Quantil.
    11. Dashti Moghaddam, M. & Serota, R.A., 2021. "Combined multiplicative–Heston model for stochastic volatility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 561(C).
    12. M. Dashti Moghaddam & Jiong Liu & R. A. Serota, 2021. "Implied and realized volatility: A study of distributions and the distribution of difference," International Journal of Finance & Economics, John Wiley & Sons, Ltd., vol. 26(2), pages 2581-2594, April.
    13. Liu, Z. & Serota, R.A., 2017. "Correlation and relaxation times for a stochastic process with a fat-tailed steady-state distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 474(C), pages 301-311.
    14. M. Dashti Moghaddam & R. A. Serota, 2018. "Combined Mutiplicative-Heston Model for Stochastic Volatility," Papers 1807.10793, arXiv.org.
    15. Jiong Liu & R. A. Serota, 2022. "Rethinking Generalized Beta Family of Distributions," Papers 2209.05225, arXiv.org.
    16. M. Dashti Moghaddam & Zhiyuan Liu & R. A. Serota, 2019. "Distribution of Historic Market Data ¨C Implied and Realized Volatility," Applied Economics and Finance, Redfame publishing, vol. 6(5), pages 104-130, September.
    17. Geoffrey Ducournau, 2021. "Bayesian inference and superstatistics to describe long memory processes of financial time series," Papers 2105.04171, arXiv.org.
    18. Dashti Moghaddam, M. & Mills, Jeffrey & Serota, R.A., 2020. "From a stochastic model of economic exchange to measures of inequality," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 559(C).
    19. Z. Liu & J. G. Holden & R. A. Serota, 2016. "Probability Density Of Response Times And Neurophysiology Of Cognition," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 19(04n05), pages 1-17, June.

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