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Volatility in Equilibrium: Asymmetries and Dynamic Dependencies

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  • Tim Bollerslev
  • Natalia Sizova
  • George Tauchen

Abstract

Stock market volatility clusters in time, appears fractionally integrated, carries a risk premium, and exhibits asymmetric leverage effects relative to returns. At the same time, the volatility risk premium, defined by the difference between the risk-neutral and objective expectations of the volatility, is distinctly less persistent and appears short-memory. This paper develops the first internally consistent equilibrium based explanation for all of these empirical facts. The model is cast in continuous-time and entirely self-contained, involving non-separable recursive preferences. Our empirical investigations are made possible through the use of newly available high-frequency intra-day data for the VIX volatility index, along with corresponding high-frequency data for the S&P 500 aggregate market portfolio. We show that the qualitative implications from the new theoretical model match remarkably well with the distinct shapes and patterns in the sample auto-correlations and dynamic cross-correlations in the returns and volatilities observed in the data.

Suggested Citation

  • Tim Bollerslev & Natalia Sizova & George Tauchen, 2010. "Volatility in Equilibrium: Asymmetries and Dynamic Dependencies," Working Papers 10-34, Duke University, Department of Economics.
    • Handle: RePEc:duk:dukeec:10-34
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    Cited by:

    1. Kanniainen, Juho & Piché, Robert, 2013. "Stock price dynamics and option valuations under volatility feedback effect," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(4), pages 722-740.
    2. Andersen, Torben G. & Varneskov, Rasmus T., 2022. "Testing for parameter instability and structural change in persistent predictive regressions," Journal of Econometrics, Elsevier, vol. 231(2), pages 361-386.
    3. Manabu Asai & Michael McAleer & Marcelo C. Medeiros, 2012. "Asymmetry and Long Memory in Volatility Modeling," Journal of Financial Econometrics, Oxford University Press, vol. 10(3), pages 495-512, June.
    4. Stanislav Khrapov, 2011. "Pricing Central Tendency in Volatility," Working Papers w0168, Center for Economic and Financial Research (CEFIR).
    5. Manabu Asai & Michael McAleer, 2017. "A fractionally integrated Wishart stochastic volatility model," Econometric Reviews, Taylor & Francis Journals, vol. 36(1-3), pages 42-59, March.
    6. Ilze Kalnina & Dacheng Xiu, 2017. "Nonparametric Estimation of the Leverage Effect: A Trade-Off Between Robustness and Efficiency," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(517), pages 384-396, January.
    7. Jovanović, Mario, 2011. "Does Monetary Policy Affect Stock Market Uncertainty? – Empirical Evidence from the United States," Ruhr Economic Papers 240, RWI - Leibniz-Institut für Wirtschaftsforschung, Ruhr-University Bochum, TU Dortmund University, University of Duisburg-Essen.
    8. Christian Schlag & Michael Semenischev & Julian Thimme, 2021. "Predictability and the Cross-Section of Expected Returns: A Challenge for Asset Pricing Models," Management Science, INFORMS, vol. 67(12), pages 7932-7950, December.
    9. Asai, Manabu & McAleer, Michael, 2015. "Forecasting co-volatilities via factor models with asymmetry and long memory in realized covariance," Journal of Econometrics, Elsevier, vol. 189(2), pages 251-262.
    10. Flavia Barsotti, 2012. "Optimal Capital Structure with Endogenous Default and Volatility Risk," Working Papers - Mathematical Economics 2012-02, Universita' degli Studi di Firenze, Dipartimento di Scienze per l'Economia e l'Impresa.
    11. Aït-Sahalia, Yacine & Fan, Jianqing & Li, Yingying, 2013. "The leverage effect puzzle: Disentangling sources of bias at high frequency," Journal of Financial Economics, Elsevier, vol. 109(1), pages 224-249.
    12. Mario Jovanovic, 2011. "Does Monetary Policy Affect Stock Market Uncertainty? – Empirical Evidence from the United States," Ruhr Economic Papers 0240, Rheinisch-Westfälisches Institut für Wirtschaftsforschung, Ruhr-Universität Bochum, Universität Dortmund, Universität Duisburg-Essen.
    13. Izhakian, Yehuda, 2020. "A theoretical foundation of ambiguity measurement," Journal of Economic Theory, Elsevier, vol. 187(C).
    14. Senarathne Chamil W. & Šoja Tijana, 2019. "Heteroskedasticity in Excess Bitcoin Return Data: Google Trend vs. Garch Effects," Financial Sciences. Nauki o Finansach, Sciendo, vol. 24(3), pages 35-45, September.
    15. Bollerslev, Tim & Marrone, James & Xu, Lai & Zhou, Hao, 2014. "Stock Return Predictability and Variance Risk Premia: Statistical Inference and International Evidence," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 49(3), pages 633-661, June.
    16. Geert Bekaert & Eric Engstrom, 2009. "Asset Return Dynamics under Bad Environment Good Environment Fundamentals," NBER Working Papers 15222, National Bureau of Economic Research, Inc.
    17. Guglielmo Maria Caporale & Luis A. Gil-Alana & Miguel Martin-Valmayor, 2020. "Persistence in the Realized Betas: Some Evidence for the Spanish Stock Market," CESifo Working Paper Series 8171, CESifo.
    18. Tim Bollerslev & Daniela Osterrieder & Natalia Sizova & George Tauchen, 2011. "Risk and Return: Long-Run Relationships, Fractional Cointegration, and Return Predictability," CREATES Research Papers 2011-51, Department of Economics and Business Economics, Aarhus University.
    19. Schlag, Christian & Semenischev, Michael & Thimme, Julian, 2020. "Predictability and the cross-section of expected returns: A challenge for asset pricing models," SAFE Working Paper Series 289, Leibniz Institute for Financial Research SAFE.
    20. Yingying Xu & Donald Lien, 2022. "Forecasting volatilities of oil and gas assets: A comparison of GAS, GARCH, and EGARCH models," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 41(2), pages 259-278, March.
    21. Stelios Arvanitis & Tassos Magdalinos, 2018. "Mildly Explosive Autoregression Under Stationary Conditional Heteroskedasticity," Journal of Time Series Analysis, Wiley Blackwell, vol. 39(6), pages 892-908, November.
    22. Eraker, Bjørn & Wang, Jiakou, 2015. "A non-linear dynamic model of the variance risk premium," Journal of Econometrics, Elsevier, vol. 187(2), pages 547-556.
    23. repec:zbw:rwirep:0240 is not listed on IDEAS
    24. Jianjun Miao & Bin Wei & Hao Zhou, 2019. "Ambiguity Aversion and the Variance Premium," Quarterly Journal of Finance (QJF), World Scientific Publishing Co. Pte. Ltd., vol. 9(02), pages 1-36, June.

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    More about this item

    Keywords

    Equilibrium asset pricing; stochastic volatility; leverage effect; volatility feedback; option implied volatility; realized volatility; variance risk premium;
    All these keywords.

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • G14 - Financial Economics - - General Financial Markets - - - Information and Market Efficiency; Event Studies; Insider Trading

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