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A non-linear dynamic model of the variance risk premium

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  • Eraker, Bjørn
  • Wang, Jiakou

Abstract

We propose a new class of non-linear diffusion processes for modeling financial markets data. Our non-linear diffusions are obtained as transformations of affine processes. We show that asset-pricing and estimation is possible and likelihood estimation is straightforward. We estimate a non-linear diffusion model for the VIX index under both the objective measure and the risk-neutral measure where the latter is obtained from futures prices. We find evidence of significant non-linearity under both measures. We define the difference between the P and Q drift as a measure of the variance risk premium and show that it has strong predictive power for stock returns.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:econom:v:187:y:2015:i:2:p:547-556
    DOI: 10.1016/j.jeconom.2015.02.038
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    1. Yingzi Zhu & Jin E. Zhang, 2007. "Variance Term Structure And Vix Futures Pricing," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 10(01), pages 111-127.
    2. Tim Bollerslev & George Tauchen & Hao Zhou, 2009. "Expected Stock Returns and Variance Risk Premia," The Review of Financial Studies, Society for Financial Studies, vol. 22(11), pages 4463-4492, November.
    3. Bandi, Federico M., 2002. "Short-term interest rate dynamics: a spatial approach," Journal of Financial Economics, Elsevier, vol. 65(1), pages 73-110, July.
    4. Pan, Jun, 2002. "The jump-risk premia implicit in options: evidence from an integrated time-series study," Journal of Financial Economics, Elsevier, vol. 63(1), pages 3-50, January.
    5. Tim Bollerslev & Natalia Sizova & George Tauchen, 2011. "Volatility in Equilibrium: Asymmetries and Dynamic Dependencies," Review of Finance, European Finance Association, vol. 16(1), pages 31-80.
    6. Bollerslev, Tim & Gibson, Michael & Zhou, Hao, 2011. "Dynamic estimation of volatility risk premia and investor risk aversion from option-implied and realized volatilities," Journal of Econometrics, Elsevier, vol. 160(1), pages 235-245, January.
    7. Goyal, Amit & Saretto, Alessio, 2009. "Cross-section of option returns and volatility," Journal of Financial Economics, Elsevier, vol. 94(2), pages 310-326, November.
    8. Jin E. Zhang & Jinghong Shu & Menachem Brenner, 2010. "The new market for volatility trading," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 30(9), pages 809-833, September.
    9. Hui Chen & Scott Joslin, 2012. "Generalized Transform Analysis of Affine Processes and Applications in Finance," The Review of Financial Studies, Society for Financial Studies, vol. 25(7), pages 2225-2256.
    10. repec:bla:jfinan:v:59:y:2004:i:4:p:1481-1509 is not listed on IDEAS
    11. repec:bla:jfinan:v:59:y:2004:i:3:p:1367-1404 is not listed on IDEAS
    12. 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.
    13. Bandi, Federico M. & Renò, Roberto, 2012. "Time-varying leverage effects," Journal of Econometrics, Elsevier, vol. 169(1), pages 94-113.
    14. Jin E. Zhang & Yuqin Huang, 2010. "The CBOE S&P 500 three‐month variance futures," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 30(1), pages 48-70, January.
    15. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    16. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    17. Xavier Gabaix, 2012. "Variable Rare Disasters: An Exactly Solved Framework for Ten Puzzles in Macro-Finance," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 127(2), pages 645-700.
    18. Stanton, Richard, 1997. "A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk," Journal of Finance, American Finance Association, vol. 52(5), pages 1973-2002, December.
    19. Yueh‐Neng Lin, 2007. "Pricing VIX futures: Evidence from integrated physical and risk‐neutral probability measures," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 27(12), pages 1175-1217, December.
    20. Ruijun Bu & Ludovic Giet & Kaddour Hadri & Michel Lubrano, 2011. "Modeling Multivariate Interest Rates Using Time-Varying Copulas and Reducible Nonlinear Stochastic Differential Equations," Journal of Financial Econometrics, Oxford University Press, vol. 9(1), pages 198-236, Winter.
    21. Joost Driessen & Pascal J. Maenhout & Grigory Vilkov, 2009. "The Price of Correlation Risk: Evidence from Equity Options," Journal of Finance, American Finance Association, vol. 64(3), pages 1377-1406, June.
    22. Jin E. Zhang & Yingzi Zhu, 2006. "VIX futures," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 26(6), pages 521-531, June.
    23. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    24. Joshua D. Coval & Tyler Shumway, 2001. "Expected Option Returns," Journal of Finance, American Finance Association, vol. 56(3), pages 983-1009, June.
    25. Xiu, Dacheng, 2014. "Hermite polynomial based expansion of European option prices," Journal of Econometrics, Elsevier, vol. 179(2), pages 158-177.
    26. Ahn, Dong-Hyun & Gao, Bin, 1999. "A Parametric Nonlinear Model of Term Structure Dynamics," The Review of Financial Studies, Society for Financial Studies, vol. 12(4), pages 721-762.
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    Cited by:

    1. Bu, Ruijun & Jawadi, Fredj & Li, Yuyi, 2017. "An empirical comparison of transformed diffusion models for VIX and VIX futures," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 46(C), pages 116-127.
    2. Ruijun Bu & Fredj Jawadi & Yuyi Li, 2020. "A multifactor transformed diffusion model with applications to VIX and VIX futures," Econometric Reviews, Taylor & Francis Journals, vol. 39(1), pages 27-53, January.
    3. Bu, Ruijun & Hadri, Kaddour & Kristensen, Dennis, 2021. "Diffusion copulas: Identification and estimation," Journal of Econometrics, Elsevier, vol. 221(2), pages 616-643.
    4. Chang, Chia-Lin & McAleer, Michael, 2015. "Econometric analysis of financial derivatives: An overview," Journal of Econometrics, Elsevier, vol. 187(2), pages 403-407.
    5. Doshi, Hitesh & Ericsson, Jan & Fournier, Mathieu & Seo, Sang Byung, 2024. "The risk and return of equity and credit index options," Journal of Financial Economics, Elsevier, vol. 161(C).
    6. Bollerslev, Tim & Todorov, Viktor & Xu, Lai, 2015. "Tail risk premia and return predictability," Journal of Financial Economics, Elsevier, vol. 118(1), pages 113-134.
    7. José Mauricio Gil-León & Andrés Felipe Suárez-Cant, 2020. "Implications of risk premium shocks in a small and open economy," Lecturas de Economía, Universidad de Antioquia, Departamento de Economía, issue 92, pages 133-172, Enero-Jun.
    8. Zhigang Tong, 2017. "Modelling VIX and VIX derivatives with reducible diffusions," International Journal of Bonds and Derivatives, Inderscience Enterprises Ltd, vol. 3(2), pages 153-175.
    9. Chang, C-L. & McAleer, M.J., 2014. "Econometric Analysis of Financial Derivatives," Econometric Institute Research Papers EI 2015-02, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    10. McCausland, William & Miller, Shirley & Pelletier, Denis, 2021. "Multivariate stochastic volatility using the HESSIAN method," Econometrics and Statistics, Elsevier, vol. 17(C), pages 76-94.

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    More about this item

    Keywords

    VIX; Semi-nonparametric diffusion; VIX futures; GMM; MLE; Non-linear asset pricing;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

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