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Closed-form variance swap prices under general affine GARCH models and their continuous-time limits

Author

Listed:
  • Alexandru Badescu

    (University of Calgary)

  • Zhenyu Cui

    (Stevens Institute of Technology)

  • Juan-Pablo Ortega

    (Universität Sankt Gallen
    Université de Franche-Comté)

Abstract

Fully explicit closed-form expressions are developed for the fair strike prices of discrete-time variance swaps under general affine GARCH type models that have been risk-neutralized with a family of variance dependent pricing kernels. The methodology relies on solving differential recursions for the coefficients of the joint cumulant generating function of the log price and the conditional variance processes. An alternative derivation is provided in the case of Gaussian innovations. Using standard assumptions on the asymptotic behavior of the GARCH parameters as the sampling frequency increases, the diffusion limit of a Gaussian GARCH model is derived and the convergence of the variance swap prices to its continuous-time limit is further investigated. Numerical examples on the term structure of the variance swap rates and on the convergence results are also presented.

Suggested Citation

  • Alexandru Badescu & Zhenyu Cui & Juan-Pablo Ortega, 2019. "Closed-form variance swap prices under general affine GARCH models and their continuous-time limits," Annals of Operations Research, Springer, vol. 282(1), pages 27-57, November.
    • Handle: RePEc:spr:annopr:v:282:y:2019:i:1:d:10.1007_s10479-018-2941-9
    DOI: 10.1007/s10479-018-2941-9
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    References listed on IDEAS

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    1. Nelson, Daniel B., 1990. "ARCH models as diffusion approximations," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 7-38.
    2. Lian, Guanghua & Chiarella, Carl & Kalev, Petko S., 2014. "Volatility swaps and volatility options on discretely sampled realized variance," Journal of Economic Dynamics and Control, Elsevier, vol. 47(C), pages 239-262.
    3. Mencía, Javier & Sentana, Enrique, 2013. "Valuation of VIX derivatives," Journal of Financial Economics, Elsevier, vol. 108(2), pages 367-391.
    4. Anatoliy Swishchuk, 2013. "Modeling and Pricing of Swaps for Financial and Energy Markets with Stochastic Volatilities," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 8660, December.
    5. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold & Paul Labys, 2003. "Modeling and Forecasting Realized Volatility," Econometrica, Econometric Society, vol. 71(2), pages 579-625, March.
    6. Carole Bernard & Zhenyu Cui, 2014. "Prices and Asymptotics for Discrete Variance Swaps," Applied Mathematical Finance, Taylor & Francis Journals, vol. 21(2), pages 140-173, April.
    7. Peter Christoffersen & Steven Heston & Kris Jacobs, 2013. "Capturing Option Anomalies with a Variance-Dependent Pricing Kernel," The Review of Financial Studies, Society for Financial Studies, vol. 26(8), pages 1963-2006.
    8. Jinji Hao & Jin E. Zhang, 2013. "GARCH Option Pricing Models, the CBOE VIX, and Variance Risk Premium," Journal of Financial Econometrics, Oxford University Press, vol. 11(3), pages 556-580, June.
    9. Alireza Javaheri & Paul Wilmott & Espen Haug, 2004. "GARCH and Volatility swaps," Quantitative Finance, Taylor & Francis Journals, vol. 4(5), pages 589-595.
    10. Song, Zhaogang & Xiu, Dacheng, 2016. "A tale of two option markets: Pricing kernels and volatility risk," Journal of Econometrics, Elsevier, vol. 190(1), pages 176-196.
    11. Mark Broadie & Ashish Jain, 2008. "The Effect Of Jumps And Discrete Sampling On Volatility And Variance Swaps," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 11(08), pages 761-797.
    12. Fred Espen Benth & Martin Groth & Rodwell Kufakunesu, 2007. "Valuing Volatility and Variance Swaps for a Non-Gaussian Ornstein-Uhlenbeck Stochastic Volatility Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(4), pages 347-363.
    13. Duan, Jin-Chuan & Yeh, Chung-Ying, 2010. "Jump and volatility risk premiums implied by VIX," Journal of Economic Dynamics and Control, Elsevier, vol. 34(11), pages 2232-2244, November.
    14. Majewski, Adam A. & Bormetti, Giacomo & Corsi, Fulvio, 2015. "Smile from the past: A general option pricing framework with multiple volatility and leverage components," Journal of Econometrics, Elsevier, vol. 187(2), pages 521-531.
    15. Peter Carr & Roger Lee, 2009. "Volatility Derivatives," Annual Review of Financial Economics, Annual Reviews, vol. 1(1), pages 319-339, November.
    16. Christophe Chorro & Fanirisoa Rahantamialisoa H., 2016. "Option Valuation with IG_GARCH Model and an U-Shaped Pricing Kernel," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01400242, HAL.
    17. 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.
    18. Heston, Steven L & Nandi, Saikat, 2000. "A Closed-Form GARCH Option Valuation Model," The Review of Financial Studies, Society for Financial Studies, vol. 13(3), pages 585-625.
    19. Robert Jarrow & Younes Kchia & Martin Larsson & Philip Protter, 2013. "Discretely sampled variance and volatility swaps versus their continuous approximations," Finance and Stochastics, Springer, vol. 17(2), pages 305-324, April.
    20. Simon Lalancette & Jean†Guy Simonato, 2017. "The Role of the Conditional Skewness and Kurtosis in VIX Index Valuation," European Financial Management, European Financial Management Association, vol. 23(2), pages 325-354, March.
    21. Tianyi Wang & Yiwen Shen & Yueting Jiang & Zhuo Huang, 2017. "Pricing the CBOE VIX Futures with the Heston–Nandi GARCH Model," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 37(7), pages 641-659, July.
    22. Alexandru Badescu & Zhenyu Cui & Juan-Pablo Ortega, 2017. "Non-affine GARCH Option Pricing Models, Variance-Dependent Kernels, and Diffusion Limits," Journal of Financial Econometrics, Oxford University Press, vol. 15(4), pages 602-648.
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    Cited by:

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    2. Marcos Escobar-Anel & Ben Spies & Rudi Zagst, 2024. "Optimal consumption and investment in general affine GARCH models," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 46(3), pages 987-1026, September.
    3. Escobar-Anel, Marcos & Gollart, Maximilian & Zagst, Rudi, 2022. "Closed-form portfolio optimization under GARCH models," Operations Research Perspectives, Elsevier, vol. 9(C).
    4. Lazar, Emese & Qi, Shuyuan, 2022. "Model risk in the over-the-counter market," European Journal of Operational Research, Elsevier, vol. 298(2), pages 769-784.
    5. Escobar-Anel, Marcos & Rastegari, Javad & Stentoft, Lars, 2021. "Option pricing with conditional GARCH models," European Journal of Operational Research, Elsevier, vol. 289(1), pages 350-363.
    6. Escobar-Anel, Marcos & Rastegari, Javad & Stentoft, Lars, 2020. "Affine multivariate GARCH models," Journal of Banking & Finance, Elsevier, vol. 118(C).
    7. Hongkai Cao & Alexandru Badescu & Zhenyu Cui & Sarath Kumar Jayaraman, 2020. "Valuation of VIX and target volatility options with affine GARCH models," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 40(12), pages 1880-1917, December.
    8. Escobar-Anel, Marcos & Rastegari, Javad & Stentoft, Lars, 2023. "Covariance dependent kernels, a Q-affine GARCH for multi-asset option pricing," International Review of Financial Analysis, Elsevier, vol. 87(C).
    9. Sha Lin & Xin‐Jiang He, 2024. "Closed‐Form Formulae for Variance and Volatility Swaps Under Stochastic Volatility With Stochastic Liquidity Risks," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 44(8), pages 1447-1461, August.
    10. Xin-Jiang He & Sha Lin, 2024. "A probabilistic approach for the valuation of variance swaps under stochastic volatility with jump clustering and regime switching," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 10(1), pages 1-23, December.
    11. He, Xin-Jiang & Lin, Sha, 2023. "Analytically pricing variance and volatility swaps under a Markov-modulated model with liquidity risks," The North American Journal of Economics and Finance, Elsevier, vol. 67(C).
    12. Xu, De-xuan & Yang, Ben-zhang & Kang, Jian-hao & Huang, Nan-jing, 2021. "Variance and volatility swaps valuations with the stochastic liquidity risk," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 566(C).
    13. Maciej Augustyniak & Alexandru Badescu, 2021. "On the computation of hedging strategies in affine GARCH models," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 41(5), pages 710-735, May.

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