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Volatility swaps and volatility options on discretely sampled realized variance

Author

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  • Lian, Guanghua
  • Chiarella, Carl
  • Kalev, Petko S.

Abstract

Volatility swaps and volatility options are financial products written on discretely sampled realized variance. Actively traded in over-the-counter markets, these products are often priced by continuously sampled approximations to simplify the computations. This paper presents an analytical approach to efficiently and accurately price discretely sampled volatility derivatives, under a general stochastic volatility model. We first obtain an accurate approximation for the characteristic function of the discretely sampled realized variance. This characteristic function is then applied to price discrete volatility derivatives through either semi-analytical pricing formulae (up to an inverse Fourier transform) or an efficient Fourier-cosine series method. Numerical experiments show that our approximation is more accurate in comparison to the approximations in the literature. We remark that although discretely sampled variance swaps and options are usually more expensive than their continuously sampled counterparts, discretely sampled volatility swaps are more prone to be cheaper than the continuously sampled counterparts. An analysis is then provided to explain why this is the case in general for realistic contract specifications and reasonable model parameters.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:dyncon:v:47:y:2014:i:c:p:239-262
    DOI: 10.1016/j.jedc.2014.08.014
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    References listed on IDEAS

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    1. Rama Cont & Thomas Kokholm, 2013. "A Consistent Pricing Model For Index Options And Volatility Derivatives," Post-Print hal-00801536, HAL.
    2. 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.
    3. Fang, Fang & Oosterlee, Kees, 2008. "A Novel Pricing Method For European Options Based On Fourier-Cosine Series Expansions," MPRA Paper 9319, University Library of Munich, Germany.
    4. 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.
    5. Mencía, Javier & Sentana, Enrique, 2013. "Valuation of VIX derivatives," Journal of Financial Economics, Elsevier, vol. 108(2), pages 367-391.
    6. Robert Elliott & Tak Kuen Siu & Leunglung Chan, 2007. "Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(1), pages 41-62.
    7. Peter Carr & Jian Sun, 2007. "A new approach for option pricing under stochastic volatility," Review of Derivatives Research, Springer, vol. 10(2), pages 87-150, May.
    8. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    9. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    10. Peter Carr & Roger Lee & Liuren Wu, 2012. "Variance swaps on time-changed Lévy processes," Finance and Stochastics, Springer, vol. 16(2), pages 335-355, April.
    11. Martin Keller-Ressel & Johannes Muhle-Karbe, 2013. "Asymptotic and exact pricing of options on variance," Finance and Stochastics, Springer, vol. 17(1), pages 107-133, January.
    12. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    13. Andrey Itkin & Peter Carr, 2010. "Pricing swaps and options on quadratic variation under stochastic time change models—discrete observations case," Review of Derivatives Research, Springer, vol. 13(2), pages 141-176, July.
    14. Wendong Zheng & Yue Kuen Kwok, 2014. "Saddlepoint Approximation Methods for Pricing Derivatives on Discrete Realized Variance," Applied Mathematical Finance, Taylor & Francis Journals, vol. 21(1), pages 1-31, March.
    15. Peter Carr & Roger Lee, 2010. "Hedging variance options on continuous semimartingales," Finance and Stochastics, Springer, vol. 14(2), pages 179-207, April.
    16. Sam Howison & Avraam Rafailidis & Henrik Rasmussen, 2004. "On the pricing and hedging of volatility derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 11(4), pages 317-346.
    17. Alireza Javaheri & Paul Wilmott & Espen Haug, 2004. "GARCH and Volatility swaps," Quantitative Finance, Taylor & Francis Journals, vol. 4(5), pages 589-595.
    18. Peter Carr & Roger Lee, 2009. "Volatility Derivatives," Annual Review of Financial Economics, Annual Reviews, vol. 1(1), pages 319-339, November.
    19. Guang-Hua Lian & Song-Ping Zhu, 2013. "Pricing VIX options with stochastic volatility and random jumps," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 36(1), pages 71-88, May.
    20. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    21. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    22. Peter Carr & Hélyette Geman & Dilip Madan & Marc Yor, 2005. "Pricing options on realized variance," Finance and Stochastics, Springer, vol. 9(4), pages 453-475, October.
    23. Marjon Ruijter & Kees Oosterlee, 2012. "Two-dimensional Fourier cosine series expansion method for pricing financial options," CPB Discussion Paper 225, CPB Netherlands Bureau for Economic Policy Analysis.
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    Cited by:

    1. Dupret, Jean-Loup & Hainaut, Donatien, 2023. "A fractional Hawkes process for illiquidity modeling," LIDAM Discussion Papers ISBA 2023001, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Wendong Zheng & Chi Hung Yuen & Yue Kuen Kwok, 2016. "Recursive Algorithms For Pricing Discrete Variance Options And Volatility Swaps Under Time-Changed Lévy Processes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(02), pages 1-29, March.
    3. Wang, Xingchun & Fu, Jianping & Wang, Guanying & Wang, Yongjin, 2015. "Quadratic hedging strategies for volatility swaps," Finance Research Letters, Elsevier, vol. 15(C), pages 125-132.
    4. 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.
    5. Cui, Zhenyu & Lars Kirkby, J. & Nguyen, Duy, 2017. "A general framework for discretely sampled realized variance derivatives in stochastic volatility models with jumps," European Journal of Operational Research, Elsevier, vol. 262(1), pages 381-400.
    6. Li, Shaoyu & Huang, Henry H. & Zhang, Teng, 2020. "Generalized affine transform on pricing quanto range accrual note," The North American Journal of Economics and Finance, Elsevier, vol. 54(C).
    7. Li, Shaoyu & Zhang, Yuanyuan & Zhu, Chunhui, 2021. "A closed-form exact solution for pricing fixed-income variance swaps with affine-jump model," The North American Journal of Economics and Finance, Elsevier, vol. 58(C).
    8. Anqi Zou & Jiajie Wang & Chiye Wu, 2023. "Pricing Variance Swaps under MRG Model with Regime-Switching: Discrete Observations Case," Mathematics, MDPI, vol. 11(12), pages 1-30, June.

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    More about this item

    Keywords

    Realized variance; Variance swaps; Volatility swaps; Variance options; Stochastic volatility; Fourier-cosine series;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • C3 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables

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