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Prices and Asymptotics for Discrete Variance Swaps

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  • Carole Bernard
  • Zhenyu Cui

Abstract

We study the fair strike of a discrete variance swap for a general time-homogeneous stochastic volatility model. In the special cases of Heston, Hull--White and Sch�bel--Zhu stochastic volatility models, we give simple explicit expressions (improving Broadie and Jain (2008a). The effect of jumps and discrete sampling on volatility and variance swaps. International Journal of Theoretical and Applied Finance, 11 (8), 761--797) in the case of the Heston model). We give conditions on parameters under which the fair strike of a discrete variance swap is higher or lower than that of the continuous variance swap. The interest rate and the correlation between the underlying price and its volatility are key elements in this analysis. We derive asymptotics for the discrete variance swaps and compare our results with those of Broadie and Jain (2008a. The effect of jumps and discrete sampling on volatility and variance swaps. International Journal of Theoretical and Applied Finance, 11 (8), 761--797), Jarrow et al. (2013. Discretely sampled variance and volatility swaps versus their continuous approximations. Finance and Stochastics, 17 (2), 305--324) and Keller-Ressel and Griessler (2012. Convex order of discrete, continuous and predictable quadratic variation and applications to options on variance . Working paper. Retrieved from http://arxiv.org/abs/1103.2310.

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  • 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.
    • Handle: RePEc:taf:apmtfi:v:21:y:2014:i:2:p:140-173
    DOI: 10.1080/1350486X.2013.820524
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    Citations

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    Cited by:

    1. Kim, Seong-Tae & Kim, Jeong-Hoon, 2020. "Stochastic elasticity of vol-of-vol and pricing of variance swaps," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 177(C), pages 420-440.
    2. Tong, Zhigang & Liu, Allen, 2022. "Pricing variance swaps under subordinated Jacobi stochastic volatility models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 593(C).
    3. Carol Alexander & Johannes Rauch, 2017. "The Aggregation Property and its Applications to Realised Higher Moments," Papers 1709.08188, arXiv.org.
    4. Wu, Bin & Chen, Pengzhan & Ye, Wuyi, 2024. "Variance swaps with mean reversion and multi-factor variance," European Journal of Operational Research, Elsevier, vol. 315(1), pages 191-212.
    5. Jin, Xing & Hong, Yi, 2023. "Jump-diffusion volatility models for variance swaps: An empirical performance analysis," International Review of Financial Analysis, Elsevier, vol. 87(C).
    6. Yang, Ben-Zhang & Yue, Jia & Wang, Ming-Hui & Huang, Nan-Jing, 2019. "Volatility swaps valuation under stochastic volatility with jumps and stochastic intensity," Applied Mathematics and Computation, Elsevier, vol. 355(C), pages 73-84.
    7. Chi Hung Yuen & Wendong Zheng & Yue Kuen Kwok, 2015. "Pricing Exotic Discrete Variance Swaps under the 3/2-Stochastic Volatility Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(5), pages 421-449, November.
    8. Kim, See-Woo & Kim, Jeong-Hoon, 2018. "Analytic solutions for variance swaps with double-mean-reverting volatility," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 130-144.
    9. Carol Alexander & Johannes Rauch, 2014. "Model-Free Discretisation-Invariant Swaps and S&P 500 Higher-Moment Risk Premia," Papers 1404.1351, arXiv.org, revised Feb 2016.
    10. Jiling Cao & Teh Raihana Nazirah Roslan & Wenjun Zhang, 2018. "Pricing Variance Swaps in a Hybrid Model of Stochastic Volatility and Interest Rate with Regime-Switching," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1359-1379, December.
    11. Carol Alexander & Johannes Rauch, 2016. "Model-Free Discretisation-Invariant Swap Contracts," Papers 1602.00235, arXiv.org, revised Apr 2016.
    12. 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.
    13. Yuecai Han & Xudong Zheng, 2022. "Approximate Pricing of Derivatives Under Fractional Stochastic Volatility Model," Papers 2210.15453, arXiv.org.
    14. Cui, Zhenyu & Kirkby, J. Lars & Nguyen, Duy, 2021. "Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations," European Journal of Operational Research, Elsevier, vol. 290(3), pages 1046-1062.
    15. 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.
    16. Teh Raihana Nazirah Roslan & Wenjun Zhang & Jiling Cao, 2016. "Pricing variance swaps with stochastic volatility and stochastic interest rate under full correlation structure," Papers 1610.09714, arXiv.org, revised Apr 2020.
    17. Elisa Alòs & Masaaki Fukasawa, 2021. "The asymptotic expansion of the regular discretization error of Itô integrals," Mathematical Finance, Wiley Blackwell, vol. 31(1), pages 323-365, January.
    18. Cui, Zhenyu & Lars Kirkby, J. & Nguyen, Duy, 2019. "A general framework for time-changed Markov processes and applications," European Journal of Operational Research, Elsevier, vol. 273(2), pages 785-800.
    19. Coqueret, Guillaume & Tavin, Bertrand, 2016. "An investigation of model risk in a market with jumps and stochastic volatility," European Journal of Operational Research, Elsevier, vol. 253(3), pages 648-658.
    20. Alexander, Carol & Rauch, Johannes, 2021. "A general property for time aggregation," European Journal of Operational Research, Elsevier, vol. 291(2), pages 536-548.
    21. 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.
    22. Kim, See-Woo & Kim, Jeong-Hoon, 2019. "Variance swaps with double exponential Ornstein-Uhlenbeck stochastic volatility," The North American Journal of Economics and Finance, Elsevier, vol. 48(C), pages 149-169.
    23. 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.
    24. Jaehyuk Choi & Yue Kuen Kwok, 2023. "Simulation schemes for the Heston model with Poisson conditioning," Papers 2301.02800, arXiv.org, revised Nov 2023.

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