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Pricing generalized variance swaps under the Heston model with stochastic interest rates

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  • Kim, See-Woo
  • Kim, Jeong-Hoon

Abstract

Unlike vanilla variance swaps, generalized variance swaps such as gamma, corridor variance and conditional variance swaps are expected to be not free from interest rates because of their weight processes. To examine the impact of stochastic interest rates on the generalized variance swaps, this paper considers discrete sampling times and the Heston stochastic volatility model incorporated by stochastic interest rates driven by the Cox–Ingersoll–Ross process. Based on the explicit calculation of the discounted characteristic function of Duffie et al. (2000), we obtain exact solutions for the fair strike prices of the generalized variance swaps for an affine version of the hybrid model. The solutions are given in closed form expression for the vanilla variance and gamma swaps and in Fourier integral expression for the corridor and conditional variance swaps. We apply the projection techniques of Grzelak and Oosterlee (2011) to the original non-affine model with a generalized correlation structure and obtain affine approximate solutions. We show the effects of stochastic interest rates on the strike prices of the generalized variance swaps.

Suggested Citation

  • Kim, See-Woo & Kim, Jeong-Hoon, 2020. "Pricing generalized variance swaps under the Heston model with stochastic interest rates," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 168(C), pages 1-27.
    • Handle: RePEc:eee:matcom:v:168:y:2020:i:c:p:1-27
    DOI: 10.1016/j.matcom.2019.07.013
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    References listed on IDEAS

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    1. Peter Christoffersen & Steven Heston & Kris Jacobs, 2009. "The Shape and Term Structure of the Index Option Smirk: Why Multifactor Stochastic Volatility Models Work So Well," Management Science, INFORMS, vol. 55(12), pages 1914-1932, December.
    2. Kung, James J. & Wu, E-Ching, 2013. "An evaluation of some popular investment strategies under stochastic interest rates," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 94(C), pages 96-108.
    3. Rehez Ahlip & Marek Rutkowski, 2015. "Semi-analytical Pricing of Currency Options in the Heston/CIR Jump-Diffusion Hybrid Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(1), pages 1-27, March.
    4. 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..
    5. 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.
    6. 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.
    7. Cao, Jiling & Lian, Guanghua & Roslan, Teh Raihana Nazirah, 2016. "Pricing variance swaps under stochastic volatility and stochastic interest rate," Applied Mathematics and Computation, Elsevier, vol. 277(C), pages 72-81.
    8. 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.
    9. Wendong Zheng & Yue Kuen Kwok, 2014. "Closed Form Pricing Formulas For Discretely Sampled Generalized Variance Swaps," Mathematical Finance, Wiley Blackwell, vol. 24(4), pages 855-881, October.
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    Cited by:

    1. Youngin Yoon & Jeong-Hoon Kim, 2023. "A Closed Form Solution for Pricing Variance Swaps Under the Rescaled Double Heston Model," Computational Economics, Springer;Society for Computational Economics, vol. 61(1), pages 429-450, January.

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