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A Closed-Form Pricing Formula for Log-Return Variance Swaps under Stochastic Volatility and Stochastic Interest Rate

Author

Listed:
  • Chen Mao

    (College of Mathematics, Harbin Normal University, Harbin 150000, China)

  • Guanqi Liu

    (College of Mathematics, Harbin Normal University, Harbin 150000, China)

  • Yuwen Wang

    (College of Mathematics, Harbin Normal University, Harbin 150000, China)

Abstract

At present, the study concerning pricing variance swaps under CIR the (Cox–Ingersoll–Ross)–Heston hybrid model has achieved many results; however, due to the instantaneous interest rate and instantaneous volatility in the model following the Feller square root process, only a semi-closed solution can be obtained by solving PDEs. This paper presents a simplified approach to price log-return variance swaps under the CIR–Heston hybrid model. Compared with Cao’s work, an important feature of our approach is that there is no need to solve complex PDEs; a closed-form solution is obtained by applying the martingale theory and It o ^ ’s lemma. The closed-form solution is significant because it can achieve accurate pricing and no longer takes time to adjust parameters by numerical method. Another significant feature of this paper is that the impact of sampling frequency on pricing formula is analyzed; then the closed-form solution can be extended to an approximate formula. The price curves of the closed-form solution and the approximate solution are presented by numerical simulation. When the sampling frequency is large enough, the two curves almost coincide, which means that our approximate formula is simple and reliable.

Suggested Citation

  • Chen Mao & Guanqi Liu & Yuwen Wang, 2021. "A Closed-Form Pricing Formula for Log-Return Variance Swaps under Stochastic Volatility and Stochastic Interest Rate," Mathematics, MDPI, vol. 10(1), pages 1-17, December.
    • Handle: RePEc:gam:jmathe:v:10:y:2021:i:1:p:5-:d:707306
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    References listed on IDEAS

    as
    1. Aziz Issaka & Indranil SenGupta, 2017. "Analysis of variance based instruments for Ornstein–Uhlenbeck type models: swap and price index," Annals of Finance, Springer, vol. 13(4), pages 401-434, November.
    2. Steven Heston & Saikat Nandi, 2000. "Derivatives on volatility: some simple solutions based on observables," FRB Atlanta Working Paper 2000-20, Federal Reserve Bank of Atlanta.
    3. 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.
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    5. 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.
    6. Semere Habtemicael & Indranil SenGupta, 2016. "Pricing variance and volatility swaps for Barndorff-Nielsen and Shephard process driven financial markets," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(04), pages 1-35, December.
    7. Semere Habtemicael & Indranil Sengupta, 2016. "Pricing Covariance Swaps For Barndorff–Nielsen And Shephard Process Driven Financial Markets," Annals of Financial Economics (AFE), World Scientific Publishing Co. Pte. Ltd., vol. 11(03), pages 1-32, September.
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    Cited by:

    1. 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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