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Closed‐Form Formulae for Variance and Volatility Swaps Under Stochastic Volatility With Stochastic Liquidity Risks

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  • Sha Lin
  • Xin‐Jiang He

Abstract

We construct a stochastic volatility model considering stochastic liquidity risks when valuing variance and volatility swaps with discrete sampling. We base our model on Heston stochastic volatility, which is adopted for the modeling of stock prices when the market is perfectly liquid. Stock dynamics are further revised by discounting their prices through the employment of mean reverting market liquidity. We convert the stock dynamics under the physical measure into the one under a risk‐neutral measure via measure transform, with which the analytical valuation of variance and volatility swaps is realized. By taking the limit of sampling frequency, we further consider how both swaps with continuous sampling can be priced. Numerical implementation is finally carried out, with which the capability of the constructed model in capturing the influence of the two common types of financial risks can be clear.

Suggested Citation

  • 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.
    • Handle: RePEc:wly:jfutmk:v:44:y:2024:i:8:p:1447-1461
    DOI: 10.1002/fut.22531
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    References listed on IDEAS

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    1. Francis A. Longstaff & Sanjay Mithal & Eric Neis, 2005. "Corporate Yield Spreads: Default Risk or Liquidity? New Evidence from the Credit Default Swap Market," Journal of Finance, American Finance Association, vol. 60(5), pages 2213-2253, October.
    2. Bakshi, Gurdip & Cao, Charles & Chen, Zhiwu, 1997. "Empirical Performance of Alternative Option Pricing Models," Journal of Finance, American Finance Association, vol. 52(5), pages 2003-2049, December.
    3. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," The Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    4. 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.
    5. Xin‐Jiang He & Sha Lin, 2023. "Analytically pricing exchange options with stochastic liquidity and regime switching," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 43(5), pages 662-676, May.
    6. 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.
    7. Puneet Pasricha & Song-Ping Zhu & Xin-Jiang He, 2022. "A closed-form pricing formula for European options in an illiquid asset market," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 8(1), pages 1-18, December.
    8. 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.
    9. Li, Zhe & Zhang, Wei-Guo & Liu, Yong-Jun, 2018. "European quanto option pricing in presence of liquidity risk," The North American Journal of Economics and Finance, Elsevier, vol. 45(C), pages 230-244.
    10. 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.
    11. Pun, Chi Seng & Chung, Shing Fung & Wong, Hoi Ying, 2015. "Variance swap with mean reversion, multifactor stochastic volatility and jumps," European Journal of Operational Research, Elsevier, vol. 245(2), pages 571-580.
    12. Anh Le, 2015. "Separating the Components of Default Risk: A Derivative-Based Approach," Quarterly Journal of Finance (QJF), World Scientific Publishing Co. Pte. Ltd., vol. 5(01), pages 1-48.
    13. Weiyi Liu & Song‐Ping Zhu, 2019. "Pricing variance swaps under the Hawkes jump‐diffusion process," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 39(6), pages 635-655, June.
    14. Yongmin Zhang & Shusheng Ding & Meryem Duygun, 2019. "Derivatives pricing with liquidity risk," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 39(11), pages 1471-1485, November.
    15. Kian Guan Lim & Da Zhi, 2002. "Pricing options using implied trees: Evidence from FTSE‐100 options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 22(7), pages 601-626, July.
    16. Liu, Hong & Yong, Jiongmin, 2005. "Option pricing with an illiquid underlying asset market," Journal of Economic Dynamics and Control, Elsevier, vol. 29(12), pages 2125-2156, December.
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