Variance Swaps Under Multiscale Stochastic Volatility of Volatility


Many hedge funds and retail investors demand volatility and variance derivatives in order to manage their exposure to volatility and volatility-of-volatility risk associated with their trading positions. The Heston model is a standard popular stochastic volatility model for pricing volatility and variance derivatives. However, it may fail to capture some important empirical features of the relevant market data due to the fact that the elasticity of volatility of volatility of the underlying price takes a special value, i.e., 1/2, whereas it has a merit of analytical tractability. We exploit a multiscale stochastic extension of volatility of volatility to obtain a better agreement with the empirical data while taking analytical advantage of the original Heston dynamics as much as possible in the context of pricing discrete variance swaps. By using an asymptotic technique with two small parameters, we derive a quasi-closed form formula for the fair strike price of variance swap and find useful pricing properties with respect to the stochastic extension parameters.

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We thank the anonymous reviewers whose comments and suggestions helped improve and clarify this manuscript. The research of M.-K. Lee was supported by National Research Foundation of Korea NRF-2016R1D1A3B03933060 and the research of J.-H. Kim was supported by the National Research Foundation of Korea NRF-2020R1H1A2006105.

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Correspondence to Jeong-Hoon Kim.

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Appendix: VIX term structure

Appendix: VIX term structure

Table 4 2020-01-02, 2020-04-03 VIX term structure data quoted from the CBOE website (

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Lee, MK., Kim, SW. & Kim, JH. Variance Swaps Under Multiscale Stochastic Volatility of Volatility. Methodol Comput Appl Probab (2020).

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  • Variance swap
  • Stochastic volatility
  • Stochastic volatility of volatility
  • Asymptotic expansion

Mathematics Subject Classification (2010)

  • 91G20
  • 60J60
  • 35Q91