Abstract
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.
Similar content being viewed by others
References
Heston SL (1993) A closed form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6(2):327–343
Swishchuk A (2004) Modeling of variance and volatility swaps for financial markets with stochastic volatilities. Wilmott magazine, September issue, technical article 64–72
Broadie M, Jain A (2008) The effect of jumps and discrete sampling on volatility and variance swaps. Int J Theor Appl Financ 11:761–797
Sepp A (2008) Pricing options on realized variance in the Heston model with jumps in returns and volatility. J Comput Financ 11:33–70
Zhu S, Lian G (2011) A closed-form exact solution for pricing variance swaps with stochastic volatility. Math Financ 21(2):233–256
Little T, Pant V (2001) A finite-difference method for the valuation of variance swaps. J Comput Financ 5(1):81–101
Zheng W, Kwok YK (2014) Closed form pricing formulas for discretely sampled generalized variance swaps. Math Financ 24:855–881
Christoffersen P, Heston H, Jacobs K (2009) The shape and term structure of the index option smirk: Why multifactor stochastic volatility models work so well. Manag Sci 55:1914–1932
Gatheral J (2008) Consistent modeling of SPX and VIX options. In: Bachelier Congress, pp 3
Huh J, Jeon J, Kim J-H (2018) A scaled version of the double-mean-reverting model for VIX derivatives. Math Finan Econ 12(4):495–515
Kim S-W, Kim J-H (2018) Analytic solutions for variance swaps with double-mean-reverting volatility. Chaos Solitons Fractal 114:130–144
Huang D, Schlag C, Shaliastovich I, Thimme J (2018) Volatility-of-volatility risk. J Financ Quant Anal 05:1–63
Fouque J-P, Saporito Y (2018) Heston stochastic vol-of-vol model for joint calibration of VIX and S&P500 options. Quant Financ 18(6):1003–1016
Fouque J-P, Papanicolaou G, Sircar R (2000) Derivatives in financial markets with stochastic volatility. Cambridge University Press
Fouque J-P, Papanicolaou G, Sircar R, Solna K (2011) Multiscale stochastic volatility for equity, interest rate, and credit derivatives. Cambridge University Press
Oksendal B (2000) Stochastic differential equations: an introduction with applications. Springer, Berlin
Luo X, Zhang JE (2012) The term structure of VIX. J Futur Mark 32(12):1092–1123
Levenberg K (1944) A method for the solution of certain non-linear problems in least squares. Quart Appl Math 2(2):164–168
Ackerer D, Filipovic D, Pulido S (2018) The Jacobi stochastic volatility model. Finance Stoch 22(3):667–700
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: VIX term structure
Appendix: VIX term structure
Rights and permissions
About this article
Cite this article
Lee, MK., Kim, SW. & Kim, JH. Variance Swaps Under Multiscale Stochastic Volatility of Volatility. Methodol Comput Appl Probab 24, 39–64 (2022). https://doi.org/10.1007/s11009-020-09834-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-020-09834-6