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Mathematics and Financial Economics

, Volume 12, Issue 4, pp 495–515 | Cite as

A scaled version of the double-mean-reverting model for VIX derivatives

  • Jeonggyu Huh
  • Jaegi Jeon
  • Jeong-Hoon Kim
Article
  • 163 Downloads

Abstract

As the Heston model is not consistent with VIX data in real market well enough, alternative stochastic volatility models including the double-mean-reverting model of Gatheral (in: Bachelier Congress, 2008) have been developed to overcome its limitation. The double-mean-reverting model is a three factor model successfully reflecting the empirical dynamics of the variance but there is no closed form solution for VIX derivatives and SPX options and thus calibration using conventional techniques may be slow. In this paper, we propose a fast mean-reverting version of the double-mean-reverting model. We obtain a closed form approximation for VIX derivatives and show how it is effective by comparing it with the Heston model and the double-mean-reverting model.

Keywords

VIX derivatives Heston’s volatility Double-mean-reverting volatility Calibration 

JEL Classification

G12 G13 C22 C52 

Notes

Acknowledgements

We thank the anonymous reviewers and the editor for their valuable comments and suggestions to improve the paper. The author J.-H. Kim gratefully acknowledges the financial support of the National Research Foundation of Korea NRF-2017R1A2B4003226.

References

  1. 1.
    Adrian, T., Rosenberg, J.: Stock returns and volatility: pricing the short-run and long-run components of market risk. J. Finance 63(6), 2997–3030 (2008).  https://doi.org/10.1111/j.1540-6261.2008.01419.x CrossRefGoogle Scholar
  2. 2.
    Andersen, T.G., Bollerslev, T., Diebold, F.X., Ebens, H.: The distribution of realized stock return volatility. J. Financ. Econ. 61(1), 43–76 (2001).  https://doi.org/10.1016/S0304-405X(01)00055-1 CrossRefGoogle Scholar
  3. 3.
    Bates, D.S.: Maximum likelihood estimation of latent affine processes. Rev. Financ. Stud. 19(3), 909–965 (2006).  https://doi.org/10.1093/rfs/hhj022 MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bayer, C., Gatheral, J., Karlsmark, M.: Fast Ninomiya–Victoir calibration of the double-mean-reverting model. Quant. Finance 13(11), 1813–1829 (2013).  https://doi.org/10.1080/14697688.2013.818245 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brenner, M., Galai, D.: New financial instruments for hedge changes in volatility. Financ. Anal. J. 45(4), 61–65 (1989).  https://doi.org/10.2469/faj.v45.n4.61 CrossRefGoogle Scholar
  6. 6.
    Brigo, D., Mercurio, F.: Interest Rate Models-Theory and Practice: With Smile, Inflation and Credit. Springer, Heidelberg (2007)zbMATHGoogle Scholar
  7. 7.
    Brockhaus, O., Long, D.: Volatility swaps made simple. Risk 13(1), 92–95 (2000)Google Scholar
  8. 8.
    Carr, P., Madan, D.: Towards a theory of volatility trading. Volatility New Estim. Tech. Pricing Deriv. 29, 417–427 (1998)zbMATHGoogle Scholar
  9. 9.
    Chernov, M., Gallant, A.R., Ghysels, E., Tauchen, G.: Alternative models for stock price dynamics. J. Econom. 116(1), 225–257 (2003).  https://doi.org/10.1016/S0304-4076(03)00108-8 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Christoffersen, P., Heston, S., Jacobs, K.: The shape and term structure of the index option smirk: why multifactor stochastic volatility models work so well. Manag. Sci. 55(12), 1914–1932 (2009).  https://doi.org/10.1287/mnsc.1090.1065 CrossRefzbMATHGoogle Scholar
  11. 11.
    Folland, G.B.: Real Analysis: Modern Techniques and Their Applications. Wiley, New York (2013)zbMATHGoogle Scholar
  12. 12.
    Fouque, J.P., Papanicolaou, G., Sircar, R., Sølna, K.: Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
  13. 13.
    Fouque, J.P., Saporito, Y.F., Zubelli, J.P.: Multiscale stochastic volatility model for derivatives on futures. Int. J. Theor. Appl. Finance 17(07), 1450, 043 (2014).  https://doi.org/10.1142/S0219024914500435 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fouque, J.P., Lorig, M., Sircar, R.: Second order multiscale stochastic volatility asymptotics: stochastic terminal layer analysis and calibration. Finance Stochast. 20(3), 543–588 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gallant, A.R., Hsu, C.T., Tauchen, G.: Using daily range data to calibrate volatility diffusions and extract the forward integrated variance. Rev. Econ. Stat. 81(4), 617–631 (1999).  https://doi.org/10.1162/003465399558481 CrossRefGoogle Scholar
  16. 16.
    Gatheral, J.: Consistent modeling of SPX and VIX options. In: Bachelier Congress, p. 3 (2008)Google Scholar
  17. 17.
    Gauthier, P., Possamaï, D.: Efficient simulation of the double Heston model. J. Comput. Math. 4(3), 23–73 (2011).  https://doi.org/10.2139/ssrn.1434853 Google Scholar
  18. 18.
    Huang Jz, WuL: Specification analysis of option pricing models based on time-changed Levy processes. J. Finance 59(3), 1405–1439 (2004).  https://doi.org/10.1111/j.1540-6261.2004.00667.x CrossRefGoogle Scholar
  19. 19.
    Lu, Z., Zhu, Y.: Volatility components: the term structure dynamics of vix futures. J. Futures Mark. 30(3), 230–256 (2010).  https://doi.org/10.1002/fut.20415 Google Scholar
  20. 20.
    Oksendal, B.: Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin (2013)zbMATHGoogle Scholar
  21. 21.
    Papanicolaou, G., Fouque, J.P., Solna, K., Sircar, R.: Singular perturbations in option pricing. SIAM J. Appl. Math. 63(5), 1648–1665 (2003).  https://doi.org/10.1137/S0036139902401550 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rouah, F.D.: The Heston Model and Its Extensions in MATLAB and C#. Wiley, Hoboken (2013)CrossRefzbMATHGoogle Scholar
  23. 23.
    Whaley, R.E.: Derivatives on market volatility: hedging tools long overdue. J. Deriv. 1(1), 71–84 (1993).  https://doi.org/10.3905/jod.1993.407868 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsYonsei UniversitySeoulRepublic of Korea

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