Review of Derivatives Research

, Volume 20, Issue 2, pp 167–202 | Cite as

Implied volatility and skewness surface

  • Bruno Feunou
  • Jean-Sébastien Fontaine
  • Roméo Tédongap


The Homoscedastic Gamma (HG) model characterizes the distribution of returns by its mean, variance and an independent skewness parameter. The HG model preserves the parsimony and the closed form of the Black–Scholes–Merton (BSM) while introducing the implied volatility (IV) and skewness surface. Varying the skewness parameter of the HG model can restore the symmetry of IV curves. Practitioner’s variants of the HG model improve pricing (in-sample and out-of-sample) and hedging performances relative to practitioners’ BSM models, with as many or less parameters. The pattern of improvements in Delta-Hedged gains across strike prices accord with predictions from the HG model. These results imply that expanding around the Gaussian density does not offer sufficient flexibility to match the skewness implicit in options. Consistent with the model, we also find that conditioning on implied skewness increases the predictive power of the volatility spread for excess returns.


SP500 options Implied skewness Implied volatility Volatility spread Delta-hedged gains 

JEL Classification

G12 G13 

Supplementary material

11147_2016_9127_MOESM1_ESM.pdf (102 kb)
Supplementary material 1 (pdf 101 KB)


  1. Alexander, C., & Nogueira, L. M. (2005). Model-free hedge ratios and scale-invariant models. Journal of Banking and Finance, 31, 1839–1861.CrossRefGoogle Scholar
  2. Andersen, T. G., Bollerslev, T., & Diebold, F. X. (2010). Parametric and nonparametric volatility measurement. In Handbook of financial econometrics (Vol. 1, pp. 67–137). Elsevier Inc.Google Scholar
  3. Bakshi, G., Cao, C., & Chen, Z. (1997). Empirical performance of alternative option pricing models. The Journal of Finance, 51, 549–584.Google Scholar
  4. Bakshi, G., & Kapadia, N. (2003). Delta-hedged gains and the negative market volatility risk premium. Review of Financial Studies, 16, 527–566.CrossRefGoogle Scholar
  5. Bakshi, G., Kapadia, N., & Madan, D. (2003). Stock return characteristics, skew laws, and the differential pricing of individual equity options. Review of Financial Studies, 16, 101–143.CrossRefGoogle Scholar
  6. Bakshi, G., & Madan, D. (2000). Spanning and derivative-security valuation. Journal of Financial Economics, 58, 205–238.CrossRefGoogle Scholar
  7. Bakshi, G., & Madan, D. (2006). A theory of volatility spreads. Management Science, 52, 1945–1956.CrossRefGoogle Scholar
  8. Bates, D. (2000). Post-’87 crash fears in the SP500 futures option market. Journal of Econometrics, 94, 181–238.CrossRefGoogle Scholar
  9. Bates, D. (2005). Hedging the smirk. Financial Research Letters, 2, 195–200.CrossRefGoogle Scholar
  10. Bates, D. S. (1995). Testing option pricing models. Working Paper 5129. National Bureau of Economic Research.Google Scholar
  11. Bollerslev, T., Tauchen, G., & Zhou, H. (2009). Expected stock returns and variance risk premia. Review of Financial Studies, 22, 4463–4492.Google Scholar
  12. Carr, P., & Wu, L. (2009). Variance risk premiums. Review of Financial Studies, 22, 1311–1341Google Scholar
  13. Chang, B. Y., Christoffersen, P., Jacobs, K. & Vainberg, G. (2011). Option-implied measures of equity risk. Review of Finance, 16, 385–428.Google Scholar
  14. Christoffersen, P., Elkamhi, R., Feunou, B., & Jacobs, K. (2010). Option valuation with conditional heteroskedasticity and non-normality. Review of Financial Studies, 23, 2139–2183.Google Scholar
  15. Christoffersen, P., Heston, S., & Jacobs, K. (2006). Option valuation with conditional skewness. Journal of Econometrics, 131, 253–284.CrossRefGoogle Scholar
  16. Corrado, C., & Su, T. (1996). Skewness and kurtosis in SP500 index returns implied by options prices. The Journal of Financial Research, 19, 175–192.Google Scholar
  17. Dennis, P., & Mayhew, S. (2000). Risk-neutral skewness: Evidence from stock options. The Journal of Financial and Quantitative Analysis, 37, 471–493.CrossRefGoogle Scholar
  18. Dumas, B., Fleming, J., & Whaley, R. (1998). Implied volatility functions: Empirical tests. The Journal of Finance, 53, 2059–2106.CrossRefGoogle Scholar
  19. Galai, D. (1983). The components of the returns from hedging options against stocks. The Journal of Business, 56, 45–54.CrossRefGoogle Scholar
  20. Harvey, R. V., & Siddique, A. (2000). Conditional skewness in asset pricing tests. The Journal of Finance, 55, 1263–1295.CrossRefGoogle Scholar
  21. Heston, C. (1993). Invisible parameters in option pricing. The Journal of Finance, 48, 933–947.CrossRefGoogle Scholar
  22. Jarrow, R., & Rudd, A. (1982). Approximate option valuation for arbitrary stochastic processes. Financial Economics, 10, 347–369.Google Scholar
  23. Jondeau, E., & Rockinger, M. (2001). Gram-charlier densities. Journal of Economic Dynamics and Control, 25, 1457–1483.Google Scholar
  24. Kim, T.-H., & White, H. (2004). On more robust estimation of skewness and kurtosis. Finance Research Letters, 1, 56–73.Google Scholar
  25. Kraus, A., & Litzenberger, R. (1976). Skewness preference and the valuation of risk assets. The Journal of Finance, 31, 1085–1100.Google Scholar
  26. León, A., Mencía, J., & Sentana, E. (2009). Parametric properties of semi-nonparametric distributions, with applications to option valuation. Journal of Business and Economic Statistics, 27, 176–192.Google Scholar
  27. Polimenis, V. (2006). Skewness corrections for asset pricing. Working Paper.Google Scholar
  28. Potters, M., Cont, R., & Bouchaud, J.-P. (1998). Financial markets as adaptative systems. Europhysics Letters, 41, 239–244Google Scholar
  29. Rompolis, L., & Tzavalis, E. (2008). The effects of the risk-neutral skewness on implied volatility regressions. Working Paper.Google Scholar
  30. Rubinstein, M., & Jackwerth, J. (1998). Recovering probability distribution from option prices. The Journal of Finance, 51, 1611–1631.Google Scholar
  31. Zhang, J., & Xiang, Y. (2005). Implied volatility smirk. Quantitative Finance, 8, 263–284.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Bruno Feunou
    • 1
  • Jean-Sébastien Fontaine
    • 1
  • Roméo Tédongap
    • 2
  1. 1.Bank of CanadaOttawaCanada
  2. 2.ESSEC Business SchoolCergy-PontoiseFrance

Personalised recommendations