Finance and Stochastics

, Volume 21, Issue 3, pp 593–630 | Cite as

Bounds for VIX futures given S&P 500 smiles



We derive sharp bounds for the prices of VIX futures using the full information of S&P 500 smiles. To that end, we formulate the model-free sub/superreplication of the VIX by trading in the S&P 500 and its vanilla options as well as the forward-starting log-contracts. A dual problem of minimizing/maximizing certain risk-neutral expectations is introduced and shown to yield the same value.

The classical bounds for VIX futures given the smiles only use a calendar spread of log-contracts on the S&P 500. We analyze for which smiles the classical bounds are sharp and how they can be improved when they are not. In particular, we introduce a family of functionally generated portfolios which often improves the classical bounds while still being tractable; more precisely, they are determined by a single concave/convex function on the line. Numerical experiments on market data and SABR smiles show that the classical lower bound can be improved dramatically, whereas the upper bound is often close to optimal.


VIX futures Price bounds Model-free pricing Robust hedging 

Mathematics Subject Classification (2010)

91B25 60G42 49N05 

JEL Classification



  1. 1.
    Bayraktar, E., Zhao, Z.: On arbitrage and duality under model uncertainty and portfolio constraints. Math. Finance (2015). doi:10.1111/mafi.12104 Google Scholar
  2. 2.
    Beiglböck, M., Henry-Labordère, P., Penkner, F.: Model-independent bounds for option prices: a mass-transport approach. Finance Stoch. 17, 477–501 (2013) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Beiglböck, M., Juillet, N.: On a problem of optimal transport under marginal martingale constraints. Ann. Probab. 44, 42–106 (2016) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Beiglböck, M., Nutz, M., Touzi, N.: Complete duality for martingale optimal transport on the line. Ann. Probab. (2016, to appear). Available online at arXiv:1507.00671
  5. 5.
    Bouchard, B., Nutz, M.: Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab. 25, 823–859 (2015) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Carr, P., Lee, R.: Robust replication of volatility derivatives. Preprint (2009). Available online at
  7. 7.
    Carr, P., Lee, R.: Hedging variance options on continuous semimartingales. Finance Stoch. 14, 179–207 (2010) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Carr, P., Madan, D.: Towards a theory of volatility trading. In: Musiela, M., et al. (eds.) Option Pricing, Interest Rates, and Risk Management, pp. 458–476. Cambridge University Press, Cambridge (2001) CrossRefGoogle Scholar
  9. 9.
    The CBOE volatility index—VIX: (accessed on August 3, 2016)
  10. 10.
    Cox, A., Wang, J.: Root’s barrier: construction, optimality and applications to variance options. Ann. Appl. Probab. 23, 859–894 (2013) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Dellacherie, C., Meyer, P.-A.: Probabilities and Potential A. North Holland, Amsterdam (1978) MATHGoogle Scholar
  12. 12.
    De Marco, S., Henry-Labordère, P.: Linking vanillas and VIX options: a constrained martingale optimal transport problem. SIAM J. Financ. Math. 6, 1171–1194 (2015) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dupire, B.: Model free results on volatility derivatives. SAMSI, Research Triangle Park (2006). Available online at
  14. 14.
    Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 3rd edn. de Gruyter, Berlin (2011) CrossRefMATHGoogle Scholar
  15. 15.
    Galichon, A., Henry-Labordère, P., Touzi, N.: A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options. Ann. Appl. Probab. 24, 312–336 (2014) MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Henry-Labordère, P.: Automated option pricing: numerical methods. Int. J. Theor. Appl. Finance 16, 135–162 (2013) MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Hobson, D., Klimmek, M.: Model-independent hedging strategies for variance swaps. Finance Stoch. 16, 611–649 (2012) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Johansen, S.: A representation theorem for a convex cone of quasi convex functions. Math. Scand. 30, 297–312 (1972) MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Johansen, S.: The extremal convex functions. Math. Scand. 34, 61–68 (1974) MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kahalé, N.: Model-independent lower bound on variance swaps. Math. Finance 26, 939–961 (2016) MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Neuberger, A.: The log contract. J. Portf. Manag. 20, 74–80 (1994) CrossRefGoogle Scholar
  22. 22.
    Scarsini, M.: Multivariate convex orderings, dependence, and stochastic equality. J. Appl. Probab. 35, 93–103 (1998) MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Sion, M.: On general minimax theorems. Pac. J. Math. 8, 171–176 (1958) MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat. 36, 423–439 (1965) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Quantitative ResearchBloomberg L.P.New YorkUSA
  2. 2.Departments of Statistics and MathematicsColumbia UniversityNew YorkUSA

Personalised recommendations