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Root's barrier: Construction, optimality and applications to variance options


  • Alexander M. G. Cox
  • Jiajie Wang


Recent work of Dupire and Carr and Lee has highlighted the importance of understanding the Skorokhod embedding originally proposed by Root for the model-independent hedging of variance options. Root's work shows that there exists a barrier from which one may define a stopping time which solves the Skorokhod embedding problem. This construction has the remarkable property, proved by Rost, that it minimizes the variance of the stopping time among all solutions. In this work, we prove a characterization of Root's barrier in terms of the solution to a variational inequality, and we give an alternative proof of the optimality property which has an important consequence for the construction of subhedging strategies in the financial context.

Suggested Citation

  • Alexander M. G. Cox & Jiajie Wang, 2011. "Root's barrier: Construction, optimality and applications to variance options," Papers 1104.3583,, revised Mar 2013.
  • Handle: RePEc:arx:papers:1104.3583

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    References listed on IDEAS

    1. Haydyn Brown & David Hobson & L. C. G. Rogers, 2001. "Robust Hedging of Barrier Options," Mathematical Finance, Wiley Blackwell, vol. 11(3), pages 285-314.
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    Cited by:

    1. Matteo Burzoni & Marco Frittelli & Zhaoxu Hou & Marco Maggis & Jan Ob{l}'oj, 2016. "Pointwise Arbitrage Pricing Theory in Discrete Time," Papers 1612.07618,, revised Feb 2018.
    2. Y. Dolinsky & H. M. Soner, 2014. "Martingale optimal transport in the Skorokhod space," Papers 1404.1516,, revised Feb 2015.
    3. Anna Aksamit & Zhaoxu Hou & Jan Obl'oj, 2016. "Robust framework for quantifying the value of information in pricing and hedging," Papers 1605.02539,
    4. Guo, Gaoyue & Tan, Xiaolu & Touzi, Nizar, 2017. "Tightness and duality of martingale transport on the Skorokhod space," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 927-956.
    5. Julien Guyon & Romain Menegaux & Marcel Nutz, 2016. "Bounds for VIX Futures given S&P 500 Smiles," Papers 1609.05832,, revised Jun 2017.
    6. repec:spr:finsto:v:21:y:2017:i:3:d:10.1007_s00780-017-0334-6 is not listed on IDEAS
    7. David Hobson & Martin Klimmek, 2015. "Robust price bounds for the forward starting straddle," Finance and Stochastics, Springer, vol. 19(1), pages 189-214, January.
    8. Beatrice Acciaio & Alexander M. G. Cox & Martin Huesmann, 2016. "Model-independent pricing with insider information: a Skorokhod embedding approach," Papers 1610.09124,
    9. Gassiat, Paul & Oberhauser, Harald & dos Reis, Gonçalo, 2015. "Root’s barrier, viscosity solutions of obstacle problems and reflected FBSDEs," Stochastic Processes and their Applications, Elsevier, vol. 125(12), pages 4601-4631.
    10. repec:wsi:ijtafx:v:20:y:2017:i:02:n:s021902491750008x is not listed on IDEAS
    11. Patrick Cheridito & Michael Kupper & Ludovic Tangpi, 2016. "Duality formulas for robust pricing and hedging in discrete time," Papers 1602.06177,, revised Sep 2017.
    12. Julien Claisse & Gaoyue Guo & Pierre Henry-Labordere, 2015. "Some Results on Skorokhod Embedding and Robust Hedging with Local Time," Papers 1511.07230,, revised Oct 2017.
    13. Gaoyue Guo & Jan Obloj, 2017. "Computational Methods for Martingale Optimal Transport problems," Papers 1710.07911,
    14. Pierre Henry-Labordère & Nizar Touzi, 2016. "An explicit martingale version of the one-dimensional Brenier theorem," Finance and Stochastics, Springer, vol. 20(3), pages 635-668, July.
    15. Alexander M. G. Cox & Jiajie Wang, 2013. "Optimal robust bounds for variance options," Papers 1308.4363,
    16. Sergey Nadtochiy & Jan Obloj, 2016. "Robust Trading of Implied Skew," Papers 1611.05518,
    17. Henry-Labordère, Pierre & Tan, Xiaolu & Touzi, Nizar, 2016. "An explicit martingale version of the one-dimensional Brenier’s Theorem with full marginals constraint," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2800-2834.
    18. Mathias Beiglboeck & Alexander Cox & Martin Huesmann, 2017. "The geometry of multi-marginal Skorokhod Embedding," Papers 1705.09505,

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