Skip to main content
SpringerLink
Log in

A quasi-sure optional decomposition and super-hedging result on the Skorokhod space

  • Published:
Finance and Stochastics Aims and scope Submit manuscript

Abstract

We prove a robust super-hedging duality result for path-dependent options on assets with jumps in a continuous-time setting. It requires that the collection of martingale measures is rich enough and that the payoff function satisfies some continuity property. It is a by-product of a quasi-sure version of the optional decomposition theorem, which can also be viewed as a functional version of Itô’s lemma that applies to non-smooth functionals (of càdlàg processes) which are concave in space and nonincreasing in time, in the sense of Dupire.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Biagini, S., Bouchard, B., Kardaras, C., Nutz, M.: Robust fundamental theorem for continuous processes. Math. Finance 27, 963–987 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bouchard, B., Tan, X.: Understanding the dual formulation for the hedging of path-dependent options with price impact. Preprint (2019). arXiv:1912.03946

  3. Carlen, E., Protter, P.: On semimartingale decompositions of convex functions of semimartingales. Ill. J. Math. 36, 420–427 (1992)

    MathSciNet  MATH  Google Scholar 

  4. Cont, R., Fournié, D.-A.: Change of variable formulas for non-anticipative functionals on path space. J. Funct. Anal. 259, 1043–1072 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cont, R., Fournié, D.-A.: Functional Itô calculus and stochastic integral representation of martingales. Ann. Probab. 41, 109–133 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Delbaen, F., Schachermayer, W.: A compactness principle for bounded sequences of martingales with applications. In: Dalang, R.C., et al. (eds.) Seminar on Stochastic Analysis, Random Fields and Applications. Progress in Probability, vol. 45, pp. 137–173. Birkhäuser, Basel (1999)

    Chapter  MATH  Google Scholar 

  7. Dolinsky, Y., Soner, H.M.: Martingale optimal transport in the Skorokhod space. In: Stochastic Processes and their Applications, vol. 125, pp. 3893–3931 (2015)

    Google Scholar 

  8. Dupire, B.: Functional Itô calculus. Quant. Finance 19, 721–729 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. El Karoui, N., Quenez, M.-C.: Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33, 29–66 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. El Karoui, N., Tan, X.: Capacities, measurable selection and dynamic programming. Part I: abstract framework. Preprint (2013). arXiv:1310.3363

  11. Föllmer, H., Kabanov, Yu.M.: Optional decomposition and Lagrange multipliers. Finance Stoch. 2, 69–81 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Föllmer, H., Kramkov, D.: Optional decompositions under constraints. Probab. Theory Relat. Fields 109, 1–25 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Guo, G., Tan, X., Touzi, N.: Tightness and duality of martingale transport on the Skorokhod space. In: Stochastic Processes and their Applications, vol. 127, pp. 927–956 (2017)

    MATH  Google Scholar 

  14. Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  15. Jakubowski, A.: A non-Skorohod topology on the Skorohod space. Electron. J. Probab. 2, 1–21 (1997)

    Article  MathSciNet  Google Scholar 

  16. Kiiski, M.: The Riesz representation theorem and weak⋆ compactness of semimartingales. Finance Stoch. 24, 827–870 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Köpfer, B., Rüschendorf, L.: Comparison of path-dependent functionals of semimartingales. Preprint (2019). arXiv:1908.10076

  18. Kramkov, D.O.: Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Theory Relat. Fields 105, 459–479 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Meyer, P.-A.: Un cours sur les intégrales stochastiques. In: Meyer, P.-A. (ed.) Séminaire de Probabilités X. Lecture Notes in Mathematics, vol. 511, pp. 245–400. Springer, Berlin (1976)

    Google Scholar 

  20. Neufeld, A., Nutz, M.: Superreplication under volatility uncertainty for measurable claims. Electron. J. Probab. 18, 48 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nutz, M.: Pathwise construction of stochastic integrals. Electron. Commun. Probab. 17(24), 1–7 (2012)

    MathSciNet  MATH  Google Scholar 

  22. Nutz, M., Van Handel, R.: Constructing sublinear expectations on path space. Stoch. Process. Appl. 123, 3100–3121 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nutz, M.: Robust superhedging with jumps and diffusion. In: Stochastic Processes and their Applications, vol. 125, pp. 4543–4555 (2015)

    MATH  Google Scholar 

  24. Possamaï, D., Royer, G., Touzi, N.: On the robust superhedging of measurable claims. Electron. Commun. Probab. 18, paper no. 95, 1–13 (2013)

    MathSciNet  MATH  Google Scholar 

  25. Saporito, Y.F.: The functional Meyer–Tanaka formula. Stoch. Dyn. 18(04), 1850030 (2018)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We should like to thank Pierre Cardaliaguet who pointed out to us the identity and its proof in Remark 2.7.

Author information

Authors and Affiliations

  1. CEREMADE, Université Paris-Dauphine, PSL, CNRS, Paris, France

    Bruno Bouchard

  2. Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, Hong Kong

    Xiaolu Tan

Authors
  1. Bruno Bouchard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaolu Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bruno Bouchard.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors are grateful to two anonymous reviewers for useful comments and suggestions. The research of Xiaolu Tan is supported by CUHK Faculty of Science Direct Grant 2019-2020.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bouchard, B., Tan, X. A quasi-sure optional decomposition and super-hedging result on the Skorokhod space. Finance Stoch 25, 505–528 (2021). https://doi.org/10.1007/s00780-021-00458-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00780-021-00458-3

Keywords

Mathematics Subject Classification (2020)

JEL Classification

Search

Navigation