Skip to main content
SpringerLink
Log in

Guaranteed Deterministic Approach to Superhedging: Sensitivity of Solutions of the Bellman-Isaacs Equations and Numerical Methods

  • Published:
Computational Mathematics and Modeling Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

We consider a guaranteed deterministic formulation for the super-replication problem in discrete time: find a guaranteed coverage of a contingent claim on an option under all possible scenarios. These scenarios are specified by a priori compacta that depend on historical prices: the price increments at each instant should be in the corresponding compacta. We assume the presence of trading constraints and the absence of transaction costs. The problem is posed in a game-theoretical setting and leads to Bellman–Isaacs equations in both pure and mixed “market” strategies. In the present article, we investigate the sensitivity of the solutions to small perturbations of the compacta that describe price uncertainties over time. Numerical methods are proposed allowing for the problem’s specific features.

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. S. N. Smirnov, “Guaranteed deterministic approach to superhedging: market model, trading constraints, no-arbitrage, and the Bellman–Isaacs equations,” Matem. Teoriya Igr i Ee Prilozheniya, 10, No. 4, 59–99 (2018).

    MATH  Google Scholar 

  2. S. N. Smirnov, “Guaranteed deterministic approach to superhedging: “no arbitrage” market properties,” Matem. Teoriya Igr i Ee Prilozheniya, 11, No. 2, 68–95 (2019).

    MATH  Google Scholar 

  3. S. N. Smirnov, “Guaranteed deterministic approach to superhedging: semi-continuity and continuity properties of the solutions of the Bellman–Isaacs equations,” Matem. Teoriya Igr i Ee Prilozheniya, 11, No. 4, 87–115 (2019).

    MATH  Google Scholar 

  4. S. N. Smirnov, “Guaranteed deterministic approach to superhedging: Lipschitz properties of solutions of the Bellman–Isaacs equations,” in: Frontiers of Dynamic Games. Game Theory and Management, St. Petersburg, Cham, Birkhauser (2019), pp. 267–288.

  5. S. N. Smirnov, “Guaranteed deterministic approach to superhedging: mixed strategies and game equilibrium,” Matem. Teoriya Igr i Ee Prilozheniya, 12, No. 1, 60–90 (2020).

    MATH  Google Scholar 

  6. S. N. Smirnov, “Guaranteed deterministic approach to superhedging: the best market behavior scenarios and the moment problem,” Matem. Teoriya Igr i Ee Prilozheniya, 12 (2020), in print.

  7. S. N. Smirnov, “A guaranteed deterministic approach to superheding: case of the convex payoff functions on options,” Mathematics, 7, No. 1246, 1–19 (2019).

    Google Scholar 

  8. S. Hu and N. Papageorgiou, Handbook of Multivalued Analysis: Theory, Vol. I, Mathematics and Its Applications, Vol. 419, Springer, Berlin (1997).

  9. L. N. Polyakov, Development of MathematicalTheory and Numerical Methods for the Solution of Some Classes of Nonsmooth Optimization Problems [in Russian], Doctoral Thesis, Sankt-Peterburg (1998).

  10. E. S. Polovinkin, Multivalued Analysis and Differential Inclusions [in Russian], Nauka, Moscow (2015).

  11. R. T. Rockafellar, Convex Analysis, Princeton Univ. Press (1970).

  12. K. Leichtweiss, Convex Sets [Russian translation from the German], Nauka, Moscow (1985).

  13. A. G. Sukharev, Minimax Algorithms in Numerical Analysis [in Russian], Nauka, Moscow (1989).

  14. V. M. Tikhomirov, Some Topics in Approximation Theory [in Russian], Izd. MGU, Moscow(1976).

  15. V. N. Ushakov and P. D. Lebedev, “Optimal covering algorithms for sets on ℝ2,” Vestnik Udmurt. Univ., Mat. Mechan., Comput. Sci., 26, No. 2, 258–270 2016.

  16. T. Hales et al., “A formal proof of the Kepler conjecture,” Forum of Mathematics, Pi, Vol. 5, Cambridge Univ. Press (2017).

  17. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups, Springer, New York (1993).

    Book  Google Scholar 

  18. J. Goodman, J. O’Rourke, and C. D. Toth, Handbook of Discrete and Computational Geometry, 3rd ed., CRC Press, Boca Raton (2017).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, Russia

    S. N. Smirnov

Authors
  1. S. N. Smirnov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. N. Smirnov.

Additional information

Translated from Prikladnaya Matematika i Informatika, No. 63, 2019, pp. 82–104.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Smirnov, S.N. Guaranteed Deterministic Approach to Superhedging: Sensitivity of Solutions of the Bellman-Isaacs Equations and Numerical Methods. Comput Math Model 31, 384–401 (2020). https://doi.org/10.1007/s10598-020-09499-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10598-020-09499-3

Keywords

Search

Navigation