Skip to main content
SpringerLink
Log in

A Short Note on Super-Hedging an Arbitrary Number of European Options with Integer-Valued Strategies

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The usual theory of asset pricing in finance assumes that the financial strategies, i.e. the quantity of risky assets to invest, are real-valued so that they are not integer-valued in general, see the Black and Scholes model for instance. This is clearly contrary to what it is possible to do in the real world. Surprisingly, it seems that there are not many contributions in that direction in the literature, except for a finite number of states. In this paper, for arbitrary \(\Omega \), we show that, in discrete-time, it is possible to evaluate the minimal super-hedging price when we restrict ourselves to integer-valued strategies. To do so, we only consider terminal claims that are continuous piecewise affine functions of the underlying asset. We formulate a dynamic programming principle that can be directly implemented on historical data and which also provides the optimal integer-valued strategy. The problem with general payoffs remains open but should be solved with the same approach.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Baptiste, J., Carassus, L., Lepinette, E.: Pricing without Martingale Measures. To appear in ESAIM Proceedings MAS (2022). https://hal.science/hal-01774150v4

  2. Bienstock, D.: Computational study of a family of mixed-integer quadratic programming problems. Math. Progr. 74, 121–140 (1996)

    Article  MathSciNet  Google Scholar 

  3. Bonami, P., Lejeune, M.A.: An exact solution approach for portfolio optimization problems under stochastic and integer constraints. Oper. Res. 57, 650–670 (2009)

    Article  MathSciNet  Google Scholar 

  4. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Political Econ. 81(3), 637–659 (1973)

    Article  MathSciNet  Google Scholar 

  5. Baumann, P., Trautmann, N.: Portfolio-optimization models for small investors. Math. Methods Oper. Res. 77, 345–356 (2013)

    Article  MathSciNet  Google Scholar 

  6. Carassus, L., Lepinette, E.: Pricing without no-arbitrage condition in discrete-time. J. Math. Anal. Appl. 505(1), 125441 (2022)

    Article  MathSciNet  Google Scholar 

  7. Deng, X., Li, Z., Wang, S.: Computational complexity of arbitrage in frictional security market. Int. J. Found. Comput. Sci. 13, 681–684 (2002)

    Article  Google Scholar 

  8. Dalang, E.C., Morton, A., Willinger, W.: Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stoch. Stoch. Reports 29, 185–201 (1990)

    Article  MathSciNet  Google Scholar 

  9. Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Annal. 300, 463–520 (1994)

    Article  MathSciNet  Google Scholar 

  10. Delbaen, F., Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Annal. 312, 215–250 (1996)

    Article  MathSciNet  Google Scholar 

  11. Delbaen, F., Schachermayer, W.: The mathematics of arbitrage. Springer Finance, Springer Berlin, Heidelberg (2006). https://doi.org/10.1007/978-3-540-31299-4

  12. Gerhold, S., Krühner, P.: Dynamic trading under trading constraints. Finance Stoch. 22, 919–957 (2018)

    Article  MathSciNet  Google Scholar 

  13. Meriem, El Mansour, Lepinette, E.: Conditional interior and conditional closure of a random sets. J. Optim. Theory Appl. 5051(187), 356–369 (2020)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Sciences of Tunis, Tunis, Tunisia

    Dorsaf Cherif & Meriam El Mansour

  2. Ceremade, UMR CNRS 7534, Paris-Dauphine University, PSL, Place du Maréchal De Lattre De Tassigny, 75775, Paris cedex 16, France

    Meriam El Mansour & Emmanuel Lepinette

Authors
  1. Dorsaf Cherif
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Meriam El Mansour
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Emmanuel Lepinette
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emmanuel Lepinette.

Additional information

Communicated by Klaus Reiner Schenk-Hoppe.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cherif, D., El Mansour, M. & Lepinette, E. A Short Note on Super-Hedging an Arbitrary Number of European Options with Integer-Valued Strategies. J Optim Theory Appl (2024). https://doi.org/10.1007/s10957-024-02409-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10957-024-02409-2

Keywords

Mathematics Subject Classification

Search

Navigation