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

Author

Listed:
  • Dorsaf Cherif

    (Faculty of Sciences of Tunis)

  • Meriam El Mansour

    (Faculty of Sciences of Tunis
    Paris-Dauphine University, PSL)

  • Emmanuel Lepinette

    (Paris-Dauphine University, PSL)

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 $$\Omega $$ Ω , 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.

Suggested Citation

  • Dorsaf Cherif & Meriam El Mansour & Emmanuel Lepinette, 2024. "A Short Note on Super-Hedging an Arbitrary Number of European Options with Integer-Valued Strategies," Journal of Optimization Theory and Applications, Springer, vol. 201(3), pages 1301-1312, June.
    • Handle: RePEc:spr:joptap:v:201:y:2024:i:3:d:10.1007_s10957-024-02409-2
    DOI: 10.1007/s10957-024-02409-2
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    References listed on IDEAS

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    1. P. Bonami & M. A. Lejeune, 2009. "An Exact Solution Approach for Portfolio Optimization Problems Under Stochastic and Integer Constraints," Operations Research, INFORMS, vol. 57(3), pages 650-670, June.
    2. Meriam El Mansour & Emmanuel Lépinette, 2020. "Conditional Interior and Conditional Closure of Random Sets," Journal of Optimization Theory and Applications, Springer, vol. 187(2), pages 356-369, November.
    3. Stefan Gerhold & Paul Krühner, 2018. "Dynamic trading under integer constraints," Finance and Stochastics, Springer, vol. 22(4), pages 919-957, October.
    4. Philipp Baumann & Norbert Trautmann, 2013. "Portfolio-optimization models for small investors," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(3), pages 345-356, June.
    5. Pierre Bonami & Miguel A. Lejeune, 2009. "An Exact Solution Approach for Integer Constrained Portfolio Optimization Problems Under Stochastic Constraints," Post-Print hal-00421756, HAL.
    6. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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