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Dynamic trading under integer constraints

Author

Listed:
  • Stefan Gerhold

    (TU Wien)

  • Paul Krühner

    (University of Liverpool)

Abstract

In this paper, we investigate discrete-time trading under integer constraints, that is, we assume that the offered goods or shares are traded in integer quantities instead of the usual real quantity assumption. For finite probability spaces and rational asset prices, this has little effect on the core of the theory of no-arbitrage pricing. For price processes not restricted to the rational numbers, a novel theory of integer-arbitrage-free pricing and hedging emerges. We establish an FTAP, involving a set of absolutely continuous martingale measures satisfying an additional property. The set of prices of a contingent claim is not necessarily an interval, but is either empty or dense in an interval. We also discuss superhedging with integer-valued portfolios.

Suggested Citation

  • Stefan Gerhold & Paul Krühner, 2018. "Dynamic trading under integer constraints," Finance and Stochastics, Springer, vol. 22(4), pages 919-957, October.
    • Handle: RePEc:spr:finsto:v:22:y:2018:i:4:d:10.1007_s00780-018-0369-3
    DOI: 10.1007/s00780-018-0369-3
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    References listed on IDEAS

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    1. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
    2. 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.
    3. Pierre Bonami & Miguel A. Lejeune, 2009. "An Exact Solution Approach for Integer Constrained Portfolio Optimization Problems Under Stochastic Constraints," Post-Print hal-00421756, HAL.
    4. Laurence Carassus & Huye^n Pham & Nizar Touzi, 2001. "No Arbitrage in Discrete Time Under Portfolio Constraints," Mathematical Finance, Wiley Blackwell, vol. 11(3), pages 315-329, July.
    5. Föllmer, Hans & Kramkov, D. O., 1997. "Optional decompositions under constraints," SFB 373 Discussion Papers 1997,31, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    6. 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.
    7. Christian Bender & Michael Kohlmann, 2008. "Optimal Superhedging Under Non-Convex Constraints — A Bsde Approach," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 11(04), pages 363-380.
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    Citations

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    Cited by:

    1. Dorsaf Cherif & Meriam El Mansour & Emmanuel Lepinette, 2023. "A short note on super-hedging an arbitrary number of European options with integer-valued strategies," Papers 2311.08871, arXiv.org.
    2. 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.

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    More about this item

    Keywords

    Arbitrage; Hedging; Integer constraints;
    All these keywords.

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools

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