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Duality theory for portfolio optimisation under transaction costs

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  • Czichowsky, Christoph
  • Schachermayer, Walter

Abstract

We consider the problem of portfolio optimisation with general càdlàg price processes in the presence of proportional transaction costs. In this context, we develop a general duality theory. In particular, we prove the existence of a dual optimiser as well as a shadow price process in an appropriate generalised sense. This shadow price is defined by means of a "sandwiched" process consisting of a predictable and an optional strong supermartingale, and pertains to all strategies that remain solvent under transaction costs. We provide examples showing that, in the general setting we study, the shadow price processes have to be of such a generalised form.

Suggested Citation

  • Czichowsky, Christoph & Schachermayer, Walter, 2016. "Duality theory for portfolio optimisation under transaction costs," LSE Research Online Documents on Economics 63362, London School of Economics and Political Science, LSE Library.
    • Handle: RePEc:ehl:lserod:63362
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    References listed on IDEAS

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    Cited by:

    1. Palma, Nuno, 2018. "Money and modernization in early modern England," Financial History Review, Cambridge University Press, vol. 25(3), pages 231-261, December.
    2. Christoph Czichowsky & Rémi Peyre & Walter Schachermayer & Junjian Yang, 2018. "Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs," Post-Print hal-02373296, HAL.
    3. Christoph Kuhn, 2023. "The fundamental theorem of asset pricing with and without transaction costs," Papers 2307.00571, arXiv.org, revised Aug 2024.
    4. Erhan Bayraktar & Leonid Dolinskyi & Yan Dolinsky, 2020. "Extended weak convergence and utility maximisation with proportional transaction costs," Finance and Stochastics, Springer, vol. 24(4), pages 1013-1034, October.
    5. Christoph Czichowsky & Raphael Huwyler, 2022. "Robust utility maximisation under proportional transaction costs for c\`adl\`ag price processes," Papers 2211.00532, arXiv.org, revised Aug 2024.
    6. Lingqi Gu & Yiqing Lin & Junjian Yang, 2017. "Utility maximization problem under transaction costs: optimal dual processes and stability," Papers 1710.04363, arXiv.org.
    7. Peter Bank & David Besslich, 2018. "Modelling information flows by Meyer-$\sigma$-fields in the singular stochastic control problem of irreversible investment," Papers 1810.08495, arXiv.org, revised Mar 2020.
    8. E. Babaei & I.V. Evstigneev & K.R. Schenk-Hoppé, 2019. "Log-Optimal and Rapid Paths in von Neumann-Gale Dynamical Systems," Economics Discussion Paper Series 1902, Economics, The University of Manchester.
    9. Erhan Bayraktar & Christoph Czichowsky & Leonid Dolinskyi & Yan Dolinsky, 2021. "A Note on Utility Maximization with Proportional Transaction Costs and Stability of Optimal Portfolios," Papers 2107.01568, arXiv.org, revised Sep 2021.
    10. Kim Weston, 2017. "Existence of a Radner equilibrium in a model with transaction costs," Papers 1702.01706, arXiv.org, revised Feb 2018.
    11. Yiqing Lin & Junjian Yang, 2016. "Utility maximization problem with random endowment and transaction costs: when wealth may become negative," Papers 1604.08224, arXiv.org, revised Sep 2016.
    12. Jin Hyuk Choi & Tae Ung Gang, 2021. "Optimal investment in illiquid market with search frictions and transaction costs," Papers 2101.09936, arXiv.org, revised Aug 2021.
    13. Christoph Belak & Jörn Sass, 2019. "Finite-horizon optimal investment with transaction costs: construction of the optimal strategies," Finance and Stochastics, Springer, vol. 23(4), pages 861-888, October.
    14. Christoph Czichowsky & Rémi Peyre & Walter Schachermayer & Junjian Yang, 2018. "Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs," Finance and Stochastics, Springer, vol. 22(1), pages 161-180, January.
    15. Christoph Kühn & Alexander Molitor, 2022. "Semimartingale price systems in models with transaction costs beyond efficient friction," Finance and Stochastics, Springer, vol. 26(4), pages 927-982, October.
    16. Albert Altarovici & Max Reppen & H. Mete Soner, 2016. "Optimal Consumption and Investment with Fixed and Proportional Transaction Costs," Papers 1610.03958, arXiv.org.
    17. Christoph Czichowsky & R'emi Peyre & Walter Schachermayer & Junjian Yang, 2016. "Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs," Papers 1608.01415, arXiv.org.
    18. Jan Obloj & Johannes Wiesel, 2021. "Distributionally robust portfolio maximisation and marginal utility pricing in one period financial markets," Papers 2105.00935, arXiv.org, revised Nov 2021.
    19. Eberhard Mayerhofer, 2024. "Almost Perfect Shadow Prices," JRFM, MDPI, vol. 17(2), pages 1-18, February.
    20. Tae Ung Gang & Jin Hyuk Choi, 2024. "Unified Asymptotics For Investment Under Illiquidity: Transaction Costs And Search Frictions," Papers 2407.13547, arXiv.org.
    21. Miklos Rasonyi, 2017. "On utility maximization without passing by the dual problem," Papers 1702.00982, arXiv.org, revised Mar 2018.

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    More about this item

    Keywords

    utility maximisation; proportional transaction costs; convex duality; shadow prices; supermartingale deflators; optional strong supermartingales; predictable strong supermartingales; logarithmic utility;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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