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Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs

Author

Listed:
  • Christoph Czichowsky

    (London School of Economics and Political Science, Columbia House)

  • Rémi Peyre

    (Universität Wien
    CNRS – Université de Lorraine)

  • Walter Schachermayer

    (Universität Wien
    ETH Zürich)

  • Junjian Yang

    (École Polytechnique)

Abstract

The present paper accomplishes a major step towards a reconciliation of two conflicting approaches in mathematical finance: on the one hand, the mainstream approach based on the notion of no arbitrage (Black, Merton & Scholes), and on the other hand, the consideration of non-semimartingale price processes, the archetype of which being fractional Brownian motion (Mandelbrot). Imposing (arbitrarily small) proportional transaction costs and considering logarithmic utility optimisers, we are able to show the existence of a semimartingale, frictionless shadow price process for an exponential fractional Brownian financial market.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:finsto:v:22:y:2018:i:1:d:10.1007_s00780-017-0351-5
    DOI: 10.1007/s00780-017-0351-5
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    References listed on IDEAS

    as
    1. Christoph Czichowsky & Walter Schachermayer & Junjian Yang, 2017. "Shadow Prices For Continuous Processes," Mathematical Finance, Wiley Blackwell, vol. 27(3), pages 623-658, July.
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    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. 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.
    2. Paolo Guasoni & Yuliya Mishura & Miklós Rásonyi, 2021. "High-frequency trading with fractional Brownian motion," Finance and Stochastics, Springer, vol. 25(2), pages 277-310, April.
    3. Christian Bender & Sebastian Ferrando & Alfredo Gonzalez, 2021. "Model-Free Finance and Non-Lattice Integration," Papers 2105.10623, arXiv.org.
    4. Garnier, Josselin & Solna, Knut, 2019. "Emergence of turbulent epochs in oil prices," Chaos, Solitons & Fractals, Elsevier, vol. 122(C), pages 281-292.
    5. 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.
    6. Josselin Garnier & Knut Solna, 2018. "Emergence of Turbulent Epochs in Oil Prices," Papers 1808.09382, arXiv.org, revised Apr 2019.

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    More about this item

    Keywords

    Proportional transaction costs; Fractional Brownian motion; Shadow prices; Two-way crossing; Logarithmic utility;
    All these keywords.

    JEL classification:

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

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