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

Author

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  • Czichowsky, Christoph Johannes
  • Peyre, Rémi
  • Schachermayer, Walter
  • Yang, Junjian

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

  • Czichowsky, Christoph Johannes & Peyre, Rémi & Schachermayer, Walter & Yang, Junjian, 2018. "Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs," LSE Research Online Documents on Economics 85230, London School of Economics and Political Science, LSE Library.
    • Handle: RePEc:ehl:lserod:85230
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    References listed on IDEAS

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    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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    3. Griselda Deelstra & Huyên Pham & Nizar Touzi, 2001. "Dual formulation of the utility maximisation problem under transaction costs," ULB Institutional Repository 2013/7596, ULB -- Universite Libre de Bruxelles.
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    6. Paolo Guasoni, 2006. "No Arbitrage Under Transaction Costs, With Fractional Brownian Motion And Beyond," Mathematical Finance, Wiley Blackwell, vol. 16(3), pages 569-582, July.
    7. Jan Kallsen & Johannes Muhle-Karbe, 2011. "Existence of shadow prices in finite probability spaces," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(2), pages 251-262, April.
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    10. Luciano Campi & Mark Owen, 2011. "Multivariate utility maximization with proportional transaction costs," Finance and Stochastics, Springer, vol. 15(3), pages 461-499, September.
    11. Czichowsky, Christoph & Schachermayer, Walter & Yang, Junjian, 2017. "Shadow prices for continuous processes," LSE Research Online Documents on Economics 63370, London School of Economics and Political Science, LSE Library.
    12. Benoit Mandelbrot, 1967. "The Variation of Some Other Speculative Prices," The Journal of Business, University of Chicago Press, vol. 40, pages 393-393.
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    Cited by:

    1. 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.
    2. Josselin Garnier & Knut Solna, 2018. "Emergence of Turbulent Epochs in Oil Prices," Papers 1808.09382, arXiv.org, revised Apr 2019.
    3. 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.
    4. Christian Bender & Sebastian Ferrando & Alfredo Gonzalez, 2021. "Model-Free Finance and Non-Lattice Integration," Papers 2105.10623, arXiv.org.
    5. Garnier, Josselin & Solna, Knut, 2019. "Emergence of turbulent epochs in oil prices," Chaos, Solitons & Fractals, Elsevier, vol. 122(C), pages 281-292.
    6. 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.

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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:

    • 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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