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Optimal Trading with Linear and (small) Non-Linear Costs

Author

Listed:
  • A. Rej
  • R. Benichou
  • J. de Lataillade
  • G. Z'erah
  • J. -Ph. Bouchaud

Abstract

We reconsider the problem of optimal trading in the presence of linear and quadratic costs, for arbitrary linear costs but in the limit where quadratic costs are small. Using matched asymptotic expansion techniques, we find that the trading speed vanishes inside a band that is narrower than in the absence of quadratic costs, by an amount that scales as the one-third power of quadratic costs. Outside of the band, we find three regimes: a small boundary layer where the velocity vanishes linearly with the distance to the band, an intermediate region where the velocity behaves as a square-root of that distance, and a far region where it becomes linear. Our solution is consistent with available numerical results. We determine the conditions in which our expansion is useful in practical applications, and generalize our solution to other forms of non-linear costs.

Suggested Citation

  • A. Rej & R. Benichou & J. de Lataillade & G. Z'erah & J. -Ph. Bouchaud, 2015. "Optimal Trading with Linear and (small) Non-Linear Costs," Papers 1511.07359, arXiv.org, revised Nov 2016.
    • Handle: RePEc:arx:papers:1511.07359
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    References listed on IDEAS

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    1. Nicolae Gârleanu & Lasse Heje Pedersen, 2013. "Dynamic Trading with Predictable Returns and Transaction Costs," Journal of Finance, American Finance Association, vol. 68(6), pages 2309-2340, December.
    2. J. Donier & J. Bonart & I. Mastromatteo & J.-P. Bouchaud, 2015. "A fully consistent, minimal model for non-linear market impact," Quantitative Finance, Taylor & Francis Journals, vol. 15(7), pages 1109-1121, July.
    3. Jonathan Donier & Julius Bonart & Iacopo Mastromatteo & Jean-Philippe Bouchaud, 2014. "A fully consistent, minimal model for non-linear market impact," Papers 1412.0141, arXiv.org, revised Mar 2015.
    4. Obizhaeva, Anna A. & Wang, Jiang, 2013. "Optimal trading strategy and supply/demand dynamics," Journal of Financial Markets, Elsevier, vol. 16(1), pages 1-32.
    5. Joachim de Lataillade & Cyril Deremble & Marc Potters & Jean-Philippe Bouchaud, 2012. "Optimal Trading with Linear Costs," Papers 1203.5957, arXiv.org.
    6. Bence Toth & Yves Lemperiere & Cyril Deremble & Joachim de Lataillade & Julien Kockelkoren & Jean-Philippe Bouchaud, 2011. "Anomalous price impact and the critical nature of liquidity in financial markets," Papers 1105.1694, arXiv.org, revised Nov 2011.
    7. Richard Martin & Torsten Schoneborn, 2011. "Mean Reversion Pays, but Costs," Papers 1103.4934, arXiv.org.
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