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Multivariate Transient Price Impact and Matrix-Valued Positive Definite Functions

Author

Listed:
  • Aurélien Alfonsi

    () (Université Paris-Est, CERMICS, Projet MathRisk ENPC-INRIA-UMLV, Ecole des Ponts, 77455 Marne La Vallée, France)

  • Florian Klöck

    () (Department of Mathematics, University of Mannheim, A5, 6, 68131 Mannheim, Germany)

  • Alexander Schied

    () (Department of Mathematics, University of Mannheim, A5, 6, 68131 Mannheim, Germany)

Abstract

We consider a model for linear transient price impact for multiple assets that takes cross-asset impact into account. Our main goal is to single out properties that need to be imposed on the decay kernel so that the model admits well-behaved optimal trade execution strategies. We first show that the existence of such strategies is guaranteed by assuming that the decay kernel corresponds to a matrix-valued positive definite function. An example illustrates, however, that positive definiteness alone does not guarantee that optimal strategies are well-behaved. Building on previous results from the one-dimensional case, we investigate a class of nonincreasing, non-negative, and convex decay kernels with values in a space of symmetric matrices. We show that these decay kernels are always positive definite and characterize when they are even strictly positive definite, a result that may be of independent interest. Optimal strategies for kernels from this class are particularly well-behaved if one requires that the decay kernel is also commuting. We show how such decay kernels can be constructed by means of matrix functions and provide a number of examples. In particular, we completely solve the case of matrix exponential decay.

Suggested Citation

  • Aurélien Alfonsi & Florian Klöck & Alexander Schied, 2016. "Multivariate Transient Price Impact and Matrix-Valued Positive Definite Functions," Mathematics of Operations Research, INFORMS, vol. 41(3), pages 914-934, August.
  • Handle: RePEc:inm:ormoor:v:41:y:2016:i:3:p:914-934
    DOI: 10.1287/moor.2015.0761
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    File URL: http://dx.doi.org/10.1287/moor.2015.0761
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Mehdi Tomas & Iacopo Mastromatteo & Michael Benzaquen, 2020. "How to build a cross-impact model from first principles: Theoretical requirements and empirical results," Papers 2004.01624, arXiv.org, revised Sep 2020.
    2. Luis Carlos Garc'ia del Molino & Iacopo Mastromatteo & Michael Benzaquen & Jean-Philippe Bouchaud, 2018. "The Multivariate Kyle model: More is different," Papers 1806.07791, arXiv.org, revised Dec 2018.
    3. Ulrich Horst & Xiaonyu Xia, 2019. "Multi-dimensional optimal trade execution under stochastic resilience," Finance and Stochastics, Springer, vol. 23(4), pages 889-923, October.
    4. Mehdi Tomas & Iacopo Mastromatteo & Michael Benzaquen, 2020. "How to build a cross-impact model from first principles: Theoretical requirements and empirical results," Working Papers hal-02567489, HAL.
    5. L. C. Garcia Del Molino & I. Mastromatteo & Michael Benzaquen & J.-P. Bouchaud, 2020. "The Multivariate Kyle model: More is different," Post-Print hal-02323433, HAL.
    6. Dirk Becherer & Todor Bilarev & Peter Frentrup, 2018. "Optimal liquidation under stochastic liquidity," Finance and Stochastics, Springer, vol. 22(1), pages 39-68, January.
    7. L. C. Garcia Del Molino & I. Mastromatteo & Michael Benzaquen & J.-P. Bouchaud, 2019. "The Multivariate Kyle model: More is different," Working Papers hal-02323433, HAL.

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