Multivariate transient price impact and matrix-valued positive definite functions

Aurélien Alfonsi (), Alexander Schied and Florian Klöck
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Aurélien Alfonsi: CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique - ENPC - École nationale des ponts et chaussées, MATHRISK - Mathematical Risk Handling - UPEM - Université Paris-Est Marne-la-Vallée - ENPC - École nationale des ponts et chaussées - Centre Inria de Paris - Inria - Institut National de Recherche en Informatique et en Automatique
Alexander Schied: Department of Mathematics and Computer Science [Mannheim] - University of Mannheim = Universität Mannheim
Florian Klöck: Department of Mathematics and Computer Science [Mannheim] - University of Mannheim = Universität Mannheim

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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, nonnegative and convex decay kernels with values in the symmetric $K\times K$ 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 well-behaved when 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.

Keywords: Multivariate price impact; matrix-valued positive de nite function; optimal trade execution; optimal portfolio liquidation; matrix function (search for similar items in EconPapers)
Date: 2016-03-01
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Citations: View citations in EconPapers (12)

Published in Mathematics of Operations Research, 2016, ⟨10.1287/moor.2015.0761⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-00919895

DOI: 10.1287/moor.2015.0761

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