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# Multivariate transient price impact and matrix-valued positive definite functions

2 MATHRISK - Mathematical Risk Handling
UPEM - Université Paris-Est Marne-la-Vallée, ENPC - École des Ponts ParisTech, Inria de Paris
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.
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https://hal-enpc.archives-ouvertes.fr/hal-00919895
Contributeur : Aurélien Alfonsi <>
Soumis le : mardi 17 décembre 2013 - 14:49:58
Dernière modification le : mercredi 26 février 2020 - 19:06:17

### Citation

Aurélien Alfonsi, Alexander Schied, Florian Klöck. Multivariate transient price impact and matrix-valued positive definite functions. Mathematics of Operations Research, INFORMS, 2016, ⟨10.1287/moor.2015.0761⟩. ⟨hal-00919895⟩

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