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Multivariate transient price impact and matrix-valued positive definite functions

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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Soumis le : mardi 17 décembre 2013 - 14:49:58
Dernière modification le : vendredi 8 juillet 2022 - 10:09:38

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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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