HAL CCSD
Multivariate transient price impact and matrix-valued positive definite functions
Alfonsi, Aurélien
Schied, Alexander
Klöck, Florian
Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS) ; École des Ponts ParisTech (ENPC)
Mathematical Risk Handling (MATHRISK) ; Université Paris-Est Marne-la-Vallée (UPEM)-École des Ponts ParisTech (ENPC)-Inria de Paris ; Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
Department of Mathematics and Computer Science [Mannheim] ; Universität Mannheim
International audience
ISSN: 0364-765X
EISSN: 1526-5471
Mathematics of Operations Research
INFORMS
hal-00919895
https://enpc.hal.science/hal-00919895
https://enpc.hal.science/hal-00919895
Mathematics of Operations Research, 2016, ⟨10.1287/moor.2015.0761⟩
ARXIV: 1310.4471
info:eu-repo/semantics/altIdentifier/arxiv/1310.4471
DOI: 10.1287/moor.2015.0761
info:eu-repo/semantics/altIdentifier/doi/10.1287/moor.2015.0761
en
Multivariate price impact
matrix-valued positive de nite function
optimal trade execution
optimal portfolio liquidation
matrix function
[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
[QFIN.TR]Quantitative Finance [q-fin]/Trading and Market Microstructure [q-fin.TR]
info:eu-repo/semantics/article
Journal articles
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.
2016-03-01