Multivariate transient price impact and matrixvalued positive definite functions
Abstract
We consider a model for linear transient price impact for multiple assets that takes crossasset 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 wellbehaved optimal trade execution strategies. We first show that the existence of such strategies is guaranteed by assuming that the decay kernel corresponds to a matrixvalued positive definite function. An example illustrates, however, that positive definiteness alone does not guarantee that optimal strategies are wellbehaved. Building on previous results from the onedimensional 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 wellbehaved 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2013
 DOI:
 10.48550/arXiv.1310.4471
 arXiv:
 arXiv:1310.4471
 Bibcode:
 2013arXiv1310.4471A
 Keywords:

 Quantitative Finance  Trading and Market Microstructure;
 Mathematics  Optimization and Control;
 42A82;
 90C20;
 91G80;
 91G10