Multivariate transient price impact and matrix-valued positive definite functions
AbstractWe 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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Bibliographic InfoPaper provided by arXiv.org in its series Papers with number 1310.4471.
Date of creation: Oct 2013
Date of revision: Oct 2013
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Web page: http://arxiv.org/
This paper has been announced in the following NEP Reports:
- NEP-ALL-2013-10-25 (All new papers)
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