Multivariate Transient Price Impact and Matrix-Valued Positive Definite Functions

• 成果类型: Article
• 署名作者: Alfonsi, Aurelien; Kloeck, Florian; Schied, Alexander
• 署名单位: Institut Polytechnique de Paris; Ecole des Ponts ParisTech; Universite Gustave-Eiffel; University of Mannheim
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2015.0761
• 发表日期: 2016
• 页码: 914-934
• 关键词: optimal execution; ORDER

摘要

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, non-negative, and convex decay kernels with values in a space of symmetric 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 particularly well-behaved if 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.