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 × 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 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.

Cross-impact and no-dynamic-arbitrage

“…in agreement with Proposition 3.7. (b) of Alfonsi et al [2016] and equivalent to the condition for a symmetric 2 × 2 matrix to be positive-semidefinite.…”

“…Note that this excludes kernels where for some ij G ij (0) = 0 but G ij (τ ) > 0 for some τ > 0. Indeed Example 3 in Alfonsi et al [2016] has a kernel that is asymmetric for τ > 0 and which does not allow price manipulation.…”

The Multivariate Kyle Model: More is Different

“…The first two properties are extremely important in the calibration of cross-impact models, as shown in [AKS16,SL18]: to perform pricing within a cross-impact setup, the matrix Λ should be SPD in order to ensure absence of price manipulation. This is not the case for a MLE, which is thus not suitable for practical purposes.…”

Multi-dimensional optimal trade execution under stochastic resilience

“…Existence of an optimal liquidation strategy in their model is guaranteed if the decay kernel corresponds to a matrix-valued positive definite function but additional assumptions on the decay kernel are required for the optimal liquidation strategies to be well-behaved. In their recent paper, Schneider and Lillo [20] established necessary conditions on the size and impact of cross-impact for the absence of dynamic arbitrage in a continuous time version of [1] that can be directly verified on data.…”

“…He characterized the value function as the solution to the two-dimensional PDEs and showed that the optimal portfolio process depends only on the co-variance and crossasset market impact of the assets in his setting. Alfonsi et al [1] considered a discrete-time model of optimal basket liquidation with linear transient price impact and general deterministic resilience. Existence of an optimal liquidation strategy in their model is guaranteed if the decay kernel corresponds to a matrix-valued positive definite function but additional assumptions on the decay kernel are required for the optimal liquidation strategies to be well-behaved.…”