Optimal execution with deterministically time varying liquidity: well posedness and price manipulation

TL;DR

This paper analyzes the well-posedness and price manipulation issues in optimal trade execution models with time-varying market impact parameters, providing conditions for unique solutions and manipulation prevention.

Contribution

It offers new conditions ensuring well-posedness and no price manipulation in models with deterministic impact parameter changes over time.

Findings

Established sufficient conditions for unique solutions.

Derived second-order optimality conditions.

Identified scenarios preventing transaction-triggered manipulation.

Abstract

We investigate the well-posedness in the Hadamard sense and the absence of price manipulation in the optimal execution problem within the Almgren-Chriss framework, where the temporary and permanent impact parameters vary deterministically over time. We present sufficient conditions for the existence of a unique solution and provide second-order conditions for the problem, with a particular focus on scenarios where impact parameters change monotonically over time. Additionally, we establish conditions to prevent transaction-triggered price manipulation in the optimal solution, i.e. the occurence of buying and selling in the same trading program. Our findings are supported by numerical analyses that explore various regimes in simple parametric settings for the dynamics of impact parameters.