Optimal Trade Execution with Instantaneous Price Impact and Stochastic Resilience

TL;DR

This paper develops a mathematical framework for optimal trade execution in illiquid markets considering both immediate and lasting price impacts, stochastic resilience, and continuous trading strategies, characterized by a complex BSDE system.

Contribution

It introduces a novel BSDE-based model for optimal execution with stochastic resilience and proves existence and uniqueness of solutions, enabling precise strategy characterization.

Findings

Existence and uniqueness of solutions to the BSDE system.

Characterization of the value function and optimal strategy.

Asymptotic expansion method at terminal time.

Abstract

We study an optimal execution problem in illiquid markets with both instantaneous and persistent price impact and stochastic resilience when only absolutely continuous trading strategies are admissible. In our model the value function can be described by a three-dimensional system of backward stochastic differential equations (BSDE) with a singular terminal condition in one component. We prove existence and uniqueness of a solution to the BSDE system and characterize both the value function and the optimal strategy in terms of the unique solution to the BSDE system. Our existence proof is based on an asymptotic expansion of the BSDE system at the terminal time that allows us to express the system in terms of a equivalent system with finite terminal value but singular driver.