Since January 2024, the German Academic Exchange Service has supported the project “Modeling and Simulation of Illiquid Financial Markets” (ILLMAR), led by Thomas Kruse (Chair of Applied and Computational Mathematics at the University of Wuppertal), through its program for project-related personnel exchange with France (PROCOPE). The project’s French partner is Alexandre Popier’s research group at the Laboratoire Manceau de Mathématiques at Le Mans University. The funded project focuses on developing risk management strategies in illiquid financial markets.
Price Impact and Execution Costs
When a large institutional investor seeks to buy or sell a significant amount of an asset, it can significantly impact market prices. This is especially true in markets with limited liquidity, such as certain energy markets. If the available supply (or demand) at the current price is insufficient, the investor creates price impact and must execute the remainder of the position at progressively less favorable prices. As a result, their own trading pushes the price against them, generating what are known as execution costs.
To mitigate these costs, the investor may choose to space out transactions over a set period rather than executing the entire position at once. Dividing a large order into a sequence of smaller trades often reduces market impact and achieves a better overall execution price. This naturally leads to the optimal trade execution problem: How should the timing and size of each trade be scheduled to minimize total execution costs while fulfilling the desired buy-or-sell objective within a given time frame?
Stochastic price impact:
Liquidity in financial markets is not constant, but rather evolves randomly over time. In the context of trade execution, there has recently been a growing interest in the scientific literature on the effects of stochastic liquidity on optimal trade execution strategies. In this context, backward stochastic differential equations (BSDEs) have emerged as a powerful mathematical tool for characterizing optimal trade execution strategies.
Project goals with the Le Mans team
The project, funded by the DAAD, aims to analyze two important aspects of optimal trade execution that are currently under-explored in the scientific literature. These aspects will be examined in collaboration with the partner team in Le Mans. First, we aim to set up multidimensional trade execution problems with stochastic liquidity parameters. Second, we will investigate how investors with positions in multiple illiquid assets use the interdependencies between the liquidity structures of individual assets to mitigate their overall liquidity risk. Specifically, we use recently developed tools from the theory of BSDEs to characterize multi-asset optimal liquidation strategies and reveal cross-hedging effects.
Second, we develop efficient numerical schemes tailored for approximating the BSDEs that arise in optimal trade execution problems. Designing such algorithms is challenging since many BSDEs in optimal trade execution have singular terminal conditions, complicating their analysis. Recently, machine learning methods have shown remarkable success in solving partial differential equations and their associated BSDEs, particularly in high-dimensional settings. In this project, we apply the group’s expertise to improve numerical methods for BSDEs using state-of-the-art machine learning techniques. This paves the way for a simulation-based assessment of liquidity risks in the context of trade execution.
The final phase
The project has progressed very successfully, with several key objectives already achieved. In particular, several junior scientists gained valuable international experience through research stays at the partner university. Additionally, new research questions have emerged, paving the way for continued collaboration between the teams and fostering a partnership between the two institutions.
In the final phase of the project, members are synthesizing their research findings. The figure above illustrates some of these results by presenting simulation results for a specific trade execution problem with stochastic price impact. The orange line shows one realization of the stochastic price impact process. The blue line depicts the corresponding solution path of the BSDE with a singular terminal condition, and the green line represents the associated optimal position path.