# Mean-Field Game Strategies for Optimal Execution

Applied Mathematical Finance

30 Pages Posted: 16 Mar 2015 Last revised: 9 Apr 2019

See all articles by Xuancheng Huang

## Xuancheng Huang

university of Toronto

## Sebastian Jaimungal

University of Toronto - Department of Statistics

## Mojtaba Nourian

Bank of Montreal

Date Written: March 15, 2015

### Abstract

Algorithmic trading strategies for execution often focus on the individual agent who is liquidating/acquiring shares. When generalized to multiple agents, the resulting stochastic game is notoriously difficult to solve in closed-form. Here, we circumvent the difficulties by investigating a mean-field game framework containing (i) a major agent who is liquidating a large number of shares, (ii) a number of minor agents (high-frequency traders (HFTs)) who detect and trade against the liquidator, and (iii) noise traders who buy and sell for exogenous reasons. Our setup accounts for permanent price impact stemming from all trader types inducing an interaction between major and minor agents. Both optimizing agents trade against noise traders as well as one another. This stochastic dynamic game contains couplings in the price and trade dynamics, and we use a mean-field game approach to solve the problem. We obtain a set of decentralized feedback trading strategies for the major and minor agents, and express the solution explicitly in terms of a deterministic fixed point problem. For a finite $N$ population of HFTs, the set of major-minor agent mean-field game strategies is shown to have an $\epsilon_N$-Nash equilibrium property where $\epsilon_N\to0$ as $N\to\infty$.

Keywords: Algorithmic trading, high-frequency trading, optimal execution, stochastic optimal control, mean-field games, market microstructure

JEL Classification: C06, C61, C63, G01, G10, G12, G17, G60

Suggested Citation

Huang, Xuancheng and Jaimungal, Sebastian and Nourian, Mojtaba, Mean-Field Game Strategies for Optimal Execution (March 15, 2015). Applied Mathematical Finance, Available at SSRN: https://ssrn.com/abstract=2578733 or http://dx.doi.org/10.2139/ssrn.2578733