Stochastic Differential Game in High Frequency Market

Abstract

This paper presents an application of a linear quadratic stochastic differential game to a financial model that describes trading behaviors of different types of players in a high frequency stock market. Stability of the stock market in a high frequency environment is a central issue in financial markets. Building a model that expresses the trading behaviors of the different types of players and the price actions in a high frequency market helps introduce better regulations for a healthy market. Firstly, we represent trading behaviors of the three types of players, algorithmic traders, general traders, and market makers as well as the mid-price process of a risky asset by a linear quadratic stochastic differential game. Secondly, we obtain a Nash equilibrium for open loop admissible strategies by solving a forward-backward stochastic differential equation (FBSDE) derived from the stochastic maximum principle. Finally, we present numerical examples of the Nash equilibrium for open loop admissible strategies and the corresponding price action of the risky asset, which agree with the empirical findings on the mechanism of high frequency markets. This model can be used to investigate the impact of regulation changes on the market stability as well as trading strategies of the players.