Reinforcement Learning in High-frequency Market Making

TL;DR

This paper provides a theoretical analysis of reinforcement learning in high-frequency market making, exploring the effects of sampling frequency and convergence to continuous-time equilibria, supported by Monte Carlo simulations.

Contribution

It bridges RL theory with continuous-time financial models and analyzes the tradeoff between error and complexity as sampling frequency varies, including multi-agent convergence results.

Findings

Smaller time increments reduce error but increase complexity.

Nash equilibrium converges to continuous-time equilibrium as sampling frequency increases.

Monte Carlo simulations support the theoretical findings.

Abstract

This paper establishes a new and comprehensive theoretical analysis for the application of reinforcement learning (RL) in high-frequency market making. We bridge the modern RL theory and the continuous-time statistical models in high-frequency financial economics. Different with most existing literature on methodological research about developing various RL methods for market making problem, our work is a pilot to provide the theoretical analysis. We target the effects of sampling frequency, and find an interesting tradeoff between error and complexity of RL algorithm when tweaking the values of the time increment $\Delta$ $-$ as $\Delta$ becomes smaller, the error will be smaller but the complexity will be larger. We also study the two-player case under the general-sum game framework and establish the convergence of Nash equilibrium to the continuous-time game equilibrium as $\Delta\rightarrow0$. The Nash Q-learning algorithm, which is an online multi-agent RL method, is applied to solve the equilibrium. Our theories are not only useful for practitioners to choose the sampling frequency, but also very general and applicable to other high-frequency financial decision making problems, e.g., optimal executions, as long as the time-discretization of a continuous-time markov decision process is adopted. Monte Carlo simulation evidence support all of our theories.