Reinforcement Learning for Optimal Market Making with the Presence of Rebate

Abstract

We propose a reinforcement learning (RL) framework to solve the HJB equations of optimal market making with the presence of rebate. As a numerical solution, the RL algorithm successfully mirrors the analytical solutions under the scheme of no rebate and constant rebate. Under the time-dependent rebate scheme, there is no closed form and RL provides a plausible solution. We investigate the numerical performance of the RL solutions in simulations, which show that the RL solutions deliver stable accuracy in various situations and are robust to estimation errors. Moreover, the RL solutions demonstrate the impact of rebate on the behaviour of market makers (MMs) and the quality of market. In particular, the presence of a rebate stimulates MM to quote with narrower spreads on both sides of order books and the rebate is fully transferred to the end customers, which is consistent with the theoretical results in the analytical solutions. It also improves market quality by increasing the total trading volume and providing more terminal wealth to MMs. Finally, the time-dependent rebate scheme is found to be more cost efficient than a constant rebate.

Keywords: Optimal market making, Rebate, Reinforcement learning

JEL Classification: C6, G1