Optimal market making with competition

Abstract

Competition between market makers, which considers the impacts on trading strat egy of individual and liquidity of whole market resulting from multiple market mak ers competing for order flow and market maker incentives, was not properly studied in the literature of optimal market making problem. This thesis is devoted to the optimal market making problem, with competition between market makers. Three main topics are studied in this thesis. In the first topic, we consider the price competition between market makers. We discuss optimal market marking with price competition and incomplete information, which results in a looping dependence structure among market makers. We solve the problem with the non-zero-sum stochastic differential game approach and char acterize the equilibrium value function with a coupled system of nonlinear ordinary differential equations. We prove, do not assume a priori, that the Issac condition is satisfied, which ensures the existence and uniqueness of Nash equilibrium. We also perform some numerical tests that show our model produces tighter bid/ask spread than a benchmark model without price competition and improves market liquidity. In the second topic, we consider market makers competing for the market maker incentive reward proposed by exchange, which depends on their trading volume ranking. We model the competition as a stochastic mean field game, which can be further reduced to a finite state mean field game, whose equilibrium is characterized by a forward backward ODE systems. We numerically solve the equilibrium with the deep neural network approach proposed in our third topic, and perform some numerical tests to compare bid/ask spread under different types market maker in centive reward. It is suggested that the introduction of incentive can reduce the implicit trading cost, and rank-based reward, compared with the linear trading vol ume reward, can produce lower best bid/ask spread. In the third topic, we discuss the deep neural network approach for solving the forward backward ODE system corresponding to a more general class of finite state mean field game, and the game in the second topic is just a special case of it. We prove that the error between true solution and our approximation is linear to the square root of loss function of our deep neural network.