Understanding the worst-kept secret of high-frequency trading

Volume imbalance in a limit order book is often considered as a reliable indicator for predicting future price movements. In this study, we confirm this statement by analyzing an optimal control problem in which a market maker controls volumes in the limit order book of a large-tick stock and quotes prices at a half-tick distance from the mid-price. We model the mid-price, which is not a controlled variable, using uncertainty zones. The market maker has information about the underlying efficient price and consequently of the probability of a price jump in the future. By using this information, it is optimal for the market maker to create imbalances which are predictive of price movements. The value function of the market maker's control problem can be understood as a family of functions, indexed by the level of the market maker's inventory, solving a coupled system of PDEs. We show existence and uniqueness of smooth solutions for this coupled system of equations. In the case of a continuous inventory, we also prove the uniqueness of the market maker's optimal control policy. This is joint work with Sergio Pulido and Mathieu Rosenbaum.