Understanding the worst-kept secret of high-frequency trading

TL;DR

This paper models the relationship between volume imbalance and price movements in high-frequency trading, revealing that optimal quoting strategies are predictive of future price jumps and aiding platform regulation.

Contribution

It introduces a market-making model linking volume imbalance to optimal quoting strategies and provides mathematical proofs for the existence and uniqueness of solutions to the associated PDE system.

Findings

Optimal quoting strategies are predictive of future price jumps.

The model confirms that quoting a predictive imbalance is optimal.

Existence and uniqueness of solutions to the PDE system are established.

Abstract

Volume imbalance in a limit order book is often considered as a reliable indicator for predicting future price moves. In this work, we seek to analyse the nuances of the relationship between prices and volume imbalance. To this end, we study a market-making problem which allows us to view the imbalance as an optimal response to price moves. In our model, there is an underlying efficient price driving the mid-price, which follows the model with uncertainty zones. A single market maker knows the underlying efficient price and consequently the probability of a mid-price jump in the future. She controls the volumes she quotes at the best bid and ask prices. Solving her optimization problem allows us to understand endogenously the price-imbalance connection and to confirm in particular that it is optimal to quote a predictive imbalance. Our model can also be used by a platform to select a suitable tick size, which is known to be a crucial topic in financial regulation. The value function of the market maker's control problem can be viewed 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 classical solutions to this coupled system of equations. In the case of a continuous inventory, we also prove uniqueness of the market maker's optimal control policy.