Understanding the worst-kept secret of high-frequency trading

ArXiv ID: 2307.15599 “View on arXiv”

Authors: Unknown

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.

Keywords: limit order book, volume imbalance, stochastic control, coupled PDEs, Equities / Market Making

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper involves advanced stochastic control, PDE theory with existence/uniqueness proofs, and heavy use of measure theory and functional analysis, placing math complexity very high; however, the empirical section relies on numerical approximations of the HJB equation without backtests, real data, or implementation details, resulting in low empirical rigor.

  flowchart TD
    A["Research Goal:<br>Analyze the relationship between<br>volume imbalance & price moves"] --> B["Methodology:<br>Market-Making Optimization Problem"]

    B --> C["Key Inputs & Model:<br>• Underlying efficient price<br>• Uncertainty zones for mid-price<br>• Market maker's inventory level"]

    C --> D["Computational Process:<br>Solve Coupled PDE System<br>(Value function V(q,t) for inventory q)"]

    D --> E["Theoretical Proof:<br>Existence & Uniqueness of<br>Classical Solutions"]

    D --> F["Optimal Policy Derivation:<br>Control of bid/ask quote volumes"]

    E --> G["Key Outcome 1:<br>Price-Imbalance Relationship<br>Derived endogenously from optimization"]
    F --> H["Key Outcome 2:<br>Predictive Imbalance<br>Confirmed as optimal strategy"]
    F --> I["Key Outcome 3:<br>Tick Size Selection Model<br>Framework for regulators/platforms"]