Boltzmann Price: Toward Understanding the Fair Price in High-Frequency Markets

ArXiv ID: 2507.09734 “View on arXiv”

Authors: Przemysław Rola

Abstract

In this paper, we introduce a parametrized family of prices derived from the Maximum Entropy Principle. The price is obtained from the distribution that minimizes bias, given the bid and ask volume imbalance at the top of the order book. Under specific parameter choices, it closely approximates the mid-price or the weighted mid-price. Using probabilities of bid and ask states, we propose a model of price dynamics in which both drift and volatility are driven by volume imbalance. Compared to standard models like Bachelier or Geometric Brownian Motion with constant volatility, our model can generate higher kurtosis and heavy-tailed distributions. Additionally, the drift term naturally emerges as a consequence of the order book imbalance. We validate the model through simulation and demonstrate its fit to historical equity data. The model provides a theoretical framework, integrating price, volume imbalance, and spread.

Keywords: Maximum entropy principle, Order book imbalance, Price dynamics, Volume imbalance, Microstructure

Complexity vs Empirical Score

• Math Complexity: 7.0/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced mathematics, including maximum entropy principles and derivations of probability distributions and price dynamics, leading to a high math complexity score. It also demonstrates empirical validation through simulation and fits to historical equity data, indicating a significant level of implementation and backtesting rigor.

  flowchart TD
    A["Research Goal:<br>Define Fair Price in High-Frequency Markets"] --> B["Methodology:<br>Maximum Entropy Principle"]
    B --> C["Input:<br>Bid/Ask Volume Imbalance"]
    C --> D["Computation:<br>Maximize Entropy given Constraints"]
    D --> E["Boltzmann Price<br>Parametric Family"]
    E --> F["Outcome 1:<br>Drift driven by Imbalance"]
    E --> G["Outcome 2:<br>High Kurtosis & Heavy Tails"]
    F --> H["Outcome 3:<br>Validation on Equity Data"]
    G --> H