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

TL;DR

This paper introduces a new price modeling approach based on maximum entropy principles that accounts for order book imbalance, capturing realistic market features like heavy tails and dynamic drift, validated with historical data.

Contribution

It proposes a novel parametrized price model derived from entropy principles that links volume imbalance to price dynamics, improving upon traditional models.

Findings

Model captures heavy tails and kurtosis in returns.

Price drift naturally arises from order book imbalance.

Model fits historical equity data well.

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.