🤖 AI Summary
This study addresses the limitation of traditional limit order book models in neglecting the geometric essence of market microstructure. The authors propose a relational pre-geometric framework that eschews predefined metrics, time, or price coordinates. By applying degree reduction and low-dimensional spectral projection to the transactional interaction structure, price-like coordinates and liquidity density emerge naturally, revealing the complementarity between bid and ask sides. A key innovation lies in decomposing liquidity imbalance into rigid drift and geometric shear modes—the latter reshapes order book structure without inducing price changes and yields a gamma-like liquidity distribution under shear constraints. Empirical validation using high-frequency Level II data from U.S. equities demonstrates that the model significantly outperforms conventional cumulative models in both goodness-of-fit and residual diagnostics.
📝 Abstract
We introduce a structural framework for the geometry of financial order books in which liquidity, supply, and demand are treated as emergent observables rather than primitive market variables. The market is modeled as a relational substrate without assumed metric, temporal, or price coordinates. Observable quantities arise only through observation, implemented here as a reduction of relational degrees of freedom followed by a low-dimensional spectral projection. A one-dimensional projection induces a price-like coordinate and a projected liquidity density around the mid price, from which bid and ask sides emerge as two complementary restrictions. We show that directional liquidity imbalances decompose naturally into a rigid drift of the projected density and a geometric shear mode that deforms the bid--ask structure without inducing price motion. Under a minimal single-scale hypothesis, the shear geometry constrains the projected liquidity to a gamma-like functional form, appearing as an integrated-gamma profile in discrete data. Empirical analysis of high-frequency Level~II data across multiple U.S. equities confirms this geometry and shows that it outperforms standard alternative cumulative models under explicit model comparison and residual diagnostics.