🤖 AI Summary
This study challenges the conventional view that liquidity, supply, and demand are fundamental economic variables, arguing instead that they emerge from the geometric structure induced by order book observations. By modeling the market as an expanding relational system devoid of predefined metrics, time, or price coordinates, and applying spectral embedding of the graph Laplacian to obtain a one-dimensional projection, the authors derive a price-like coordinate and a corresponding liquidity distribution. Remarkably, this approach reproduces canonical order book regularities without invoking assumptions about agent behavior. Using high-frequency Level II data from U.S. equities, the research demonstrates the cross-asset universality of a cumulative gamma-shaped liquidity profile, with information criteria confirming its superior fit compared to existing models.
📝 Abstract
We propose a structural framework for the geometry of financial order books in which liquidity, supply, and demand are treated as emergent observables rather than primitive economic variables. The market is modeled as an inflationary relational system without assumed metric, temporal, or price coordinates. Observable quantities arise only through projection, implemented here via spectral embeddings of the graph Laplacian. A one-dimensional projection induces a price-like coordinate, while the projected density defines liquidity profiles around the mid price. Under a minimal single-scale hypothesis -- excluding intrinsic length scales beyond distance to the mid and finite visibility -- we show that projected supply and demand are constrained to gamma-like functional forms. In discrete data, this prediction translates into integrated-gamma cumulative profiles. We test these results using high-frequency Level~II data for several U.S. equities and find robust agreement across assets and intraday windows. Explicit comparison with alternative cumulative models using information criteria demonstrates a systematic preference for the integrated-gamma geometry. A minimal simulation of inflationary relational dynamics reproduces the same structure without invoking agent behavior or price formation mechanisms. These results indicate that key regularities of order-book liquidity reflect geometric constraints induced by observation rather than detailed microstructural dynamics. Supplementary Material is available at the arXiv submission.