Paper: arXiv 2610.07985

Authors: Christopher Angstmann, Derick Diana, Tim Gebbie

Abstract

We give a unified analytic account of a finite bid–ask spread in a two-field reaction–diffusion order book. The model retains separate bid and ask densities in operational time, with diffusion, cancellation, reaction and external order creation. On the symmetric equal-coefficient branch, the imbalance field determines the reaction price, while the total standing density carries the spread geometry. A market-making source is introduced by withholding order placement over a finite interval around the reaction boundary, representing round-trip replenishment displaced from the quote. Under a separated, quasistationary and weak-overlap approximation, transport penetrates inward from the placement edges over the cancellation length, giving the quoted spread as the sub-threshold core of the placement-source-free interval. Market orders remove standing density at the observed quote, whereas replenishment is delivered outside it and must diffuse back across the penetration depth. We derive the resulting delayed and attenuated quote-level response in operational time. On that branch, finite spread and price impact arise from distinct response sectors, while their calendar-time appearance requires a separate observation clock.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly mathematical and theoretical model of bid-ask spread dynamics using reaction-diffusion equations. While the mathematical framework is sophisticated and novel, there is no empirical validation or backtesting presented, placing it firmly in the ‘Lab Rats’ quadrant. The clarity is good for a highly technical paper, but the lack of empirical rigor limits its immediate practical applicability.

Research Flowchart

  flowchart TD
    A[Research Goal: Explain Finite Bid-Ask Spread] --> B{Methodology: Reaction-Diffusion Order Book Model};
    B -- Model Inputs --> C[Inputs: Separate Bid/Ask Densities, Diffusion, Cancellation, Reaction, External Order Creation, Market-Making Source];
    C -- Computational Processes --> D{Processes: Symmetric Equal-Coefficient Branch, Quasistationary/Weak-Overlap Approx.};
    D -- Outcomes --> E[Outcomes: Finite Bid-Ask Spread, Delayed/Attenuated Quote-Level Response, Price Impact];