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];

Cited References (47)

  1. Event-Time Order-Flow Memory, Operational-Time Impact, and Subordinated Market Observables (2026) — link
  2. Revisiting Trade-sign Long-memory and Square-root Law price impact (2026) — link
  3. Non-unique time and market incompleteness (2026) — link
  4. Scaling Laws for the Market Microstructure of the Interdealer Broker Markets (1998) — link
  5. Analyzing and modelling 1+1d markets (2001) — link
  6. How does latent liquidity get revealed in the limit order book? (2019) — link
  7. Anomalous Price Impact and the Critical Nature of Liquidity in Financial Markets (2011) — link
  8. A fully consistent, minimal model for non-linear market impact (2015) — link
and 39 more
  1. Market impact with multi-timescale liquidity (2018) — link
  2. Agent-based models for latent liquidity and concave price impact (2014) — link
  3. Anomalous diffusion and price impact in the fluid-limit of an order book — → 2310.06079
  4. A simple microstructural explanation of the concavity of price impact (2022) — link
  5. Trades, Quotes and Prices: Financial Markets Under the Microscope (2018) — link
  6. Price variations in a stock market with many agents (1997) — link
  7. Statistical theory of the continuous double auction (2003) — link
  8. Statistical properties of stock order books: empirical results and models (2002) — link
  9. Relation between bid–ask spread, impact and volatility in order-driven markets (2008) — link
  10. The Supply of Dealer Services in Securities Markets (1978) — link
  11. Dealership market: Market-making with inventory (1980) — link
  12. Optimal dealer pricing under transactions and return uncertainty (1981) — link
  13. Bid, ask and transaction prices in a specialist market with heterogeneously informed traders (1985) — link
  14. Liquidity and Market Structure (1988) — link
  15. The Components of the Bid-Ask Spread: A General Approach (1997) — link
  16. High-frequency trading in a limit order book (2008) — link
  17. Dealing with the inventory risk: a solution to the market making problem (2013) — link
  18. Fluctuations and response in financial markets: The subtle nature of random price changes (2004) — link
  19. No-dynamic-arbitrage and market impact (2010) — link
  20. Market impact and trading profile of hidden orders in stock markets (2009) — link
  21. The non-linear market impact of large trades: Evidence from buy-side order flow (2013) — link
  22. How markets slowly digest changes in supply and demand (2009)
  23. Theory for long memory in supply and demand (2005) — link
  24. Inferring Microscopic Financial Information from the Long Memory in Market-Order Flow: A Quantitative Test of the Lillo–Mike–Farmer Model (2023) — link
  25. Exact Solution to a Generalised Lillo–Mike–Farmer Model with Heterogeneous Order-Splitting Strategies — → 2306.13378
  26. The Long Memory of the Efficient Market (2004) — link
  27. How efficiency shapes market impact (2013) — link
  28. The Long Memory of Order Flow in the Foreign Exchange Spot Market (2016) — link
  29. A discrete time random walk model for anomalous diffusion (2015) — link
  30. Stochastic Models for Fractional Calculus (2012)
  31. Random Walks on Lattices. II (1965) — link
  32. The Random Walk’s Guide to Anomalous Diffusion: A Fractional Dynamics Approach (2000) — link
  33. Limit Theorems for Continuous-Time Random Walks with Infinite Mean Waiting Times (2004) — link
  34. Limit Theorems for Coupled Continuous Time Random Walks (2004) — link
  35. The M -Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey (2010) — link
  36. A subordinated stochastic process model with finite variance for speculative prices (1973) — link
  37. Order Flow, Transaction Clock, and Normality of Asset Returns (2000) — link
  38. The impact of asynchronous trading on Epps effect. Comparative study on Warsaw Stock Exchange and Vienna Stock Exchange (2016) — link
  39. Revisiting the Epps effect using volume time averaging: An exercise in R (2020) — link

2 of 47 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.