This paper develops a comprehensive statistical theory of the continuous double auction, the mechanism underlying most modern financial markets. The authors model order flow as independent Poisson processes for market orders, limit orders, and cancellations, with constant order size and logarithmic prices. Using dimensional analysis, they reduce the five model parameters (market order rate μ, limit order rate α, cancellation rate δ, tick size dp, and order size σ) to two nondimensional parameters: the granularity ϵ = 2δσ/μ and the nondimensional tick size dp/pc = 2α dp/μ. This reduction reveals that order granularity—the relative size of orders compared to the characteristic number of shares—is a more significant determinant of market behavior than tick size.

The study combines simulations, dimensional analysis, and mean-field approximations to derive testable predictions for key market properties. The average depth profile of the limit order book decays exponentially away from the midpoint, with asymptotic depth d = α/δ. The bid-ask spread scales as s ≈ μ/α, modified by a function of ϵ and tick size. Price diffusion is anomalous: the short-time diffusion coefficient scales as D₀ ∝ ϵ⁻⁰·⁵, while the long-time diffusion scales as D∞ ∝ ϵ⁰·⁵, indicating that granularity strongly influences volatility. The price impact function is found to be concave, consistent with empirical observations, and the model provides an explanation for this concavity through the granular nature of order flow. Additionally, the probability and time to fill limit orders are derived as functions of distance from the midpoint and granularity.

The work demonstrates that zero-intelligence stochastic models, despite their simplicity, can generate strong, falsifiable predictions about market microstructure. By splitting the problem into order flow determination and price formation, the model provides a foundation for understanding how market institutions affect liquidity, volatility, and transaction costs. The authors emphasize that all predictions are based on directly measurable quantities, leaving no free parameters, and suggest that the model can be extended to incorporate more realistic behaviors such as correlated order flow or strategic order placement.