[Paper Review] Numerical analysis of the Minimal and Two-Liquid models of the Market Microstructure

David L. C. Chan, David Eliezer+1 more|arXiv (Cornell University)|Jan 31, 2001

Complex Systems and Time Series Analysis14 references8 citations

TL;DR

This paper presents numerical simulations of the Minimal and Two-Liquid models of market microstructure, treating buyers and sellers as diffusing particles that annihilate upon meeting. It confirms scaling laws for midmarket variance, bid-offer spread, and time to midmarket sale, showing they scale as $ D/J $, with a logarithmic correction for midmarket variance, and validates analytical predictions for asymmetric fluxes and spread dynamics in the two-liquid model.

ABSTRACT

We present results of numerical analysis of several simple models for the microstructure of a double auction market without intermediaries which were introduced in cond-mat/9808240. We perform computer simulations of the minimal model in order to verify liquidity scaling laws. A logarithmic correction to the scaling law for midmarket variance is observed, but not for bid-offer spread or its fluctuation, because they are fundamentally different quantities. Time to midmarket sale ($τ_S$) is found to scale as 1/J while its fluctuation goes as $0.73/J$. A ``reduced'' time ($τ_{reduced}$) is also studied, and found to scale in a non-trivial way. Asymmetric fluxes are introduced to the minimal model and analytical result derived earlier for the speed of the moving midmarket agrees with numerical results. Simulation of the two-liquid model which describes a market with both market order and limit order traders, reveals widening of the bid-offer spread when the flux of market order traders exceeds that of limit order traders. The variation of the spread with the fraction of market-order traders is investigated. The formula for asymmetric fluxes is applied to the two-liquid model and its predictions are found to agree with experiment. The critical point is approximately determined, and the ratio of the midmarkets for $f = 0.0$ and $f = 0.5$ (where $f$ is the fraction of market-order traders) is calculated.

Motivation & Objective

• To verify proposed liquidity scaling laws in the Minimal model of market microstructure through numerical simulations.
• To investigate the dynamics of bid-offer spread, midmarket variance, and time to midmarket sale in a diffusion-annihilation framework.
• To extend the model to include both limit and market order traders via the Two-Liquid model and analyze spread behavior under varying order fluxes.
• To test analytical predictions for asymmetric fluxes and midmarket drift in both models.
• To determine critical points and scaling parameters in the transition between market states.

Proposed method

• Modeling the market as a one-dimensional system of diffusing buyers and sellers with annihilation upon meeting.
• Using Monte Carlo simulations to numerically solve the diffusion-annihilation process with time-continuous dynamics.
• Defining key observables: midmarket variance, bid-offer spread, time to midmarket sale ($ \tau_S $), and reduced time ($ \tau_{\text{reduced}} $).
• Introducing asymmetric fluxes to simulate momentum trading and testing analytical drift predictions.
• Applying the master equation to derive the diffusion coefficient $ D = a^2 p / (2\tau) $, with $ D = 1/2 $ in simulations for $ a = \tau = 1 $.
• Using ensemble averaging and variance reconstruction from averaged data to compute statistical moments accurately.

Experimental results

Research questions

• RQ1 How do midmarket variance, bid-offer spread, and their fluctuations scale with diffusion coefficient $ D $ and deal rate $ J $?
• RQ2 What is the functional form of the time to midmarket sale ($ \tau_S $) and its fluctuation in the Minimal model?
• RQ3 How does the introduction of asymmetric fluxes affect the drift of the midmarket price, and does it match analytical predictions?
• RQ4 How does the bid-offer spread in the Two-Liquid model vary with the fraction of market-order traders?
• RQ5 What is the critical point in the Two-Liquid model, and how does the midmarket ratio change between $ f = 0.0 $ and $ f = 0.5 $?

Key findings

• Midmarket variance scales as $ D/J $ with a logarithmic correction, while bid-offer spread and its fluctuation scale as $ D/J $ without logarithmic corrections.
• Time to midmarket sale ($ \tau_S $) scales as $ 1/J $, and its fluctuation scales as $ 0.73/J $, indicating non-trivial temporal dynamics.
• Reduced time ($ \tau_{\text{reduced}} $) exhibits non-trivial scaling, suggesting complex temporal structure in market efficiency.
• The analytical prediction for midmarket drift under asymmetric fluxes is confirmed numerically, validating the model's predictive power.
• In the Two-Liquid model, the bid-offer spread widens when the flux of market-order traders exceeds that of limit-order traders.
• The critical point is approximately determined, and the ratio of midmarket values at $ f = 0.0 $ and $ f = 0.5 $ is calculated, showing a significant shift in market equilibrium.