Exact solution to a generalised Lillo-Mike-Farmer model with heterogeneous order-splitting strategies

ArXiv ID: 2306.13378 “View on arXiv”

Authors: Unknown

Abstract

The Lillo-Mike-Farmer (LMF) model is an established econophysics model describing the order-splitting behaviour of institutional investors in financial markets. In the original article (LMF, Physical Review E 71, 066122 (2005)), LMF assumed the homogeneity of the traders’ order-splitting strategy and derived a power-law asymptotic solution to the order-sign autocorrelation function (ACF) based on several heuristic reasonings. This report proposes a generalised LMF model by incorporating the heterogeneity of traders’ order-splitting behaviour that is exactly solved without heuristics. We find that the power-law exponent in the order-sign ACF is robust for arbitrary heterogeneous intensity distributions. On the other hand, the prefactor in the ACF is very sensitive to heterogeneity in trading strategies and is shown to be systematically underestimated in the original homogeneous LMF model. Our work highlights that the ACF prefactor should be more carefully interpreted than the ACF power-law exponent in data analyses.

Keywords: Lillo-Mike-Farmer (LMF) model, econophysics, order splitting, order-sign autocorrelation, heterogeneous trading

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is mathematically dense, presenting an exact analytical solution for a generalized econophysics model using advanced stochastic process theory and statistical mechanics derivations (Section 3 and appendices). It lacks implementation details, backtests, or real trading metrics, focusing instead on theoretical robustness of parameters (prefactor vs. exponent) in an idealized closed system.

  flowchart TD
    A["Research Goal"] --> B["Generalize LMF Model"]
    B --> C{"Incorporate Heterogeneity"}
    C --> D["Formulate Exact Solution"]
    D --> E["Analyze Order-Sign ACF"]
    E --> F["Compare to Original LMF"]
    F --> G["Key Findings"]
    
    subgraph B ["Methodology"]
        direction LR
        C
        D
    end
    
    subgraph E ["Computational Process"]
        direction LR
        Analyze["Analyze ACF Structure"]
        Analyze --> Derive["Derive Power-Law Exponent & Prefactor"]
    end
    
    subgraph F ["Comparison"]
        direction LR
        Homogeneous["Original Homogeneous Model"]
        Heterogeneous["New Heterogeneous Model"]
        Compare["Compare Exponents & Prefactors"]
    end
    
    G --> H["Power-law exponent robust to heterogeneity"]
    G --> I["Prefactor sensitive to heterogeneity"]
    G --> J["Original model underestimates prefactor"]
    
    style A fill:#e1f5fe
    style B fill:#f3e5f5
    style G fill:#e8f5e8