[Paper Review] Exact solution to a generalised Lillo-Mike-Farmer model with heterogeneous order-splitting strategies

Yuki Sato, Kiyoshi Kanazawa|arXiv (Cornell University)|Jun 23, 2023

Complex Systems and Time Series AnalysisEconomics, Econometrics and Finance21 references3 citations

TL;DR

This paper presents an exact solution to a generalized Lillo-Mike-Farmer (LMF) model that incorporates heterogeneous order-splitting strategies among traders, moving beyond the original model's assumption of homogeneity. By analytically solving the model without heuristic approximations, it demonstrates that the power-law exponent in the order-sign autocorrelation function (ACF) is robust to heterogeneity, while the ACF prefactor is highly sensitive and systematically underestimated in the original homogeneous model.

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.

Motivation & Objective

• To address the limitation of the original Lillo-Mike-Farmer (LMF) model, which assumes homogeneous order-splitting strategies and uses heuristic approximations.
• To develop an exactly solvable generalization of the LMF model that incorporates realistic heterogeneity in traders' order-splitting behaviors.
• To rigorously assess how heterogeneity in trading strategies affects the power-law exponent and prefactor of the order-sign autocorrelation function (ACF).
• To provide a theoretically grounded interpretation of empirical ACF data, particularly distinguishing between robust and sensitive components of the ACF.

Proposed method

• The model generalizes the original LMF framework by introducing a heterogeneous distribution of metaorder lengths and splitting intensities across traders.
• It employs a stochastic process with $ M $ traders, each characterized by a unique metaorder length $ L^{(i)} $ and splitting intensity $ ho^{(i)} $, drawn from a power-law distribution.
• The exact solution is derived using a generating function approach and the inverse transform method to deterministically allocate $ L^{*(i)} $ values to match a desired power-law distribution of metaorder lengths.
• The ACF is computed exactly via the superposition of individual trader contributions, accounting for the full distribution of $ L^{(i)} $ and $ ho^{(i)} $, avoiding mean-field or heuristic approximations.
• The analysis includes a systematic comparison with the original homogeneous LMF model, highlighting discrepancies in the prefactor and robustness of the exponent.
• An alternative scenario is explored where exponential splitters with power-law intensity distributions also yield power-law ACF decay, confirming the robustness of the exponent under different dynamical assumptions.

Experimental results

Research questions

• RQ1 Does the power-law exponent $ \gamma $ in the order-sign ACF remain robust when traders' order-splitting strategies are heterogeneous?
• RQ2 How does heterogeneity in metaorder length and splitting intensity affect the ACF prefactor $ c_0 $, and is it systematically underestimated in the original homogeneous LMF model?
• RQ3 Can an exact analytical solution be derived for the generalized LMF model without relying on heuristic approximations?
• RQ4 What is the role of the distribution of splitting intensities in shaping the long-range correlation structure of order flows?
• RQ5 Under what conditions does the ACF exhibit power-law decay independent of the specific metaorder length distribution?

Key findings

• The power-law exponent $ \gamma $ in the order-sign ACF is robust to arbitrary heterogeneous intensity distributions and remains equal to $ \alpha - 1 $, matching the original LMF prediction.
• The ACF prefactor $ c_0 $ is highly sensitive to heterogeneity in trading strategies and is systematically underestimated in the original homogeneous LMF model, which assumes uniform intensity.
• The exact solution confirms that the original LMF model's formula $ c_0 = 1/(\alpha M^{2-\alpha}) $ is inaccurate when heterogeneity is present, especially for finite systems.
• The study demonstrates that the ACF decay exponent is robust even under non-homogeneous splitting dynamics, supporting the order-splitting hypothesis as a fundamental origin of long-range correlations.
• An alternative scenario with exponential splitters and power-law intensity distributions also yields power-law ACF decay with exponent $ \gamma = 2 - \beta $, showing that the exponent is robust across different dynamical mechanisms.
• Numerical validation confirms that the theoretical predictions match simulated ACFs, particularly in the intermediate asymptotic regime where the power-law behavior dominates.