From informal markets to Limit Order Book dynamics: a mean field connection

Abstract

We propose a unified mean-field framework that bridges the dynamics of informal financial markets and formal markets governed by Limit Order Books (LOBs). Both settings are modeled as interacting particle systems on a 1D price lattice, with temporal evolution described by master equations that account for new entries, cancellations, and executions. The key insight is the introduction of a preferential interaction parameter $Ψ$, which modulates the likelihood of transactions based on price compatibility: when $Ψ=0$, interactions are random and uncoordinated, reproducing the structure of informal markets; as $Ψ\to \infty$, only optimal (most mutually attractive) trades occur, recovering LOB-like dynamics. A grand-canonical interpretation is used to identify effective thermodynamic quantities -such as interaction energy and price-dependent chemical potentials -that underlie both systems. Most results are validated through numerical integration and simulations, although an analytical solution is shown to exist at least for the symmetric stationary case of the informal market.

Figures (13)

• Figure 1: Illustration of the processes that define order flow by price range. Entries are defined as the arrival of limit or market orders, while exits occur due to spontaneous voluntary cancellations and order executions.
• Figure 2: Information flow in the Cuban informal foreign exchange market. Reference price is established daily by elTOQUE platform using messages containing buy/order intentions gathered from social media and digital trade platforms.
• Figure 3: Average total buy and sell orders as a function of time, obtained through stochastic simulation and numerical integration of the master equation \ref{['ec: ecuacion maestra dinamica LOB Lage']}. The error bands corresponding to one standard deviation are also shown.
• Figure 4: Averaged stationary profiles of $n_a(p)$ and $n_b(p)$ obtained through stochastic simulation and integration of the master equation \ref{['ec: ecuacion maestra dinamica LOB Lage']}.
• Figure 5: Illustration of the informal market dynamics as a gas of agents with buy or sell intentions. Three processes occur: entry, cancellation, and execution, according to the prices and volumes of the orders.