[Paper Review] From asymptotic properties of general point processes to the ranking of financial agents

Othmane Mounjid, Mathieu Rosenbaum+1 more|arXiv (Cornell University)|Jun 12, 2019

Point processes and geometric inequalities37 references5 citations

TL;DR

This paper proposes a general non-linear limit order book model that captures high-frequency market makers' individual behaviors using point process theory. It establishes ergodicity and diffusivity under mild conditions and derives closed-form expressions for key market quantities—spread, liquidity fluctuations, and price volatility—enabling a quantitative ranking of market makers based on their impact on market stability and efficiency.

ABSTRACT

We propose a general non-linear order book model that is built from the individual behaviours of the agents. Our framework encompasses Markovian and Hawkes based models. Under mild assumptions, we prove original results on the ergodicity and diffusivity of such system. Then we provide closed form formulas for various quantities of interest: stationary distribution of the best bid and ask quantities, spread, liquidity fluctuations and price volatility. These formulas are expressed in terms of individual order flows of market participants. Our approach enables us to establish a ranking methodology for the market makers with respect to the quality of their trading.

Motivation & Objective

• To develop a general, non-Markovian limit order book model that captures the heterogeneous behaviors of high-frequency market makers.
• To analyze the ergodic and diffusive properties of the system under mild assumptions, ensuring long-term stability and convergence.
• To derive closed-form expressions for key market quantities—such as spread, liquidity fluctuations, and price volatility—in terms of individual agent order flows.
• To enable a quantitative ranking of market makers based on their contribution to market quality, particularly volatility and stability.
• To provide regulators and exchanges with a tool to assess individual market maker effects, especially in scenarios where one participant exits the market.

Proposed method

• Model the limit order book as a general point process where each event (order submission, cancellation, trade) is characterized by a random variable encoding its type and size.
• Incorporate non-linear, history-dependent dynamics inspired by Hawkes processes and queue-reactive models, allowing agents to react to both market state and past actions.
• Establish ergodicity and diffusivity of the system using stochastic process theory, ensuring the existence of a unique stationary distribution.
• Derive closed-form expressions for the stationary distribution of the best bid and ask quantities, spread, and price volatility in terms of individual agent intensities and order flow characteristics.
• Use perturbation analysis to assess the impact of removing a single market maker: compare the resulting volatility to the original to determine stabilizing or destabilizing effects.
• Define a ranking criterion based on whether a market maker's removal increases or decreases overall market volatility, with the condition $\lambda^{+,a}/\lambda^{-,a} \geq \lambda^{+}/\lambda^{-}$ ensuring stabilizing behavior.

Experimental results

Research questions

• RQ1 How can the individual contributions of high-frequency market makers to market volatility be disentangled from their collective impact?
• RQ2 What conditions ensure that a new market maker stabilizes the market, particularly in terms of reducing price volatility?
• RQ3 Can a general non-Markovian order book model with history-dependent agent behavior still exhibit ergodic and diffusive properties?
• RQ4 What closed-form expressions can be derived for key market quality metrics—such as spread and liquidity fluctuations—based on individual agent order flows?
• RQ5 How does the removal of a single market maker affect the long-term volatility of the order book, and what does this imply for their ranking by market impact?

Key findings

• The model proves ergodicity and diffusivity under mild assumptions, ensuring the existence of a unique stationary distribution for the order book state.
• Closed-form expressions are derived for the stationary distribution of best bid and ask quantities, spread, and price volatility, all expressed in terms of individual agent order flow intensities.
• The long-term price volatility is shown to decrease when a new market maker with $\lambda^{+,a}/\lambda^{-,a} \geq \lambda^{+}/\lambda^{-}$ is introduced, indicating a stabilizing effect.
• The removal of a market maker leads to a measurable increase in volatility if their $\lambda^{+,a}/\lambda^{-,a}$ ratio is below the market average, providing a quantitative criterion for destabilizing behavior.
• Numerical results for EssilorLuxottica, Michelin, and Orange show distinct patterns in liquidity provision and consumption intensities, with corresponding changes in long-term volatility when a participant is removed.
• The framework enables a ranking of market makers based on their impact on volatility: those with higher insertion-to-consumption ratios relative to the market are more likely to stabilize the system.