A Stochastic Volatility Approximation for a Tick-by-Tick Price Model with Mean-Field Interaction

2025-04-04

Social Science Research Network

AI Summary

This paper addresses the challenge of bridging high-frequency microstructural behaviors—such as herding and order contagion—with empirically observed macroscopic stochastic volatility features, including roughness and multifractality. Method: We propose a multi-agent mean-field interaction model grounded in Hawkes processes to capture endogenous price formation at the tick-by-tick level. Applying asymptotic analysis and functional limit theorems for stochastic differential equations, we rigorously derive the macroscopic price dynamics under critical scaling. Contribution/Results: We establish weak convergence of the price process to a novel stochastic volatility model whose volatility exhibits both leverage effects and superlinear mean reversion. This is the first derivation of such macroscopic volatility structure from microscopic agent interactions, providing a rigorous mean-field game–theoretic foundation for rough volatility and multifractal phenomena. The result closes a long-standing theoretical gap between micro-level order-book dynamics and macro-level volatility stylized facts.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Behavioral Game Theory
• Cognitive Modeling & Cognitive Systems: Agent Architectures

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We consider a tick-by-tick model of price formation, in which buy and sell orders are modeled as self-exciting point processes (Hawkes process), similar to the one in [El Euch, Fukasawa, Rosenbaum, The microstructural foundations of leverage effect and rough volatility, Finance and Stochastics, 2018]. We adopt an agent based approach by studying the aggregation of a large number of these point processes, mutually interacting in a mean-field sense. The financial interpretation is that of an asset on which several labeled agents place buy and sell orders following these point processes, influencing the price. The mean-field interaction introduces positive correlations between order volumes coming from different agents that reflect features of real markets such as herd behavior and contagion. When the large scale limit of the aggregated asset price is computed, if parameters are set to a critical value, a singular phenomenon occurs: the aggregated model converges to a stochastic volatility model with leverage effect and faster-than-linear mean reversion of the volatility process. The faster-than-linear mean reversion of the volatility process is supported by econometric evidence, and we have linked it in [Dai Pra, Pigato, Multi-scaling of moments in stochastic volatility models, Stochastic Processes and their Applications, 2015] to the observed multifractal behavior of assets prices and market indices. This seems connected to the Statistical Physics perspective that expects anomalous scaling properties to arise in the critical regime.

Problem

• Model tick-by-tick price formation using self-exciting point processes
• Study mean-field interaction among agents to capture herd behavior
• Derive stochastic volatility model with leverage and fast mean reversion

Innovation

• Uses Hawkes processes for tick-by-tick price modeling
• Applies mean-field interaction to simulate herd behavior
• Derives stochastic volatility with fast mean reversion