Mean-Field Hawkes Order Flow and Critical Stochastic Volatility
Summary
The document describes an agent-based model of tick-by-tick price formation. Each agent’s buy and sell orders follow mutually exciting Hawkes processes, while mean-field interaction links activity across agents and creates correlated order volumes. This interaction is intended to represent effects such as herding and contagion in market order flow.
The main result concerns the large-population limit at a critical parameter setting: the aggregate price is approximated by a stochastic volatility model with a leverage effect and volatility that mean-reverts faster than linearly. The authors connect this faster mean reversion to prior econometric evidence and to multifractal behavior reported in earlier work. The account is theoretical and offers no specific market data, calibration, or trading test, so it explains a possible mechanism rather than demonstrating a deployable strategy.
Key ideas
- Mutually exciting Hawkes processes represent agents’ buy and sell order arrivals.
- Mean-field interaction creates cross-agent correlations that can reflect herding and contagion.
- At a critical parameter setting, the aggregate price limit has stochastic volatility and a leverage effect.
- The resulting volatility process has faster-than-linear mean reversion, which the authors relate to multifractal price behavior.
- The document presents a modeling result rather than a calibrated or tested trading strategy.
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Full text
# A stochastic volatility approximation for a tick-by-tick price model with mean-field interaction # A stochastic volatility approximation for a tick-by-tick price model with mean-field interaction 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 [Bacry, Delattre, Hoffmann, Muzy, Modelling microstructure noise with mutually exciting point processes, Quantitative Finance, 2013] and [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 of the model 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.
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