Paper: arXiv 2504.03445

Authors: Paolo Dai Pra, Paolo Pigato

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 [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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical derivation linking microscopic Hawkes processes with mean-field interactions to a stochastic volatility model featuring nonlinear mean reversion and leverage effects. While high in mathematical complexity and theoretical novelty, it lacks empirical backtesting or real-world financial data validation.

Research Flowchart

  flowchart TD
    A[Goal: Derive macro stochastic volatility from micro tick-by-tick price models] --> B[Model Setup: Buy/Sell orders via self-exciting Hawkes processes]
    B --> C[Input: Large-scale agent aggregation with mean-field interaction]
    C --> D[Scaling Limit: Parameter tuning to critical regime]
    D --> E[Outcome 1: Convergence to Stochastic Volatility model with Leverage Effect]
    D --> F[Outcome 2: Faster-than-linear Mean Reversion explaining multifractal behavior]