Paper: arXiv 2412.16436

Authors: Ulrich Horst, Wei Xu, Rouyi Zhang

Abstract

We consider a microstructure foundation for rough volatility models driven by Poisson random measures. In our model the volatility is driven by self-exciting arrivals of market orders as well as self-exciting arrivals of limit orders and cancellations. The impact of market order on future order arrivals is captured by a Hawkes kernel with power law decay, and is hence persistent. The impact of limit orders on future order arrivals is temporary, yet possibly long-lived. After suitable scaling the volatility process converges to a fractional Heston model driven by an additional Poisson random measure. The random measure generates occasional spikes and clusters of spikes in the volatility process. Our results are based on novel existence and uniqueness of solutions results for stochastic path-dependent Volterra equations driven by Poisson random measures.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework unifying rough volatility models and self-exciting jump/spike dynamics via fractional Volterra equations driven by Poisson random measures. However, it is purely theoretical and probability-focused, lacking any empirical backtesting or real-world data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Microstructure foundation for rough volatility driven by Poisson random measures] --> B[Data Inputs: Order Book Arrivals<br/>Market Orders, Limit Orders & Cancellations]
    B --> C[Methodology: Limit Order Book Model<br/>Hawkes Kernel with Power Law Decay]
    C --> D[Mathematical Scaling: Large-time & High-frequency Asymptotic Rescaling]
    D --> E[Computational Process: Solve Path-dependent Fractional Volterra Equations]
    E --> F[Outcomes: Convergence to Fractional Heston Model<br/>Volatility Spikes & Clusters from Poisson Measures]