Microstructural Foundation for the Rough Hawkes--Heston Model

2026-08-07

AI Summary

This work establishes a microstructural foundation for the variance dynamics and price–volatility co-jump mechanism in the rough Hawkes–Heston model. By introducing a nearly unstable heavy-tailed Hawkes order flow marked with a Poisson process, ordinary events generate rough continuous volatility and leverage effects, while rare marked events induce synchronous jumps in both price and volatility. In the scaling limit, the rescaled price–variance–jump system converges in distribution to the unique weak solution of the canonical rough Hawkes–Heston model. The macroscopic coefficients are explicitly determined by the microscopic parameters, yielding both a Mittag–Leffler Volterra representation and a Riemann–Liouville fractional form. Numerical experiments confirm the theoretical convergence, providing the first complete microstructural justification for the rough Hawkes–Heston framework incorporating common jumps.

Abstract

Hawkes-based microstructural foundations for rough volatility, leverage, and rough Heston-type limits were developed by El Euch et al. (2018, Finance Stoch., 22(2), 241--280) and connected to the affine rough Heston framework of El Euch and Rosenbaum (2019, Math. Finance, 29(1), 3--38). The rough Hawkes--Heston model with common price--volatility jumps of Bondi et al. (2024, Math. Finance, 34(4), 1197--1241) extends this framework by adding state-dependent common jumps to rough affine volatility. We provide a microstructural foundation for its variance and common-jump mechanism by constructing a Poisson-embedded marked Hawkes order-flow model. Ordinary arrivals generate rough continuous volatility and leverage through a nearly unstable heavy-tailed Hawkes mechanism, while rare marked arrivals represent common shock events that produce simultaneous price jumps and volatility excitation. Under the nearly unstable scaling and the reduced-form admissibility conditions, the complete rescaled price/variance/jump system converges along the full sequence to the unique complete canonical rough Hawkes--Heston weak solution. The Hawkes renewal structure yields a Mittag--Leffler Volterra representation, which is then rewritten in Riemann--Liouville fractional form. The limiting coefficients are expressed explicitly in terms of the microscopic parameters. The construction provides a microstructural foundation for the variance and common-jump mechanism of the rough Hawkes--Heston model. Numerical experiments illustrate the convergence of our microstructural foundation to the rough Hawkes-Heston model.

Problem

• rough Hawkes–Heston model
• microstructural foundation
• common jumps
• volatility
• Hawkes process

Innovation

• rough Hawkes--Heston
• microstructural foundation
• common jumps
• nearly unstable Hawkes
• fractional volatility