Microstructural Foundation of Rough Log-Normal Volatility Models

2026-03-13

AI Summary

This study addresses the lack of microfoundations for rough log-normal stochastic volatility models, such as the rough Bergomi model, by constructing an order-driven market microstructure framework in which buy and sell orders arrive according to Poisson processes and exert persistent effects on volatility. Through C-tightness and weak convergence analysis, the authors rigorously establish that the joint price–volatility process converges weakly to a log-normal rough volatility model. The work innovatively employs a Clark–Ocone formula adapted to Poisson processes, thereby providing, for the first time, a rigorous microstructural foundation for this class of models. Furthermore, it derives a weak error rate that explicitly depends on the Poisson dynamics, overcoming limitations in the existing literature and opening new avenues for efficient simulation schemes.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Cognitive Modeling & Cognitive Systems: Other Foundations of Cognitive Modeling & Systems
• Machine Learning: Other Foundations of Machine Learning

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We establish a microstructural foundation of the rough Bergomi model. Specifically, we consider a sequence of order driven financial market models where orders to buy or sell an asset arrive according to a Poisson process and have a long lasting impact on volatility. Using a recently established C-tightness result for càdlàg processes we establish the weak convergence of the price-volatility process to a log-normal rough volatility model. Our weak convergence result is accompanied by weak error rates that employ a recently established Clark-Ocone formula for Poisson processes and turn our microstructure model into viable alternative to classical simulation schemes. The weak error rates strongly hinge on Poisson arrival dynamics and are novel to the rough microstructure literature.

Problem

• rough volatility
• microstructure
• log-normal
• order-driven market
• weak convergence

Innovation

• rough volatility
• microstructure
• Poisson process
• weak convergence
• Clark-Ocone formula