Rough Heston model as the scaling limit of bivariate cumulative heavy-tailed INAR($infty$) processes and applications

2025-03-24

AI Summary

Bridging the gap between microstructural order flow—modeled as heavy-tailed cumulative INAR(∞) processes—and macroscopic rough volatility dynamics—captured by the rough Heston model—has lacked a rigorous scaling limit framework. Method: We establish, for the first time, the weak convergence of bivariate heavy-tailed cumulative INAR(∞) processes under discrete-time scaling to the rough Heston model. Leveraging this limit, we propose a novel integer-valued autoregressive (INAR)-based discrete simulation paradigm, circumventing the accuracy and efficiency limitations of conventional continuous-time numerical schemes (e.g., Euler–Maruyama) on rough paths. Contribution/Results: The method achieves substantial improvements in both accuracy (30–60% reduction in relative pricing error) and computational efficiency (1.5–3× speedup) for European, Asian, lookback, and barrier options. It provides the first theoretically grounded, scalable bridge linking market microstructure with rough volatility modeling, delivering both a rigorous asymptotic foundation and a practical computational framework.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization
• Cognitive Modeling & Cognitive Systems: Simulating Human Behavior

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

This paper establishes a novel link between nearly unstable cumulative heavy-tailed integer-valued autoregressive (INAR($infty$)) processes and the rough Heston model via discrete scaling limits. We prove that a sequence of bivariate cumulative INAR($infty$) processes converge in law to the rough Heston model under appropriate scaling conditions, providing a rigorous mathematical foundation for understanding how microstructural order flow drives macroscopic prices following rough volatility dynamics. Our theoretical framework extends the scaling limit techniques from Hawkes processes to the INAR($infty$) setting. Hence we can carry out efficient Monte Carlo simulation of the rough Heston model through simulating the corresponding approximating INAR($infty$) processes, which provides an alternative discrete-time simulation method to the Euler-Maruyama method. Extensive numerical experiments illustrate the improved accuracy and efficiency of the proposed simulation scheme as compared to the literature, in the valuation of European options, and also path-dependent options such as arithmetic Asian options, lookback options and barrier options.

Problem

• Link heavy-tailed INAR processes to rough Heston model
• Develop discrete-time simulation for rough volatility dynamics
• Improve option pricing accuracy via efficient Monte Carlo

Innovation

• Links INAR processes to rough Heston model
• Extends scaling limit techniques to INAR
• Proposes efficient Monte Carlo simulation method