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

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.