Rough Heston model as the scaling limit of bivariate cumulative heavy-tailed INAR processes: Weak-error bounds and option pricing

TL;DR

The paper builds a rigorous bridge between discrete microstructure dynamics and the continuous rough Heston model by proving that nearly unstable, heavy-tailed bivariate INAR($\infty$) processes converge to the rough Heston system. It then delivers an FFT-enhanced, Monte Carlo pricing framework that operates on the INAR prelimit, enabling efficient pricing of European and path-dependent options while preserving the rough-volatility characteristics observed in markets. The authors establish finite-horizon weak-error bounds for option prices and demonstrate that the roughness parameter $\alpha<1$ reproduces key features of the implied-volatility surface, including a steep short-maturity ATM skew with power-law decay. The work provides both a microstructural interpretation of rough volatility and a practical, scalable pricing tool applicable to multi-asset and exotic-payoff contexts, with performance that approaches classical Heston benchmarks as $\alpha\to1$.

Abstract

This paper links nearly unstable, heavy-tailed \emph{bivariate cumulative} INAR($\infty$) processes to the rough Heston model via a discrete scaling limit, extending scaling-limit techniques beyond Hawkes processes and providing a microstructural mechanism for rough volatility and leverage effect. Computationally, we simulate the \emph{approximating INAR($\infty$)} sequence rather than discretizing the Volterra SDE, and implement the long-memory convolution with a \emph{divide-and-conquer FFT} (CDQ) that reuses past transforms, yielding an efficient Monte Carlo engine for \emph{European options} and \emph{path-dependent options} (Asian, lookback, barrier). We further derive finite-horizon \emph{weak-error bounds} for option pricing under our microstructural approximation. Numerical experiments show tight confidence intervals with improved efficiency; as $α\to 1$, results align with the classical Heston benchmark, where $α$ is the roughness specification. Using the simulator, we also study the \emph{implied-volatility surface}: the roughness specification ($α<1$) reproduces key empirical features -- most notably the steep short-maturity ATM skew with power-law decay -- whereas the classical model produces a much flatter skew.

Figures (2)

• Figure 1: Implied-volatility surface produced by the roughness specification ($\alpha=0.62$) with the INAR($\infty$)-FFT simulator. Grid: $T\in\{1/12,2/12,3/12,...,1\}$, $k\in[-0.2,0.2]$. All runs use a fixed time step ($\tau=320$ per year) and $10^6$ paths.
• Figure 2: ATM diagnostics under the rough specification ($\alpha=0.62$). Panel (a) shows the steepening skew toward short maturities; panel (b) reports the smoother ATM level term structure; panel (c) confirms the power-law decay $|\mathrm{skew}(T)|\propto T^{H-1/2}$.

Theorems & Definitions (54)

• Remark 2.2: Intuition behind the assumptions.
• Definition 2.3: Model Parameterization
• Remark 2.4
• Lemma 2.5: Weak Convergence of the Renewal Kernel
• proof
• Proposition 2.6
• proof
• Lemma 2.7: Lemma 3.5 of horst2023convergence
• Lemma 2.8
• proof