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

Paper: arXiv 2503.18259

Authors: Yingli Wang, Zhenyu Cui, Lingjiong Zhu

Abstract

We study nearly unstable bivariate cumulative heavy-tailed INAR($\infty$) processes and show that, under a one-factor parameterization and a suitable scaling, they converge to the rough Heston model. This yields a discrete-time microstructural route to the joint price-variance dynamics and gives explicit formulas linking the INAR asymmetry parameters to the leverage correlation and diffusion scale of the limiting volatility process. On the pricing side, we derive the exact finite-$τ$ transform recursion and reduce it, in the diffusive scaling regime, to a quadratic discrete Volterra equation. We then compare this discrete equation with the continuous fractional Riccati equation from the rough Heston model. Under an admissible-strip assumption and local-in-frequency bounds, we obtain weak-error estimates for the truncated Carr–Madan pricing functional on bounded frequency windows of the form $C_1τ^{-α}+C_2(α)τ^{-(1-α)}$, where the second branch comes from the discrete-to-continuous Volterra comparison. The coefficient $C_2(α)$ collects the vanishing contributions arising from both the weakly singular baseline quadrature and the discrete-to-continuous resolvent comparison, and satisfies $C_2(α)\to0$ as $α\uparrow1^-$. We also develop an FFT-accelerated CDQ simulator with $\mathcal O(τ\log^2τ)$ complexity per path and use it to price European and path-dependent options, examine the classical limit $α=1$, and illustrate implied-volatility diagnostics.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 4.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper establishes a rigorous theoretical bridge connecting bivariate cumulative INAR(∞) count processes to the rough Heston model, backed by weak-error bounds and an efficient FFT-accelerated simulation scheme. The mathematical depth is exceptionally high with formal limit theorems and Volterra analysis, though empirical rigor remains primarily focused on synthetic pricing and simulation diagnostics rather than extensive real-market backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Derive microstructural foundation & weak-error bounds for Rough Heston model via INAR(∞) processes"] --> B["Input: Bivariate cumulative heavy-tailed INAR(∞) process & one-factor parameterization"]
    B --> C["Convergence Analysis: Scale INAR(∞) process under nearly unstable regime to continuous limit"]
    C --> D["Pricing Derivation: Reduce exact finite-τ transform to a quadratic discrete Volterra equation"]
    D --> E["Error Estimation & Simulation: Compare discrete Volterra with continuous fractional Riccati via FFT-accelerated CDQ"]
    E --> F["Key Outcomes: Explicit leverage/volatility formulas, weak-error bound C₁τ⁻ᵃ + C₂\(α\)τ⁻⁽¹⁻ᵃ⁾, & efficient option pricing"]

Cited References (124)

1. Central limit theorems for local martingales (1980)
2. Option Valuation Using the Fast Fourier Transform (1999)
3. Option Pricing by Transform Methods: Extensions, Unification, and Error Control (2004)
4. A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions (2006)
5. Option Pricing Formulae using Fourier Transform: Theory and Application (2010)
6. A Fast and Accurate FFT-Based Method for Pricing Early-Exercise Options under L 'e vy Processes (2008)
7. Fractional Linear Multistep Methods for Abel–Volterra Integral Equations of the Second Kind (1985)
8. A Stability Analysis of Convolution Quadratures for Abel–Volterra Integral Equations (1986)

4 of 124 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.