Rough Heston model as the scaling limit of bivariate cumulative heavy-tailed INAR processes: Weak-error bounds and option pricing
Pith reviewed 2026-05-22 23:36 UTC · model grok-4.3
The pith
Nearly unstable bivariate heavy-tailed INAR processes converge to the rough Heston model under one-factor scaling.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Under a one-factor parameterization and a suitable scaling, nearly unstable bivariate cumulative heavy-tailed INAR(∞) processes 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, the exact finite-τ transform recursion reduces to a quadratic discrete Volterra equation; under an admissible-strip assumption and local-in-frequency bounds, weak-error estimates for the truncated Carr-Madan functional take the form C₁τ^{-α} + C₂(α)τ^{-(1-α)}, where C₂(α)→0 asα
What carries the argument
The one-factor parameterization of the bivariate cumulative heavy-tailed INAR(∞) processes combined with the diffusive scaling regime, which produces convergence to the rough Heston dynamics and permits direct comparison of the discrete quadratic Volterra equation against the continuous fractional Riccati equation.
If this is right
- Explicit algebraic maps connect INAR asymmetry parameters directly to leverage correlation and diffusion scale in the rough Heston limit.
- The pricing transform reduces exactly to a quadratic discrete Volterra equation whose difference from the continuous fractional Riccati equation is bounded by the stated weak-error expression.
- An FFT-accelerated CDQ simulator achieves O(τ log² τ) complexity per path for both European and path-dependent options.
- The classical Heston limit is recovered when α=1 and the implied-volatility surface can be examined numerically under the same scaling.
Where Pith is reading between the lines
- The discrete INAR construction could be used to generate sample paths of rough Heston dynamics without solving fractional SDEs directly.
- Similar scaling arguments might map other heavy-tailed integer-valued autoregressive schemes onto different rough-volatility models by altering the tail index or the one-factor loading.
- The explicit parameter maps allow calibration of the discrete INAR model to observed leverage effects before taking the continuous limit.
- The vanishing of C₂(α) as α approaches 1 suggests that the error bound becomes sharper precisely when the model approaches the classical diffusive case.
Load-bearing premise
The bivariate INAR processes must obey the one-factor parameterization and the diffusive scaling regime for the convergence and the subsequent Volterra comparison to hold.
What would settle it
A simulation in which the finite-dimensional distributions of the scaled INAR processes fail to approach those of the rough Heston model as the time step τ tends to zero would refute the claimed convergence.
read the original 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-$\tau$ 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 $C_1\tau^{-\alpha}+C_2(\alpha)\tau^{-(1-\alpha)}$, where the second branch comes from the discrete-to-continuous Volterra comparison. The coefficient $C_2(\alpha)$ collects the vanishing contributions arising from both the weakly singular baseline quadrature and the discrete-to-continuous resolvent comparison, and satisfies $C_2(\alpha)\to0$ as $\alpha\uparrow1^-$. We also develop an FFT-accelerated CDQ simulator with $\mathcal O(\tau\log^2\tau)$ complexity per path and use it to price European and path-dependent options, examine the classical limit $\alpha=1$, and illustrate implied-volatility diagnostics.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript shows that nearly unstable bivariate cumulative heavy-tailed INAR(∞) processes converge to the rough Heston model under a one-factor parameterization and diffusive scaling. This yields explicit mappings from INAR asymmetry parameters to leverage correlation and diffusion scale in the limit. For pricing, an exact finite-τ transform recursion is derived and reduced to a quadratic discrete Volterra equation, which is compared to the fractional Riccati equation of the rough Heston model. Under an admissible-strip assumption and local-in-frequency bounds, weak-error estimates C₁τ^{-α} + C₂(α)τ^{-(1-α)} (with C₂(α)→0 as α↑1) are obtained for the truncated Carr-Madan functional. An FFT-accelerated CDQ simulator of complexity O(τ log²τ) is developed and used to price options and examine implied-volatility diagnostics, including the α=1 limit.
Significance. If the stated convergence, Volterra comparison, and error bounds hold, the work supplies a concrete discrete-time, heavy-tailed microstructural derivation of the rough Heston dynamics together with explicit parameter links and quantitative pricing error rates. The simulator and the explicit form of the two-term error bound (with the vanishing C₂(α) term) constitute practical and theoretical strengths.
minor comments (2)
- The admissible-strip assumption is invoked for the error bounds but its precise statement and verification for the INAR processes would benefit from an explicit display (e.g., as a numbered assumption or lemma) rather than a parenthetical reference.
- The reduction from the exact finite-τ recursion to the quadratic discrete Volterra equation is central; a short displayed equation or diagram showing the truncation step would improve readability.
Simulated Author's Rebuttal
We thank the referee for the positive and accurate summary of the manuscript, the favorable significance assessment, and the recommendation for minor revision. No specific major comments appear in the report.
Circularity Check
No significant circularity; derivation self-contained from INAR inputs
full rationale
The paper begins with bivariate cumulative heavy-tailed INAR(∞) processes, imposes an explicit one-factor parameterization plus diffusive scaling as assumptions, and derives the rough Heston limit together with the discrete Volterra recursion and its comparison to the fractional Riccati equation. The weak-error bound C₁τ^{-α} + C₂(α)τ^{-(1-α)} is obtained by direct comparison under the admissible-strip assumption; neither the limit nor the error coefficients are defined in terms of the target quantities they recover. No self-citation is invoked as a load-bearing uniqueness theorem, no fitted parameter is relabeled as a prediction, and the derivation chain remains independent of the final rough-Heston objects.
Axiom & Free-Parameter Ledger
free parameters (1)
- asymmetry parameters
axioms (2)
- standard math Standard weak-convergence results for suitably scaled integer-valued autoregressive processes hold under the stated near-instability and heavy-tail conditions.
- domain assumption An admissible-strip assumption together with local-in-frequency bounds on the characteristic function.
discussion (0)
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