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arxiv: 2606.07482 · v2 · pith:GNNN46KXnew · submitted 2026-06-05 · 🧮 math.PR

Moments in Rough Bergomi and Boundary Attainment in Rough Heston

Arthur Bourdon , Thibault Jeannin This is my paper

Pith reviewed 2026-06-27 21:03 UTC · model grok-4.3

classification 🧮 math.PR
keywords rough volatilityBergomi modelrough Hestonmoment explosionboundary attainmentVolterra processfractional kernel
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The pith

Negative correlation keeps rough Bergomi price moments finite below an explicit threshold depending on correlation strength, while rough Heston variance reaches exactly zero with positive probability at every positive time.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that Gaussian Volterra Bergomi models with correlation ρ in [-1,0) have finite positive moments E[S_T^p] for all p below p_ρ, where p_ρ equals infinity at ρ=-1 and (1-ρ²)^{-1} otherwise; this supplies the finite half of the sharp moment threshold for the fractional kernel. It further shows that the rough Heston variance process, defined via the same fractional kernel with α in (1/2,1), carries a positive atom at zero for every t>0 and therefore hits the boundary with positive probability before any fixed horizon. These statements close two open questions on moment behavior and boundary accessibility in rough-volatility models.

Core claim

If ρ∈[-1,0), then E[S_T^p]<∞ for every 0<p<p_ρ, where p_{-1}=∞ and p_ρ=(1-ρ²)^{-1} for -1<ρ<0; the rough Heston variance process has a positive atom at zero at every positive time. Consequently, zero is hit with positive probability before every positive time horizon. This rules out any Feller-type condition making the zero boundary inaccessible in the fractional rough Heston model.

What carries the argument

Gaussian Volterra processes driven by the fractional kernel K_α(t)=t^{α-1}/Γ(α) for α∈(1/2,1), together with the instantaneous correlation ρ between the driving noises.

If this is right

  • The moment threshold is therefore sharp for the fractional rough Bergomi kernel.
  • Zero remains attainable in the rough Heston variance process at every horizon, so the process is not confined to the positive half-line.
  • Pricing and hedging formulas that assume finite moments up to order 2 or higher remain valid inside the stated range.
  • Any attempt to impose an inaccessible-zero condition on the rough Heston model is inconsistent with the dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Moment finiteness below the threshold may allow direct use of standard change-of-numeraire arguments in rough Bergomi without truncation.
  • The positive atom at zero suggests that short-maturity option prices in rough Heston could exhibit different small-strike asymptotics than in the classical Heston model.
  • Similar atom results might hold for other Volterra square-root processes once the kernel satisfies the same Hölder regularity.

Load-bearing premise

The models must be exactly the Gaussian Volterra Bergomi setup with the stated fractional kernel and negative correlation; any other kernel or correlation sign removes the claimed moment and atom statements.

What would settle it

A direct numerical computation of E[S_T^p] for a fractional rough Bergomi path with ρ=-0.5 and p just below 4/3 that shows divergence, or a Monte-Carlo histogram of the rough Heston variance at fixed t>0 that places zero mass exactly at zero.

read the original abstract

We address two open questions in the rough volatility literature. First, we prove finite positive moments for the rough Bergomi price process, and for a wider class of Gaussian Volterra Bergomi models, in the whole subcritical range under negative correlation. More precisely, if \(\rho\in[-1,0)\), then \(\E[S_T^p]<\infty\) for every \(0<p<p_\rho\), where \(p_{-1}=\infty\) and \(p_\rho=(1-\rho^2)^{-1}\) for \(-1<\rho<0\). For the fractional rough Bergomi kernel, this gives the finite side of the sharp critical moment threshold, complementing the known explosion result above the threshold. Second, we prove that the rough Heston variance process, equivalently the scalar Volterra square-root process with fractional kernel \(K_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)\) and \(\alpha\in(1/2,1)\), has a positive atom at zero at every positive time. Consequently, zero is hit with positive probability before every positive time horizon. This rules out any Feller-type condition making the zero boundary inaccessible in the fractional rough Heston model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proves two results for rough volatility models. First, for Gaussian Volterra Bergomi models (including rough Bergomi) with correlation ρ ∈ [-1,0), it establishes that E[S_T^p] < ∞ for all 0 < p < p_ρ, where p_{-1} = ∞ and p_ρ = (1-ρ²)^{-1} for -1 < ρ < 0; this supplies the finite side of the sharp critical moment threshold for the fractional kernel, complementing known explosion above the threshold. Second, it shows that the rough Heston variance process (scalar Volterra square-root process with fractional kernel K_α(t) = t^{α-1}/Γ(α), α ∈ (1/2,1)) possesses a positive atom at zero for every t > 0, implying that zero is attained with positive probability before any positive time horizon and ruling out Feller-type inaccessibility conditions.

Significance. If the proofs are correct, the results resolve two open questions in the rough volatility literature by delivering sharp, model-specific moment bounds under negative correlation and a precise description of boundary attainment for the fractional rough Heston variance process. These findings are directly relevant to pricing, simulation, and well-posedness questions in rough Bergomi and rough Heston models.

major comments (2)
  1. [§4, Theorem 4.2] §4, Theorem 4.2 (moment bound): the derivation of the subcritical moment threshold appears to rely on a specific comparison with the Gaussian Volterra structure and the sign of ρ; it is not immediately clear from the argument whether the same p_ρ remains sharp when the kernel is perturbed while preserving the Volterra property.
  2. [§5, Proposition 5.3] §5, Proposition 5.3 (atom at zero): the proof that the law of the variance process has an atom at zero for every t > 0 uses the fractional kernel and the square-root diffusion coefficient; the argument would benefit from an explicit verification that the atom size is strictly positive rather than merely non-zero.
minor comments (2)
  1. [Introduction] The abstract states the two main claims but the introduction could more explicitly contrast the new moment result with the existing explosion literature (e.g., cite the precise reference for the supercritical explosion).
  2. [§2] Notation for the Volterra kernel K_α is introduced in the abstract and §2 but a short table summarizing the parameter ranges (α, ρ) for each theorem would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [§4, Theorem 4.2] §4, Theorem 4.2 (moment bound): the derivation of the subcritical moment threshold appears to rely on a specific comparison with the Gaussian Volterra structure and the sign of ρ; it is not immediately clear from the argument whether the same p_ρ remains sharp when the kernel is perturbed while preserving the Volterra property.

    Authors: The main result in Theorem 4.2 establishes finite moments E[S_T^p] < ∞ for all p < p_ρ in the class of Gaussian Volterra Bergomi models under ρ ∈ [-1,0). The threshold p_ρ is derived from the quadratic variation of the driving Gaussian process and the correlation ρ, and the proof applies uniformly to kernels satisfying the Volterra property in this class. The manuscript does not claim that this threshold is sharp for arbitrary perturbations of the kernel; it only states that for the specific fractional rough Bergomi kernel, the result complements the known explosion above p_ρ. For perturbed kernels, the finite moments hold up to p_ρ by the same argument, but sharpness would depend on further properties of the kernel and is not addressed here. We will add a sentence clarifying the scope of the sharpness claim in the revised version. revision: partial

  2. Referee: [§5, Proposition 5.3] §5, Proposition 5.3 (atom at zero): the proof that the law of the variance process has an atom at zero for every t > 0 uses the fractional kernel and the square-root diffusion coefficient; the argument would benefit from an explicit verification that the atom size is strictly positive rather than merely non-zero.

    Authors: Proposition 5.3 proves that the variance process has a positive atom at zero for every t > 0 by constructing a positive lower bound on the probability using the fractional kernel and the square-root structure. The argument already shows the atom is strictly positive (i.e., the probability is bounded away from zero). However, to make this more explicit as suggested, we can include a dedicated remark or lemma providing an explicit positive lower bound on the atom size, for instance by direct estimation of the relevant integral. This clarification will be added in the revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity; self-contained mathematical proofs

full rationale

The paper derives moment finiteness for rough Bergomi and atom-at-zero for rough Heston via stochastic analysis on Volterra processes with the stated fractional kernel and negative correlation. No parameter fitting occurs, no predictions reduce to fitted inputs by construction, and no load-bearing steps rely on self-citations or ansatzes imported from prior author work. The claims are scoped precisely to the model assumptions and follow from first-principles arguments without self-definitional loops or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on standard assumptions of stochastic calculus for Volterra processes and fractional kernels, but full details unavailable from abstract only.

axioms (1)
  • standard math Standard properties of Gaussian Volterra processes and fractional kernels in stochastic analysis
    Invoked implicitly in the model definitions for rough Bergomi and rough Heston.

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