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REVIEW 4 major objections 4 minor 56 references

Rough Bergomi turns grey

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The grey Bergomi model replaces rough Bergomi's fractional Brownian motion with generalised grey Brownian motion, relaxing the log-normal VIX assumption and supplying closed-form small-time smile formulas for joint SPX/VIX calibration.

desk verdict Genuine new model and useful asymptotics, but the calibration evidence does not support the 'Holy Grail' claim; the paper is stronger as a model-formula contribution than as an empirical demonstration. read the letter →

arxiv 2505.08623 v1 pith:PAD42SYL submitted 2025-05-13 q-fin.PR math.PR

Antoine Jacquier , Adriano Oliveri Orioles , Zan Zuric This is my paper · ORCID
classification q-fin.PRmath.PR MSC 60G1560G2260H0791G20
keywords RoughvolatilitygreyBrownianmotionMittag-LefflerfunctionM-WrightdistributionVIXoptionsSPXMalliavincalculusasymptoticimplied
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces the grey Bergomi (gBergomi) model, a tractable extension of rough Bergomi in which the fractional Brownian motion driving the variance is replaced by a generalised grey Brownian motion. Because conditioning on an independent M-Wright variable turns the grey noise into a fractional Brownian motion, the variance becomes a randomised volatility-of-volatility process and the log-normal constraint of rough Bergomi is relaxed, so VIX marginals can carry skew and heavier tails. The paper derives semi-closed expressions for the VIX and forward variance, small-time asymptotic formulas for the VIX and SPX implied-volatility smiles, and a calibration exercise on SPX and VIX options data. A reader should care because the joint calibration of SPX and VIX options is a long-standing open problem, and the model is designed to keep rough volatility's tractability while allowing upward-sloping VIX smiles.

What carries the argument

The central object is the generalised grey Brownian motion $B^{\beta,\alpha}$, a self-similar process with stationary increments whose finite-dimensional characteristic function is $E_\beta(-\tfrac12 u^\top\Sigma_\alpha u)$, with $E_\beta$ the Mittag-Leffler function and $\Sigma_\alpha$ the covariance matrix of a fractional Brownian motion with Hurst parameter $\alpha/2$. Its load-bearing property is Lemma 2.3, $B^{\beta,\alpha}_t \stackrel{d}{=} \sqrt{Y_\beta}B^{\alpha/2}_t$ with $Y_\beta$ an independent one-sided M-Wright random variable: this representation turns the grey model into a randomised volatility-of-volatility rough Bergomi model, makes the Malliavin derivatives computable from those of the underlying fBm, and converts VIX squared into the explicit integral $\mathrm{VIX}_T^2 = \int_T^{T+\Delta}\frac{\xi_0(s)}{E_\beta(bs^{2H})}\zeta_T(s)E_\beta(b(s-T)^{2H})ds$ with $\zeta_T(s)=\sum_{k\ge0}\frac{(\eta c)^k}{k!}\frac{\Gamma(1+k/2)}{\Gamma(1+\beta k/2)}(V^T_s)^k$. This is what carries the paper's semi-closed pricing formulas and the small-time smile asymptotics.

What would settle it

Simulate the calibrated gBergomi model, for example with $(H,\beta,\eta,\rho)=(0.015,0.11,0.42,-1)$, at $T=0.094$ using the paper's own VIX and spot algorithms, compute VIX and SPX implied volatilities by Monte Carlo, and compare them with the limits from Propositions 4.2 and 4.6; if the gaps exceed the market bid-ask spread or the fitted skew and curvature differences, the reported parameters and the joint-calibration conclusion are artefacts of applying $T\downarrow0$ limits at one-month maturity.

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Extended reading notes

Core claim

The central claim is that the system (2.9), with $dS_t/S_t = \sqrt{V_t}(\rho dB_t + \sqrt{1-\rho^2}dW_t)$ and $V_t = \xi_0(t)\,\mathcal{E}^{\lozenge}(\eta\sqrt{Y_\beta}B^H_t)$, where $B^H$ is a Riemann-Liouville fractional Brownian motion and $Y_\beta$ an independent M-Wright variable, preserves self-similarity and stationary increments while letting VIX marginals depart from log-normality. By Lemma 2.3 the grey noise admits the representation $B^{\beta,\alpha}_t \stackrel{d}{=} \sqrt{Y_\beta}B^{\alpha/2}_t$, so the model is exactly rough Bergomi with volatility of volatility randomised by $Y_\beta$. The paper proves the stock price is a true martingale for $\rho\le 0$, derives explicit VIX and VIX-futures expressions and bounds, computes the Malliavin derivatives that drive the asymptotics, and obtains small-time at-the-money limits for the VIX level, skew and curvature (Proposition 4.2) and for the SPX skew (Proposition 4.6). It then calibrates to market data and reports that the ATM VIX moments alone yield $\beta^*=1$, while a full-smile grid search gives $(H,\beta,\eta)=(0.015,0.11,2)$ for the VIX and $\eta=0.42$, $\rho=-1$ for the joint fit, which the authors describe as not particularly accurate, with the short-dated SPX skew the main miss.

Load-bearing premise

The load-bearing premise is that the small-time asymptotic formulas (Propositions 4.2 and 4.6, derived as $T\downarrow0$) are accurate enough at the one-month market maturity $T=0.094$ used in the calibration, since the paper provides no finite-maturity Monte Carlo validation of those formulas.

Editorial extensions

If this is right

  • The VIX smile under gBergomi can slope upward, because VIX squared is no longer a near-log-normal conditional expectation and the M-Wright mixing injects skewness and excess kurtosis into VIX marginals.
  • The small-time limits in Propositions 4.2 and 4.6 give a parameter recipe: VIX ATM level, skew and curvature fix $(H,\beta,\eta)$ through $J_1,J_2,J_3$, and the SPX ATM skew then fixes $\rho$, allowing calibration without Monte Carlo.
  • Since the discounted stock price is a true martingale for $\rho\le 0$ and its running supremum has finite expectation, European option prices under (2.9) are arbitrage-free and numerically well behaved.
  • The paper's own calibration shows two regimes: ATM VIX moments alone return $\beta^*=1$ (the log-normal case), while a full VIX-smile search gives a strongly non-Gaussian $\beta=0.11$; the joint SPX/VIX fit is not yet accurate, with the short-dated SPX skew the main miss.
  • The VIX-futures upper and lower bounds in Proposition 2.10, together with the Markovian approximation in Appendix A, provide practical pricing shortcuts for longer maturities.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because $\beta$ and $\eta$ both scale the effective volatility of volatility, ATM level-skew-curvature data cannot separate them; a natural fix is to calibrate $\beta$ from the full VIX smile or from tail statistics such as those in Appendix B.
  • The representation $B^{\beta,\alpha}_t=\sqrt{Y_\beta}B^{\alpha/2}_t$ suggests a latent-factor reading: $Y_\beta$ is a static volatility-of-volatility shock, so the model should be testable by comparing option-implied moments of VIX with the moment structure $\Gamma(1+\beta k/2)$ inherited from the M-Wright law.
  • The short-maturity SSR limit $H+\tfrac32$ derived in Section 2.4 offers an out-of-sample check: if observed skew-stickiness at short maturities contradicts the calibrated $H$, the model cannot simultaneously match level and dynamics, pointing to the same tension seen in multifactor rough models.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper introduces the 'grey Bergomi' (gBergomi) model, in which the Gaussian driver of rough Bergomi is replaced by a generalized grey Brownian motion, adding a parameter β that makes the VIX marginals non-log-normal while retaining self-similarity and stationary increments. The authors derive the VIX representation, VIX futures bounds, Malliavin-derivative-based small-time asymptotics for both VIX and SPX implied volatilities, and numerical schemes based on truncated Cholesky and Markovian approximations. They then calibrate the model to SPX and VIX options data with maturity T=0.094 and claim that the model is a viable candidate for the joint SPX/VIX calibration problem. The mathematical framework and formulae are the main contribution; the empirical support for the headline claim is the weakest part of the manuscript.

Significance. If the model and its formulae are correct, this is a useful contribution to the rough-volatility literature: it provides a tractable non-log-normal generalization of rough Bergomi, with semi-closed VIX expressions, explicit VIX futures bounds, and Malliavin-based asymptotic smile formulas that reduce to known rough Bergomi results when β=1. The derivation of the VIX formula in Proposition 2.8 and the asymptotic formulas in Propositions 4.2 and 4.6 are consistent with the cited framework and appear algebraically sound. The calibration section, however, does not support the paper's central claim that gBergomi is a viable joint SPX/VIX calibration model: the unconstrained fit returns β*=1, the non-log-normal fit is obtained only after fixing parameters, and the SPX fit is acknowledged to be poor. The manuscript also contains the kernel of a finite-T validation problem, since small-time asymptotic limits are applied at a one-month maturity without checking their accuracy. These issues are fixable, but they currently prevent the empirical claim from being established.

major comments (4)
  1. [Section 5, Propositions 4.2 and 4.6] The calibration is built on T↓0 asymptotic limits applied to market options with maturity T=0.094, but the paper provides no finite-T Monte Carlo comparison, error bound, or convergence-rate estimate for these asymptotics. The reported optima (H*,β*,η*,ρ*)=(0.054,1,0.468,-1) and the later grid values (H,β,η,ρ)=(0.015,0.11,0.42,-1) are obtained entirely from these limits, so the empirical support for the joint-calibration claim depends on an unvalidated bridge between the asymptotics and the data. Remark 5.1 itself acknowledges that the low fitted η may be an artefact of not calibrating to one-day VIX options, which reinforces the need for a finite-T check.
  2. [Section 5 and Abstract] The abstract's claim that gBergomi 'breaks away from the log-Normal assumption' and is a viable joint SPX/VIX calibration model is not supported by the paper's own calibration results. The unconstrained optimization returns β*=1, which is exactly the log-normal rough Bergomi case, and the only non-log-normal calibration (β=0.11) is obtained by fixing η and grid-searching (H,β). After that procedure, Figure 8 shows a visibly poor SPX fit and the text states that the model 'does not achieve a particularly accurate fit, especially with respect to the short-dated skew'. The empirical evidence therefore does not establish the central claim; at most it shows that a generalized specification can produce a non-log-normal VIX smile when the parameters are constrained.
  3. [Remark 5.2] The paper acknowledges that β and η play similar roles and that one can fit the SPX smile with several (β,η) combinations if the VIX smile calibration is ignored. This implies that β is not identified from SPX options alone, which weakens the paper's claim that the non-log-normal feature is a genuine and separately estimable advantage of gBergomi. The manuscript should either provide an identification analysis, for example through the joint VIX and SPX fit, or explicitly limit the claim to the VIX marginals rather than to the joint calibration problem.
  4. [Section 4.2, proof of Proposition 4.6] The proof invokes 'reverse Fatou's lemma' to conclude that condition (iv), limsup_{T↓0} E[(sqrt(V_T/V_0)-1)^2]=0, follows from almost-sure continuity of V_t. Reverse Fatou requires a uniform integrability or domination argument, and none is supplied. Since condition (iv) is one of the hypotheses imported from [44, Proposition 5.1] that justify the SPX skew asymptotics used in the calibration, this is a load-bearing gap in the proof as written; please add the missing L^2 domination estimate or replace the argument with a direct bound.
minor comments (4)
  1. [Figure 2] Both panels of Figure 2 use the legend entry 'ggBergomi' where 'gBergomi' is intended; this typo appears twice.
  2. [Algorithm 3.1, step (iv)] The trapezoidal sum in step (iv) uses Q^2_{T,τ_j} but the grid {τ_j}_{j=0,...,N} is introduced only implicitly in step (i); the notation should be aligned so that the index range and the quadrature weights are unambiguous.
  3. [Section 2.2, around (2.9)] The correspondence between the generalized grey Brownian motion parameter α and the Hurst parameter H (α=2H) is stated only after the Riemann-Liouville replacement; stating it explicitly at the point of the replacement would improve readability.
  4. [Appendix B] The empirical normality tests are applied to VIX log-returns, while the model implication discussed in the introduction concerns the squared VIX being close to log-normal. The connection between these two objects should be stated explicitly so that the reader can see why the tests are evidence against the rBergomi VIX marginal assumption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the gBergomi pricing and asymptotic results are derived from the model dynamics, and Section 5 is an honest calibration rather than a prediction forced by construction.

full rationale

The derivation chain is self-contained with respect to the model specification. System (2.9) defines the gBergomi variance as V_t = ξ0(t) E♦(η sqrt(Yβ) B^H_t); Proposition 2.8 derives the VIX squared as an integral of conditional expectations of this variance, Proposition 2.9 gives the forward variance, and Proposition 2.10 provides futures bounds. The small-time asymptotic results in Propositions 4.2 and 4.6 are not assumed: they are obtained by checking the general Malliavin-derivative assumptions from the published framework [44] and computing the gBergomi-specific quantities J1, J2, J3 and E[D_s V_u] from the model's Malliavin derivatives (4.1). The citation of [44] is load-bearing in the proof but is an independent general theorem whose assumptions here are verified in Lemmas 4.4 and 4.5, so it does not convert the argument into a self-citation loop. The calibration in Section 5 fits model-implied ATM level, skew, and curvature to market VIX data and then to SPX skew; Figures 7 and 8 display in-sample fits, and the paper explicitly reports the unconstrained optimum beta*=1 and the poor SPX fit. Weak empirical support for the joint-calibration claim, and the use of T down to 0 asymptotics at T=0.094 without finite-T validation, are correctness and robustness concerns, not circularity. No fitted quantity is relabelled as a prediction, and no result is equivalent to its inputs by construction.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The model's output depends on cited representations, the Malliavin framework, and market data smoothing. The free parameters H, beta, eta, rho are fitted in multiple stages, and the paper itself notes beta and eta are partly redundant (Remark 5.2). The grey extension introduces no new physical entity beyond the known M-Wright mixture.

free parameters (5)
  • H (Hurst exponent) = 0.054 (first calibration), 0.015 (VIX smile grid search)
    Controls the roughness of the fractional driver; fitted to VIX ATM data and then grid-searched to match the VIX smile.
  • beta (grey parameter) = 1 (first calibration), 0.11 (grid search)
    Controls non-log-normality; fitted to market data, but collapses to 1 in the ATM calibration, meaning the model reduces to rough Bergomi.
  • eta (vol-of-vol) = 0.468 (first calibration), 2 (grid search), 0.4 to 0.42 (joint fit)
    Volatility-of-volatility coefficient; fitted and re-fitted under different calibration stages.
  • rho (correlation) = -1
    Correlation between stock and vol Brownian motions; calibrated by matching the SPX skew.
  • Arctan smoothing parameters (a,b,c,d) = 1.913, 0.746, -2.113, 0.761
    Parameters of the arctan smoothing function used to extract VIX level, skew and curvature from market data; these affect the calibration targets.
assumptions (5)
  • standard math Lemma 2.3: the generalised grey Brownian motion satisfies B^{beta,alpha}_t = sqrt(Y_beta) B^{alpha/2}_t with Y_beta an independent M-Wright random variable (Proposition 3 in [50]).
    The entire model, the VIX formula, and the asymptotic computations rely on this representation; the paper cites but does not prove it.
  • domain assumption The variance process V is in the domain D of Malliavin calculus and satisfies the Lp bounds and continuity assumptions of [44, Proposition 1].
    Used in the proofs of Propositions 4.2 and 4.6; Lemmas 4.4 and 4.5 sketch the verification, but full details are delegated to [44].
  • standard math Clark-Ocone formula gives the martingale representation used for VIX and SPX futures in Section 4.
    Standard Malliavin calculus result from [51]; the paper assumes the integrability conditions hold.
  • domain assumption Risk-neutral valuation: VIX squared equals the conditional expectation of integrated variance over one month, and the discounted stock price is a true martingale for rho <= 0.
    This underpins all option pricing and VIX futures bounds; the martingale proof is given only for rho <= 0, which includes the calibrated rho = -1.
  • domain assumption The Yahoo Finance market data from 26/10/2024 is representative, and the arctan interpolation preserves the true ATM level, skew and curvature.
    Calibration targets are computed from smoothed market data; the paper itself notes data source and election-period stress may affect the fit.

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Pith. "Pith review of Rough Bergomi turns grey." pith.science (2026). https://pith.science/paper/PAD42SYL

@misc{pith2026250508623,
  author       = {Pith},
  title        = {Pith review of: Rough Bergomi turns grey},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PAD42SYL}},
  note         = {Machine review of arXiv:2505.08623}
}
read the original abstract

We propose a tractable extension of the rough Bergomi model, replacing the fractional Brownian motion with a generalised grey Brownian motion, which we show to be reminiscent of models with stochastic volatility of volatility. This extension breaks away from the log-Normal assumption of rough Bergomi, thereby making it a viable suggestion for the Equity Holy Grail -- the joint SPX/VIX options calibration. For this new (class of) model(s), we provide semi-closed and asymptotic formulae for SPX and VIX options and show numerically its potential advantages as well as calibration results.

Figures

Figures reproduced from arXiv: 2505.08623 by the authors.

Figure 1
Figure 1. Upper and lower bounds in all three scenarios. Our lower bound here is clearly not as tight as the one in [43, Theorem 3.2] because of the difference in magnitude between Eβ [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Truncated Cholesky Monte Carlo prices and Monte Carlo standard deviations in rBergomi and gBergomi with the same parameters (except for β). 3.2. Algorithm for SPX options. Based on the above algorithm for the VIX, we now develop a numerical scheme for option prices under (2.9). From the definition of Vt in Proposition 2.8 (and the first line of its proof), then V[Vt] = 1 2H t 2H and E[VtVs] = t H+ s H− H+ 2F1  −H−,… view at source ↗
Figure 3
Figure 3. SPX Call option prices and implied volatilities. 0.3 0.2 0.1 0.0 0.1 0.2 0.3 Log-Moneyness 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 Implied Volatility ggBergomi Implied Volatility = 0 = 1 = 0.9 = 0.8 0.3 0.2 0.1 0.0 0.1 0.2 0.3 Log-Moneyness 0.16 0.18 0.20 0.22 0.24 0.26 Implied Volatility ggBergomi Implied Volatility = 0.3 = 1 = 0.9 = 0.8 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: SPX implied volatilities. following the approach in [6, Chapter 6-8]. We proceed as in [44] and consider a square￾integrable strictly positive process {At}t∈[0,T] , adapted to the filtration F introduced in Section 2.2. We further introduce the F-martingale conditional…
Figure 5
Figure 5. Figure 5: At-the-money VIX level, skew, curvature asymptotics with (H, η, ξ0) = (0.07, 1.23, 0.2352 ), taken from [43, Section 3.3.3]. Proposition 4.6. The following small maturity behaviours hold: lim T↓0 IbT = p ξ0 and lim T↓0 SbT T H+ 3 2 = ρηc √ π (2H + 1)(2H + 3)Γ(1 + 1 2 β…
Figure 6
Figure 6. Figure 6: Short-term SPX skew with (H, η, ρ) = (0.07, 1.23, −1). 5. Joint Calibration We now focus on calibrating the gBergomi model to market data. Using the for￾mulae in Section 4 we require VIX options implied volatility, skew, and curvature and SPX skew data, available on Ya…
Figure 7
Figure 7. Figure 7: VIX smile with T = 0.094, (H, β, η) = (0.015, 0.11, 2). The joint calibration process can be completed by calibrating to the SPX smile via a grid search over (η, ρ). This process results in the calibrated values (η, ρ) = (0.4, −1). Notably, the estimated volatility-of-…
Figure 8
Figure 8. Figure 8: SPX smile, T = 0.094. (H, β, η, ρ) = (0.015, 0.11, 0.42, −1) [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: Interpolated VIX Call options smile, skew, and curvature on 26/10/24 with T = 0.094 and (a, b, c, d) = 1.913, 0.746, −2.113, 0.761). References [1] E. Abi Jaber, The characteristic function of Gaussian stochastic volatility models: an analytic expression, Finance and S…
Figure 10
Figure 10. Figure 10: Gaussian fit to VIX log-returns over several time periods (CBOE data) [PITH_FULL_IMAGE:figures/full_fig_p026_10.png]
Figure 11
Figure 11. Figure 11: QQ plots of VIX log-returns vs popular distributions over several time periods [PITH_FULL_IMAGE:figures/full_fig_p027_11.png]

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Reviewed August 15, 2026 · model on record in the stance chip above.