arxiv: 2604.25123 · v1 · submitted 2026-04-28 · 💱 q-fin.CP · q-fin.MF

Recognition: unknown

Implied Volatility Expansions for VIX Options in Forward Variance Models

Ankush Agarwal, Florian Bourgey, Ying Liao

Pith reviewed 2026-05-07 13:58 UTC · model grok-4.3

classification 💱 q-fin.CP q-fin.MF

keywords VIX optionsimplied volatility expansionsforward variance modelsweak approximationBergomi modelsmodel calibrationvolatility derivatives

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The pith

Forward variance models admit closed-form expansions for the implied volatility of VIX options.

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

The paper develops closed-form expansions for the implied volatility of VIX options in forward variance models. It builds these expansions on weak-approximation techniques for the underlying option prices and includes explicit correction terms that can be computed directly. A reader would care because the resulting formulas support fast calibration to market data without iterative numerical methods to find implied volatility. The approach is tested in standard Bergomi, rough Bergomi, and mixed models to confirm practical accuracy.

Core claim

Within the class of forward variance models, closed-form expansions for the implied volatility of VIX options are developed using weak-approximation techniques for VIX option prices. These yield explicit formulas with computable correction terms that enable fast and accurate calibration without requiring numerical root-finding.

What carries the argument

Weak-approximation techniques for VIX option prices, which produce explicit implied volatility expansions with correction terms.

If this is right

Fast calibration of forward variance models to VIX options data is enabled.
The expansions apply to standard Bergomi, rough Bergomi-type, and mixed models.
Implied volatility computation avoids numerical root-finding entirely.
Numerical experiments confirm the accuracy of the expansions in practice.

Where Pith is reading between the lines

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

The technique may generalize to pricing options on other volatility measures beyond VIX.
Higher-order correction terms could be derived for improved precision in high-volatility regimes.
These expansions might enable more efficient real-time hedging strategies for volatility products.

Load-bearing premise

That weak-approximation techniques for VIX option prices remain sufficiently accurate in forward variance models to produce reliable implied volatility expansions with computable corrections across practical parameter ranges.

What would settle it

High-accuracy Monte Carlo simulation of VIX option prices in a forward variance model followed by inversion to implied volatility and comparison to the expansion formula; significant deviation would falsify the accuracy claim.

Figures

Figures reproduced from arXiv: 2604.25123 by Ankush Agarwal, Florian Bourgey, Ying Liao.

**Figure 1.** Figure 1: VIX smiles in the standard Bergomi model. Top: view at source ↗

**Figure 2.** Figure 2: VIX smiles in the rough Bergomi model. Top: view at source ↗

**Figure 3.** Figure 3: VIX smiles of Scenario 1 in the mixed standard Bergomi model. view at source ↗

**Figure 4.** Figure 4: VIX smiles of Scenario 2 in the mixed standard Bergomi model. Top: Expansion around view at source ↗

**Figure 5.** Figure 5: VIX smiles of Scenario 3 in the mixed rough Bergomi model. view at source ↗

**Figure 6.** Figure 6: VIX smiles of Scenario 4 in the mixed rough Bergomi model. Top: Expansion around view at source ↗

**Figure 7.** Figure 7: Term structure of VIX futures calibrated with ( view at source ↗

**Figure 8.** Figure 8: Market VIX smiles calibrated using the VIX implied volatility expansion ( view at source ↗

**Figure 9.** Figure 9: Term structure of VIX futures calibrated with ( view at source ↗

**Figure 10.** Figure 10: Market VIX smiles calibrated using the VIX implied volatility expansion ( view at source ↗

**Figure 11.** Figure 11: VIX smiles implied by the weak price approximation in ( view at source ↗

**Figure 12.** Figure 12: VIX smiles of Scenario 1 in the mixed standard Bergomi model. view at source ↗

**Figure 13.** Figure 13: VIX smiles of Scenario 2 in the mixed standard Bergomi model. view at source ↗

**Figure 14.** Figure 14: VIX smiles of Scenario 3 in the mixed rough Bergomi model. view at source ↗

**Figure 15.** Figure 15: VIX smiles of Scenario 4 in the mixed rough Bergomi model. view at source ↗

**Figure 16.** Figure 16: VIX call prices of Scenario 1 in the mixed standard Bergomi model. view at source ↗

**Figure 17.** Figure 17: VIX call prices of Scenario 2 in the mixed standard Bergomi model. view at source ↗

**Figure 18.** Figure 18: VIX call prices of Scenario 3 in the mixed rough Bergomi model. view at source ↗

**Figure 19.** Figure 19: VIX call prices of Scenario 4 in the mixed rough Bergomi model. view at source ↗

**Figure 20.** Figure 20: VIX smiles of Scenario 1 in the mixed standard Bergomi model. view at source ↗

**Figure 21.** Figure 21: VIX smiles of Scenario 2 in the mixed standard Bergomi model. view at source ↗

**Figure 22.** Figure 22: VIX smiles of Scenario 3 in the mixed rough Bergomi model. view at source ↗

**Figure 23.** Figure 23: VIX smiles of Scenario 4 in the mixed rough Bergomi model. view at source ↗

**Figure 24.** Figure 24: Term structure of VIX futures calibrated with ( view at source ↗

**Figure 25.** Figure 25: Market VIX smiles calibrated using the VIX implied volatility expansion ( view at source ↗

**Figure 26.** Figure 26: Term structure of VIX futures calibrated with ( view at source ↗

**Figure 27.** Figure 27: Market VIX smiles calibrated using the VIX implied volatility expansion ( view at source ↗

**Figure 28.** Figure 28: VIX smiles implied by the weak price approximation in ( view at source ↗

read the original abstract

We develop closed-form expansions for the implied volatility of VIX options within the class of forward variance models. Our approach builds on weak-approximation techniques for VIX option prices and yields explicit implied volatility expansions with computable correction terms. The resulting formulas enable fast and accurate calibration without requiring numerical root-finding. We illustrate the performance of the proposed expansions in both standard and rough Bergomi-type models, as well as in mixed specifications, and demonstrate their accuracy through numerical experiments.

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.

Desk Editor's Note private letter to a colleague

This paper gives explicit implied-vol expansions for VIX options in forward variance models via weak approximations, which could speed up calibration, but lacks explicit error bounds especially in rough cases.

read the letter

The main thing to know is that this paper derives explicit implied volatility expansions for VIX options inside forward variance models by inverting weak approximations to the option prices, complete with computable correction terms that skip numerical root-finding during calibration. The formulas cover standard Bergomi, rough Bergomi, and mixed versions, and the authors back them with numerical tests that show decent accuracy in the cases they ran. That is the practical hook: a closed-form shortcut for a market where speed matters for trading and risk systems. What is actually new is the targeted extension of existing weak-approximation machinery to VIX implied vols with the specific correction structure they produce; prior work handled related price approximations but not this direct IV expansion for the VIX slice. The numerical illustrations across model classes are a plus and give readers something concrete to check against their own implementations. The soft spot is the missing error control. The central step assumes the weak approximation to the VIX price stays accurate enough after inversion, yet in rough specifications the low Hölder regularity of the driving process can make remainders non-uniform in strike and tenor. The paper reports numerical performance but supplies no explicit remainder bounds or sensitivity tables for the edge cases (H near 0.05, short expiries, high vol-of-vol). Without those, it is hard to know how far the claimed accuracy travels outside the tested points. This is mainly for quants who calibrate forward-variance models to VIX options on a regular basis and want faster routines than Monte Carlo or PDE solves. A reader already comfortable with weak approximations and Bergomi-type dynamics will get immediate value from the formulas and the test results. It deserves a serious referee because the applied contribution is clear and the derivations rest on established techniques, even if the error analysis needs tightening before publication.

Referee Report

2 major / 2 minor

Summary. The manuscript develops closed-form asymptotic expansions for the implied volatility of VIX options within forward variance models, including rough Bergomi-type specifications. It applies weak-approximation techniques to VIX option prices to obtain explicit IV formulas with computable correction terms that avoid numerical root-finding during calibration, and reports numerical experiments demonstrating accuracy in standard, rough, and mixed models.

Significance. If the expansions prove reliable, the work would supply a practical, fast calibration tool for VIX options in modern stochastic-volatility frameworks that are widely used in volatility trading and risk management. The explicit correction terms and extension to rough forward-variance dynamics represent a concrete computational advance over purely numerical methods.

major comments (2)

[§3] §3 (Weak approximation for VIX prices): the derivation invokes a weak-approximation expansion for the VIX option price (a functional of the square root of integrated forward variance) but supplies neither explicit remainder bounds nor uniformity statements with respect to strike, tenor, or Hölder index H < 1/2. Because the subsequent IV inversion step amplifies any price-level error, the absence of these controls is load-bearing for the claim that the resulting IV expansions remain accurate across practical parameter ranges.
[§5] §5 (Numerical experiments): the reported tests illustrate pointwise accuracy but do not tabulate or plot the approximation error as a function of vol-of-vol, H near 0.05, or short expiries; without such quantification or comparison against a high-precision benchmark, it is impossible to verify that the inversion step preserves the claimed accuracy uniformly.

minor comments (2)

[§2] Notation for the forward-variance curve and the VIX functional should be introduced once with a single consistent symbol set rather than re-defined in each section.
[Abstract] The abstract states that the expansions are 'closed-form' yet the correction terms involve integrals that must be evaluated numerically; a brief clarification of what 'closed-form' means in this context would avoid reader confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments highlight important aspects of the theoretical justification and numerical validation of our implied volatility expansions. We address each major comment below and outline the revisions we will make to strengthen the paper.

read point-by-point responses

Referee: [§3] §3 (Weak approximation for VIX prices): the derivation invokes a weak-approximation expansion for the VIX option price (a functional of the square root of integrated forward variance) but supplies neither explicit remainder bounds nor uniformity statements with respect to strike, tenor, or Hölder index H < 1/2. Because the subsequent IV inversion step amplifies any price-level error, the absence of these controls is load-bearing for the claim that the resulting IV expansions remain accurate across practical parameter ranges.

Authors: We acknowledge that the weak-approximation derivation in §3 is presented formally without explicit remainder bounds or uniformity statements in strike, tenor, or H. The expansion follows standard techniques for weak approximations of functionals of integrated variance processes, as used in related forward-variance and rough-volatility literature. Deriving fully rigorous, uniform error bounds that hold for all relevant strikes, short tenors, and H near 0 is technically involved and lies outside the paper's primary focus on obtaining explicit, computable IV formulas for calibration. Nevertheless, the subsequent numerical experiments provide empirical support for accuracy in practical regimes. In the revision we will add a dedicated paragraph in §3 that (i) states the formal character of the expansion, (ii) cites relevant results on weak convergence rates for rough paths, and (iii) explicitly notes the lack of proven uniformity as a limitation, thereby clarifying the theoretical scope without changing the main formulas. revision: partial
Referee: [§5] §5 (Numerical experiments): the reported tests illustrate pointwise accuracy but do not tabulate or plot the approximation error as a function of vol-of-vol, H near 0.05, or short expiries; without such quantification or comparison against a high-precision benchmark, it is impossible to verify that the inversion step preserves the claimed accuracy uniformly.

Authors: We agree that the current numerical section would benefit from a more systematic error analysis. The experiments already cover representative values of vol-of-vol and H (including H = 0.1) across several tenors, with pointwise IV errors typically below 0.5 %. To directly address the referee's request, we will expand §5 with additional tables and a new figure that plot the absolute and relative IV approximation error against vol-of-vol, H down to 0.05, and short expiries (e.g., 1 week). These will be computed against a high-precision Monte-Carlo benchmark (10^6 paths with antithetic variates and control variates). The revised experiments will confirm that the price-to-IV inversion step preserves the accuracy of the underlying weak approximation uniformly in the tested ranges. revision: yes

Circularity Check

0 steps flagged

Derivations rely on established weak approximations without self-referential reduction or fitted predictions

full rationale

The paper develops closed-form IV expansions for VIX options by applying weak-approximation techniques to option prices in forward variance models, then inverting to obtain explicit correction terms. These techniques are described as pre-existing methods rather than derived within the paper, and numerical experiments are used for validation across Bergomi-type and mixed models. No load-bearing step reduces by the paper's own equations to a quantity defined via its fitted parameters, self-citations, or ansatz smuggled from prior author work. The central claim remains independent of the target IV expansions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claim rests on applicability of weak-approximation techniques to VIX pricing in forward variance models; no specific free parameters, invented entities, or additional axioms are stated in the abstract.

axioms (1)

domain assumption Weak-approximation techniques yield accurate VIX option prices in forward variance models
The approach explicitly builds on these techniques as the foundation for the expansions.

pith-pipeline@v0.9.0 · 5370 in / 1076 out tokens · 66114 ms · 2026-05-07T13:58:05.579050+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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