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arxiv: 2507.23392 · v4 · submitted 2025-07-31 · 💱 q-fin.MF · math.PR

Volatility Modeling with Rough Paths: A Signature-Based Alternative to Classical Expansions

Elisa Al\`os , \`Oscar Bur\'es , Rafael de Santiago , Josep Vives This is my paper

Pith reviewed 2026-05-19 02:28 UTC · model grok-4.3

classification 💱 q-fin.MF math.PR
keywords volatility modelingrough pathssignature methodimplied volatility calibrationrough BergomiHeston modelfractional Brownian motion
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The pith

Signature-based volatility modeling using truncated paths offers a flexible alternative to analytical expansions for calibrating implied volatility surfaces in both Markovian and rough settings.

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

The authors compare analytical approximations for specific volatility models with a signature-based method that expresses volatility as a linear functional of the truncated signature of a driving process. This signature approach is designed to be model-agnostic, allowing it to adapt to various dynamics without presupposing a particular stochastic process. In tests with the Heston model, it matches the accuracy of second-order expansions. For the rough Bergomi model driven by fractional Brownian motion, it performs at least as well as Markovian versions and sometimes better by capturing complex time dependencies. The work shows that while analytical methods excel when the model is correct, the signature framework provides robustness across different volatility behaviors.

Core claim

In the rough Bergomi setting with fractional Brownian motion as the primary process, the signature approach continues to perform strongly and in some cases improves upon the Markovian specification, reflecting its ability to capture more complex temporal dependencies.

What carries the argument

A linear functional applied to the truncated signature of a chosen primary stochastic process, which defines the volatility dynamics in a way that does not require specifying the full model structure in advance.

If this is right

  • When the model dynamics are known precisely, analytical expansions provide fast and accurate calibration formulas.
  • Signature-based models achieve comparable calibration accuracy to analytical methods in the Heston model without using its specific features.
  • In non-Markovian rough Bergomi models, the signature method can outperform Markovian alternatives by encoding longer-range dependencies.
  • The framework supports calibration for a wider range of volatility processes beyond standard assumptions.

Where Pith is reading between the lines

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

  • Signature methods could serve as a baseline for testing whether a proposed stochastic model fully accounts for observed market features.
  • Choosing different primary processes or higher truncation levels might allow the approach to handle even more general path-dependent volatilities.
  • Integration with machine learning techniques for selecting the linear functional could further enhance flexibility in practice.

Load-bearing premise

Volatility dynamics are accurately represented by a linear functional of the truncated signature of some primary stochastic process.

What would settle it

If fitting the signature model to option price data from a known rough Bergomi simulation yields persistently higher errors than the true model even at high truncation levels, this would indicate the linear signature approximation is insufficient.

Figures

Figures reproduced from arXiv: 2507.23392 by Elisa Al\`os, Josep Vives, \`Oscar Bur\'es, Rafael de Santiago.

Figure 1
Figure 1. Figure 1: Comparison of implied volatility surfaces: signature-based vs. second-order expansion. [PITH_FULL_IMAGE:figures/full_fig_p032_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Volatility smiles for T = 0.1 32 [PITH_FULL_IMAGE:figures/full_fig_p032_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Volatility smiles for T = 0.6 [PITH_FULL_IMAGE:figures/full_fig_p033_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Volatility smiles for T = 1.1 [PITH_FULL_IMAGE:figures/full_fig_p033_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Volatility smiles for T = 1.6 As the loss function (5.1) aggregates squared errors across all observed option prices, the global minimum reflects a trade-off: some option prices may be matched very accurately, while others may show larger deviations. To better understand this effect, it is natural to investigate whether calibration accuracy improves when the model is fit to each maturity smile independentl… view at source ↗
Figure 6
Figure 6. Figure 6: Separate smile calibration. 5.2 The Correlated Case We now consider the setting where the market exhibits correlation between the asset price and the volatility process. Specifically, we assume the asset dynamics are governed by the following Heston model dSt = rSt dt + σtSt d  ρWt + p 1 − ρ 2 Bt  dσ2 t = κ(θ − σ 2 t ) dt + ν q σ 2 t dWt, with S0 = 100, σ0 = 0.2, ν = 0.3, κ = 3, θ = 0.09, and correlation… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of implied volatility surfaces: [PITH_FULL_IMAGE:figures/full_fig_p035_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Volatility smile at T = 0.1 [PITH_FULL_IMAGE:figures/full_fig_p036_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Volatility smile at T = 0.6 [PITH_FULL_IMAGE:figures/full_fig_p036_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Volatility smile at T = 1.1. 36 [PITH_FULL_IMAGE:figures/full_fig_p036_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Volatility smile at T = 1.6. 5.3 Calibration with a Rough Bergomi Primary Process The second-order approximation in Alòs et al. (2015) provides an effective tool for calibrating implied volatility surfaces, but assumes that the market is Heston. In contrast, the signature￾based method makes no structural assumptions about the underlying stochastic volatility, offering greater flexibility and robustness ac… view at source ↗
Figure 12
Figure 12. Figure 12: Implied Volatility Surface: Rough Bergomi (market) vs. Signature Model ( [PITH_FULL_IMAGE:figures/full_fig_p038_12.png] view at source ↗
read the original abstract

We study two complementary methodologies for calibrating implied volatility surfaces: analytical approximations and data-driven models based on rough path theory. On the analytical side, we revisit a second-order asymptotic expansion for the Heston model, and we propose a new, VIX-based calibration scheme for the rough Bergomi model. Both methods yield highly accurate and computationally efficient calibration formulas when the underlying dynamics are well specified. In parallel, we develop a signature-based approach in which volatility is represented as a linear functional of the truncated signature of a primary stochastic process, providing a flexible and model-agnostic alternative. Our numerical experiments compare the two approaches across both Markovian and non-Markovian settings. In the Heston case, signature-based models achieve a level of accuracy comparable to analytical expansions. In the rough Bergomi setting, using a fractional Brownian motion as the primary process, the signature approach continues to perform strongly and in some cases improves upon the Markovian specification, reflecting its ability to capture more complex temporal dependencies. Overall, the results illustrate that analytical methods are highly effective when the model is correctly specified, while signature-based methods offer a robust and flexible framework for calibration across a wider range of volatility dynamics.

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 paper develops two complementary approaches for calibrating implied volatility surfaces: analytical approximations (a revisited second-order expansion for the Heston model and a new VIX-based scheme for rough Bergomi) and a signature-based method in which volatility is expressed as a linear functional of the truncated signature of a primary process such as fractional Brownian motion. Numerical experiments are reported to show that the signature models achieve accuracy comparable to the analytical expansions in the Heston setting and perform strongly (sometimes improving on a Markovian specification) in the rough Bergomi setting by capturing more complex temporal dependencies.

Significance. If the performance claims hold under rigorous controls, the signature framework supplies a flexible, model-agnostic alternative to classical expansions that can accommodate non-Markovian volatility dynamics without requiring a fully specified parametric model. The combination of analytical and data-driven routes is a constructive contribution to the rough-volatility literature.

major comments (2)
  1. [Numerical experiments] Numerical experiments section: the truncation order of the signature is never stated for the rough Bergomi experiments. Because a low-order truncation of the signature of fBM yields a finite-dimensional Markovian feature map, the reported improvement over the Markovian specification cannot be unambiguously attributed to rough-path structure rather than to the particular linear regression or in-sample fitting. This is load-bearing for the central claim that the method captures 'more complex temporal dependencies.'
  2. [Numerical experiments] Numerical experiments section: no error bars, cross-validation protocol, or explicit out-of-sample test set are described for the performance metrics that underpin the claim of 'comparable or superior accuracy.' Without these controls it is impossible to assess whether the reported gains survive post-hoc choices of primary process or truncation level.
minor comments (2)
  1. [Abstract] The abstract states that the signature approach 'continues to perform strongly' in rough Bergomi; the corresponding table or figure should report the precise truncation level and the exact metric (RMSE, relative error, etc.) used for each comparison.
  2. [Signature-based approach] Notation for the linear functional coefficients and the signature truncation should be introduced once in a dedicated subsection and used consistently thereafter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions regarding the numerical experiments. We agree that greater transparency on implementation details and validation protocols will strengthen the paper. We respond to each major comment below and indicate the changes we will incorporate.

read point-by-point responses

  1. Referee: Numerical experiments section: the truncation order of the signature is never stated for the rough Bergomi experiments. Because a low-order truncation of the signature of fBM yields a finite-dimensional Markovian feature map, the reported improvement over the Markovian specification cannot be unambiguously attributed to rough-path structure rather than to the particular linear regression or in-sample fitting. This is load-bearing for the central claim that the method captures 'more complex temporal dependencies.'

    Authors: We acknowledge the omission and the validity of the concern. In the revised manuscript we will explicitly state the truncation level employed for the rough Bergomi experiments and add a short paragraph clarifying that, while any finite truncation produces a finite-dimensional lifted process, the iterated integrals of the fractional Brownian motion primary process encode the specific covariance structure and roughness that are absent from a standard Brownian-motion-driven Markovian specification. This distinction underpins the reported performance differences and will be made explicit so that readers can assess the source of the gains. revision: yes

  2. Referee: Numerical experiments section: no error bars, cross-validation protocol, or explicit out-of-sample test set are described for the performance metrics that underpin the claim of 'comparable or superior accuracy.' Without these controls it is impossible to assess whether the reported gains survive post-hoc choices of primary process or truncation level.

    Authors: We agree that these controls are necessary for a rigorous assessment. The revised version will report error bars obtained from repeated simulations with varied random seeds, describe the cross-validation scheme used for hyper-parameter selection, and specify the composition and size of the held-out test set together with the corresponding out-of-sample metrics. These additions will allow direct evaluation of robustness to the choice of primary process and truncation level. revision: yes

Circularity Check

1 steps flagged

Signature linear coefficients fitted to calibration data; reported performance gains reduce to in-sample fit

specific steps
  1. fitted input called prediction [Abstract]
    "In the rough Bergomi setting, using a fractional Brownian motion as the primary process, the signature approach continues to perform strongly and in some cases improves upon the Markovian specification, reflecting its ability to capture more complex temporal dependencies."

    The linear functional coefficients are obtained by fitting to the calibration data; the quoted performance comparison and attribution to 'complex temporal dependencies' are then measured on the same fitted representation, so the claimed improvement reduces to a property of the regression fit rather than an external validation of the signature truncation or primary process.

full rationale

The paper defines volatility as a linear functional of the truncated signature and fits its coefficients to calibration data. The headline claim that this yields improvements over Markovian models in rough Bergomi (by capturing complex dependencies) is then evaluated on the same fitted quantities. This matches the fitted-input-called-prediction pattern: the reported superiority is statistically forced by the regression step rather than an independent test of the rough-path structure. No self-citation chain or definitional loop is present, so the circularity is partial and limited to the numerical claims.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claims rest on the mathematical properties of rough paths and signatures plus the modeling choice that a linear functional suffices; no new physical entities are postulated, but the linear representation and choice of primary process are key unproven assumptions for the target application.

free parameters (2)
  • linear functional coefficients
    Coefficients that define volatility as a linear combination of signature terms are determined during calibration to data.
  • signature truncation order
    The depth to which the signature is truncated is a modeling choice that controls expressivity and must be selected.
axioms (1)
  • domain assumption Volatility can be represented as a linear functional of the truncated signature of a primary stochastic process.
    This is the foundational premise of the signature-based methodology described in the abstract.

pith-pipeline@v0.9.0 · 5750 in / 1501 out tokens · 48344 ms · 2026-05-19T02:28:50.087988+00:00 · methodology

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Lean theorems connected to this paper

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    Relation between the paper passage and the cited Recognition theorem.

    volatility is represented as a linear functional of the truncated signature of a primary stochastic process... signature approach continues to perform strongly... in the rough Bergomi setting, using a fractional Brownian motion as the primary process

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Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

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