Signature-Based Calibration Method

• Signature-Based Calibration Method is a data-driven, model-agnostic approach that uses the algebraic structure of path signatures to capture complex volatility dynamics.
• It employs a linear functional on truncated signature terms with time augmentation to efficiently calibrate volatility and option prices across diverse market regimes.
• The method achieves high accuracy and robustness, outperforming classical techniques, especially in rough or non-Markovian environments.

A signature-based calibration method is a data-driven, model-agnostic approach for calibrating processes or system parameters by utilizing the algebraic structure of the signature of a primary stochastic path. The method replaces explicit parametric volatility or response models with a linear functional acting on the (possibly truncated) signature of a driving process, allowing for flexible non-linear modeling and efficient learning even in contexts exhibiting rough or non-Markovian dynamics. Its universality and robustness arise from the fundamental result that continuous functionals on path space can be approximated arbitrarily well by linear functionals on the path signature.

1. Mathematical Formulation of Signature-Based Calibration

The signature-based calibration methodology seeks to express a complex process—most notably a volatility process $\sigma_t$—as a linear combination of the signature coefficients of a primary path $X$. For a discounted asset price dynamic

$$d\widetilde{S}_t = \widetilde{S}_t\,\sigma_t\,dZ_t,$$

the instantaneous volatility is modeled as a functional: $\sigma_t = f\big((X_s)_{s\leq t}\big),$ where $f$ is to be inferred from data.

The core idea is to approximate $f$ by a linear functional on the truncated signature: $$\sigma_t(\ell) = \sum_{|I| \leq N} \ell_I\,\langle e_I, S(X)_t^{(\leq N)} \rangle,$$ where $\ell = (\ell_I)$ are the learnable parameters, $I$ are multi-indices enumerating the signature terms, $(e_I)$ is a basis for the truncated tensor algebra $T^{(N)}$, and $S(X)_t^{(\leq N)}$ is the signature truncated at level $N$. The signature $S(X)_t$ comprises all iterated integrals of the path $X$ up to time $t$.

For practical calibration, time augmentation is employed, denoting $\widehat{X}_t = (t, X_t)$, so the signature vector includes both time and state variables, with typical signature terms including $1$, $t$, $X_t$, $\int_0^t s dX_s$, etc., at the desired truncation level.

2. Calibration Workflow and Optimization Problem

The calibration procedure fits the coefficient vector $\ell$ by minimizing the discrepancy between model-derived and market-observed derivatives prices. The workflow involves:

• Simulating paths of the primary process $X$ as needed;
• Computing signature features $S(X)_t^{(\leq N)}$ along those paths;
• Expressing the process of interest (e.g., $\sigma_t$ or an option price) as a linear functional of the computed signature;
• Optimizing $\ell$ to minimize a loss, typically:

$$L(\ell) = \sum_i \gamma_i \big[ C^{\mathrm{mkt}}(K_i, T_i) - C(K_i, T_i, \ell) \big]^2,$$

where $C^{\mathrm{mkt}}$ are reference prices, $C(\cdot,\ell)$ are model prices under the signature parametrization, and $\gamma_i$ are, for example, Vega-based weights.

The mapping from signature features to volatility, and then to option prices, is linear in $\ell$, which ensures that the loss landscape is (locally) convex and amenable to gradient-based or least-squares optimization techniques.

3. Comparison with Classical Parametric and Asymptotic Methods

Traditional calibration relies on specific parametric models, such as the Heston model, and may utilize analytic formulas or asymptotic expansions (e.g., via Malliavin calculus) for implied volatility surfaces under precise model assumptions. In these methods, short-maturity or perturbative expansions yield explicit formulas relating implied volatility to Heston or similar parameters, often of the form: $$I(0, K) \approx \sigma_0 - \frac{\rho\nu}{4\sigma_0}(x-k) + \text{higher-order terms},$$ where $I$ is the implied volatility, and $(\sigma_0,\nu,\rho,\ldots)$ are model parameters.

The signature-based method, in contrast, offers:

• Model-Independence: No reliance on predetermined dynamics such as Heston; capable of fitting volatility driven by rough paths or non-Markovian sources.
• Robustness: Because any continuous functional of the path can be approximated in signature space (as per the Stone–Weierstrass theorem), the approach adapts to complex market-driven dynamics.
• Flexibility: The methodology applies as long as iterated integrals of the chosen primary process are computable or approximable, subsuming both classical and rough volatility settings.

Classical asymptotic expansions may lose accuracy outside their formal model assumptions (e.g., for pronounced rough or nonlocal volatility), whereas the signature-based alternative maintains high fidelity in these contexts.

4. Universality, Regularity, and Theoretical Underpinning

The theoretical foundation of the signature-based calibration method is the universality property: the truncated signature forms a basis capable of uniformly approximating continuous functionals on path space. This enables the model to capture and calibrate rich non-linear dynamics without prior specification of a particular structural form.

The path signature encodes all polynomially computable information about the path; in rough paths theory, signatures remain well-defined even for paths of low regularity (e.g., those with Hölder continuity below $1/2$). Numerically, low truncation levels (e.g., $N=3$) are often sufficient to achieve high calibration accuracy in practice, as demonstrated by calibration errors in implied volatility of order $10^{-3}$ or better in both classical and rough volatility regimes.

5. Numerical Experiments and Empirical Performance

Extensive experiments assess the method’s effectiveness in both canonical (Heston) and rough volatility regimes:

• In the uncorrelated case ($\rho = 0$) under Heston dynamics, signature-based calibration typically achieves a loss $L(\ell)$ of order $10^{-4}$, with differences in implied volatility vs. benchmarks less than 0.001.
• In the correlated case ($\rho = -0.5$), results remain similarly accurate, with absolute errors between signature-based and market-implied volatilities frequently in the $10^{-4}$–$10^{-3}$ range.
• When the synthetic market data is generated from a rough Bergomi volatility process (a model with fractional, rough features not captured by classical expansion), the signature method accurately recovers the implied volatility surface, achieving comparable performance.

These findings underscore that, unlike asymptotic expansions tailored to specific models, signature-based calibration offers model-independent and robust performance across market regimes.

6. Practical and Theoretical Implications

The signature-based calibration method demonstrates key strengths:

• Adaptability: Universality enables applicability to a broader class of models with minimal adaptation, including stochastic and rough volatility processes.
• Efficiency: For moderate truncation levels, the resultant calibration is computationally tractable and, for pricing, can be further accelerated by pre-computation of signature paths.
• Interpretability: The linear structure on signature space provides transparency in the mapping between path features and inferred quantities.
• Robustness: Maintains high accuracy in calibration error across traditional and rough environments, as observed in numerical comparison studies.

A plausible implication is that signature-based calibration methods could supersede model-specific approaches in applications where volatility processes are believed to be rough or non-Markovian, or where model risk is an overriding concern.

7. Limitations and Outlook

While signature-based calibration is shown to be robust and accurate, possible limitations include the dimensionality of the truncated signature for high $N$ or multidimensional $X$, which could pose computational and overfitting challenges. The selection of truncation level, the choice of primary process, and the interpretation of fitted coefficients remain areas of methodological discretion. Ongoing research addresses these challenges via regularization, efficient feature selection, and hybrid approaches integrating prior knowledge with data-driven signatures.

The methodology is poised for wider adoption in quantitative finance, particularly as empirical evidence mounts for persistent roughness and model uncertainty in financial volatility processes (Alòs et al., 31 Jul 2025).