Signature Volatility Models

• Signature volatility models are stochastic models that represent volatility as a linear functional of path signatures, extending classical frameworks like Heston and Bergomi.
• They leverage iterated integrals and the shuffle algebra to maintain tractability for Fourier pricing, Monte Carlo calibration, and quadratic hedging.
• These models enable efficient calibration and hedging in both Markovian and non-Markovian settings, offering robust tools for pricing exotic and path-dependent options.

Signature volatility models are stochastic volatility models in which the instantaneous volatility is represented by a linear functional of the signature of a time-extended primary process, typically $\widehat W_t=(t,W_t)$ or the time-augmented signature of a polynomial diffusion. In this setting, signature coordinates are iterated Stratonovich integrals, volatility is encoded as $\Sigma_t=\langle \xi_t,\sigma_t\rangle$, and the resulting asset dynamics can remain compatible with Fourier pricing, Monte Carlo calibration, quadratic hedging, and path-dependent valuation. The literature presents the framework as universal in the sense that it includes, but is not limited to, Stein–Stein, Bergomi, and some Heston-type constructions, while also extending to path-dependent and non-Markovian settings (Jaber et al., 2024).

1. Canonical definition and model architecture

The canonical construction starts from the signature of a time-extended path. For $\widehat W_t=(t,W_t)$, the signature is

$$\sigma_t = \left(1,\sigma_t^1,\sigma_t^2,\ldots\right),\qquad \sigma_t^n = \int_{0<u_1<\cdots<u_n<t} \circ d\widehat W_{u_1}\otimes\cdots\otimes \circ d\widehat W_{u_n}.$$

A signature volatility model then specifies

$$\frac{dS_t}{S_t}=\Sigma_t\, dB_t,\qquad B_t=\rho W_t+\sqrt{1-\rho^2}\,W_t^\perp,\qquad \Sigma_t=\langle \xi_t,\sigma_t\rangle,$$

with $\xi_t$ valued in the extended tensor algebra (Jaber et al., 2024).

A closely related formulation uses a time-augmented primary process $\widehat X_t=(t,X_t)$ or $\widehat X_t=(t,X_t,[X]_t)$, where $X$ is a continuous semimartingale or polynomial diffusion. In that representation, volatility is written as

$$\sigma_t^S(\ell) := \ell_{\emptyset} + \sum_{0<|I|\le n}\ell_I \langle e_I,\widehat{\mathbb X}_t\rangle,$$

so that the stock price evolves under the risk-neutral measure as

$\Sigma_t=\langle \xi_t,\sigma_t\rangle$0

This formulation makes the choice of primary process a hyperparameter, while the coefficient vector $\Sigma_t=\langle \xi_t,\sigma_t\rangle$1 is calibrated to data (Cuchiero et al., 2023).

The model class is designed to preserve linearity at the level of features while allowing nonlinear dependence on the driving path. In the more general “signature-based models” framework, the same process may be rewritten in stochastic-integral form with coefficients that are themselves linear combinations of signature coordinates of the time-extended primary process. This makes the signature expansion the central state representation, rather than an auxiliary feature map (Cuchiero et al., 2022).

2. Algebraic and probabilistic foundations

The basic tractability mechanism is the shuffle algebra. If $\Sigma_t=\langle \xi_t,\sigma_t\rangle$2 are admissible coefficient tensors, then

$\Sigma_t=\langle \xi_t,\sigma_t\rangle$3

This identity turns products of signature-linear functionals into linear functionals with shuffled coefficients. Together with signature Itô calculus,

$\Sigma_t=\langle \xi_t,\sigma_t\rangle$4

it is the main algebraic reason that volatility dynamics, integrated variance, and transform formulas remain manageable in otherwise high-dimensional models (Jaber et al., 2024).

A more structural formulation places the model on a weighted tensor algebra

$\Sigma_t=\langle \xi_t,\sigma_t\rangle$5

with volatility functional $\Sigma_t=\langle \xi_t,\sigma_t\rangle$6. Under the summability condition H1 and the exponential-integrability condition H3, the signature SDE

$\Sigma_t=\langle \xi_t,\sigma_t\rangle$7

admits a unique strong solution in $\Sigma_t=\langle \xi_t,\sigma_t\rangle$8, the solution is strictly positive almost surely, and $\Sigma_t=\langle \xi_t,\sigma_t\rangle$9. The solution is constructed as the Doléans–Dade exponential

$\widehat W_t=(t,W_t)$0

The same work separates two issues that are often conflated: H3 yields NFLVR because the stochastic exponential is a true martingale, whereas global solvability of the associated infinite-dimensional Riccati equation is an additional condition equivalent to absence of explosion for finite signature transforms (Xodarev, 16 May 2026).

Martingality imposes further constraints in finite-order models. Excluding trivial cases, if the volatility is given by a finite linear combination of signature words up to order $\widehat W_t=(t,W_t)$1 and the leading coefficient $\widehat W_t=(t,W_t)$2, then for $\widehat W_t=(t,W_t)$3 and $\widehat W_t=(t,W_t)$4 the price process is a martingale if and only if $\widehat W_t=(t,W_t)$5 is odd and $\widehat W_t=(t,W_t)$6. The same paper characterizes higher moments by the threshold

$\widehat W_t=(t,W_t)$7

under the martingale regime. A practical implication stated explicitly is that truncation order should be chosen odd if martingality is to be preserved in approximation schemes (Jaber et al., 21 Mar 2025).

3. Universality and relation to classical and non-Markovian volatility models

The framework is presented as universal in two distinct senses. First, there is an approximation-theoretic statement: continuous path functionals on compact sets can be uniformly approximated by linear functionals of signatures. In the time-extended semimartingale framework, this is obtained through a Stone–Weierstrass argument using inclusion of constants, point separation by signatures, and the algebra property induced by the shuffle product (Cuchiero et al., 2022).

Second, several classical volatility models admit exact or near-exact signature representations. For the Ornstein–Uhlenbeck process

$\widehat W_t=(t,W_t)$8

the paper gives the exact representation

$\widehat W_t=(t,W_t)$9

For the mean-reverting geometric Brownian motion

$$\sigma_t = \left(1,\sigma_t^1,\sigma_t^2,\ldots\right),\qquad \sigma_t^n = \int_{0<u_1<\cdots<u_n<t} \circ d\widehat W_{u_1}\otimes\cdots\otimes \circ d\widehat W_{u_n}.$$0

an exact signature coefficient is also provided, covering the Hull–White volatility model and Dupire’s forward variance specification as special cases. The same work treats stochastic Volterra processes, delayed equations, Riemann–Liouville fractional Brownian motion, and rough Bergomi-type volatility processes as signature-linear functionals. For the square-root CIR process

$$\sigma_t = \left(1,\sigma_t^1,\sigma_t^2,\ldots\right),\qquad \sigma_t^n = \int_{0<u_1<\cdots<u_n<t} \circ d\widehat W_{u_1}\otimes\cdots\otimes \circ d\widehat W_{u_n}.$$1

it writes a nonlinear fixed-point relation for the signature coefficient and validates the representation numerically; this is the mechanism used there for Heston-type volatility (Jaber et al., 2024).

A later non-Markovian extension reformulates option pricing with volatility process $$\sigma_t = \left(1,\sigma_t^1,\sigma_t^2,\ldots\right),\qquad \sigma_t^n = \int_{0<u_1<\cdots<u_n<t} \circ d\widehat W_{u_1}\otimes\cdots\otimes \circ d\widehat W_{u_n}.$$2 as a rough stochastic differential equation driven by

$$\sigma_t = \left(1,\sigma_t^1,\sigma_t^2,\ldots\right),\qquad \sigma_t^n = \int_{0<u_1<\cdots<u_n<t} \circ d\widehat W_{u_1}\otimes\cdots\otimes \circ d\widehat W_{u_n}.$$3

and then represents $$\sigma_t = \left(1,\sigma_t^1,\sigma_t^2,\ldots\right),\qquad \sigma_t^n = \int_{0<u_1<\cdots<u_n<t} \circ d\widehat W_{u_1}\otimes\cdots\otimes \circ d\widehat W_{u_n}.$$4 as either a linear signature functional

$$\sigma_t = \left(1,\sigma_t^1,\sigma_t^2,\ldots\right),\qquad \sigma_t^n = \int_{0<u_1<\cdots<u_n<t} \circ d\widehat W_{u_1}\otimes\cdots\otimes \circ d\widehat W_{u_n}.$$5

or a nonlinear neural map on truncated signatures. In the linear case, the truncation error exhibits factorial decay; in the nonlinear case, the pricing error is controlled by the sum of neural approximation error and signature truncation error. Numerical examples in that work report that deep nonlinear signatures are especially effective for rough Heston and rough Bergomi, while linear signature models are particularly effective for OU and mean-reverting GBM volatility (Ma et al., 21 Aug 2025).

Taken together, these results imply that signature volatility models are not confined to a single economic mechanism. They encompass affine, non-affine, rough, delayed, and path-dependent constructions, while preserving a common algebraic representation of volatility.

4. Transform methods, pricing formulas, and calibration regimes

One central analytical result is a joint characteristic functional for log-price and integrated variance. If an infinite-dimensional extended tensor algebra-valued Riccati equation admits a solution, then

$$\sigma_t = \left(1,\sigma_t^1,\sigma_t^2,\ldots\right),\qquad \sigma_t^n = \int_{0<u_1<\cdots<u_n<t} \circ d\widehat W_{u_1}\otimes\cdots\otimes \circ d\widehat W_{u_n}.$$6

with $$\sigma_t = \left(1,\sigma_t^1,\sigma_t^2,\ldots\right),\qquad \sigma_t^n = \int_{0<u_1<\cdots<u_n<t} \circ d\widehat W_{u_1}\otimes\cdots\otimes \circ d\widehat W_{u_n}.$$7. This supports Fourier inversion for European calls, geometric Asian options, and $$\sigma_t = \left(1,\sigma_t^1,\sigma_t^2,\ldots\right),\qquad \sigma_t^n = \int_{0<u_1<\cdots<u_n<t} \circ d\widehat W_{u_1}\otimes\cdots\otimes \circ d\widehat W_{u_n}.$$8-volatility swaps, and also yields quadratic hedging formulas by identifying the martingale representation from the same characteristic functional (Jaber et al., 2024).

The joint SPX/VIX calibration problem is treated in a signature-based diffusion framework where the primary process is a polynomial diffusion. Because $$\sigma_t = \left(1,\sigma_t^1,\sigma_t^2,\ldots\right),\qquad \sigma_t^n = \int_{0<u_1<\cdots<u_n<t} \circ d\widehat W_{u_1}\otimes\cdots\otimes \circ d\widehat W_{u_n}.$$9 is linear in signature coordinates, the square $$\frac{dS_t}{S_t}=\Sigma_t\, dB_t,\qquad B_t=\rho W_t+\sqrt{1-\rho^2}\,W_t^\perp,\qquad \Sigma_t=\langle \xi_t,\sigma_t\rangle,$$0 can be rewritten linearly by the shuffle identity, and the VIX is obtained in closed form as

$$\frac{dS_t}{S_t}=\Sigma_t\, dB_t,\qquad B_t=\rho W_t+\sqrt{1-\rho^2}\,W_t^\perp,\qquad \Sigma_t=\langle \xi_t,\sigma_t\rangle,$$1

The same work represents $$\frac{dS_t}{S_t}=\Sigma_t\, dB_t,\qquad B_t=\rho W_t+\sqrt{1-\rho^2}\,W_t^\perp,\qquad \Sigma_t=\langle \xi_t,\sigma_t\rangle,$$2 as a linear functional of the signature of an augmented process $$\frac{dS_t}{S_t}=\Sigma_t\, dB_t,\qquad B_t=\rho W_t+\sqrt{1-\rho^2}\,W_t^\perp,\qquad \Sigma_t=\langle \xi_t,\sigma_t\rangle,$$3. Calibration is split into offline signature sampling and online optimization, and the reported empirical result is that both SPX and VIX options can be calibrated highly accurately without adding jumps or rough volatility (Cuchiero et al., 2023).

The linearly parametrized structure also supports direct calibration to time-series and implied-volatility-surface data. In the time-series setting, once the signature path is precomputed, estimation reduces to regression in the coefficient vector $$\frac{dS_t}{S_t}=\Sigma_t\, dB_t,\qquad B_t=\rho W_t+\sqrt{1-\rho^2}\,W_t^\perp,\qquad \Sigma_t=\langle \xi_t,\sigma_t\rangle,$$4. In the implied-volatility setting, model prices are computed by Monte Carlo while reusing the same precomputed signature samples across candidate parameters. Reported experiments include calibration to Heston-generated surfaces and to SPX option data, with maturity-dependent correction terms used for slice-wise fitting; the paper describes the resulting calibrations as fast and accurate, and reports calibration times of about 4 to 15 minutes on a standard machine for Heston-generated implied-volatility surfaces, and about 1 minute 30 seconds per smile for the slice-wise calibration (Cuchiero et al., 2022).

Paper	Problem	Tractability device
(Jaber et al., 2024)	European, Asian, and volatility derivatives	Infinite-dimensional Riccati equation and Fourier inversion
(Cuchiero et al., 2023)	Joint SPX/VIX calibration	Offline signature sampling, polynomial diffusion structure, Fourier pricing
(Cuchiero et al., 2022)	Time-series and implied-volatility calibration	Linear regression on precomputed signatures and slice-wise correction terms

A recurrent theme across these papers is that linearity is the crucial tractability feature: the primary process is simulated or sampled once, while volatility, prices, and calibration losses are evaluated by linear or quadratic forms in precomputed signature coordinates.

5. American options, path-dependent hedging, and frictional control

The non-Markovian character of rough volatility makes American and Bermudan valuation a natural use case for signatures. In the primal–dual optimal stopping framework, continuation values and martingale controls are approximated by functionals of path signatures rather than current-state regressors. Under rough Bergomi and rough Heston, three variants are compared: linear signatures, deep signatures, and signature kernels. The reported conclusions are that all three methods work well for the primal problem, deep signatures are strongest for the dual problem, and for rough Bergomi with $$\frac{dS_t}{S_t}=\Sigma_t\, dB_t,\qquad B_t=\rho W_t+\sqrt{1-\rho^2}\,W_t^\perp,\qquad \Sigma_t=\langle \xi_t,\sigma_t\rangle,$$5 the best duality gaps are about $$\frac{dS_t}{S_t}=\Sigma_t\, dB_t,\qquad B_t=\rho W_t+\sqrt{1-\rho^2}\,W_t^\perp,\qquad \Sigma_t=\langle \xi_t,\sigma_t\rangle,$$6 (Bayer et al., 12 Jan 2025).

A related American-option framework adapts the underlying volatility engine to time-varying roughness. It estimates a rolling Hurst parameter from a 32-day window, forecasts $$\frac{dS_t}{S_t}=\Sigma_t\, dB_t,\qquad B_t=\rho W_t+\sqrt{1-\rho^2}\,W_t^\perp,\qquad \Sigma_t=\langle \xi_t,\sigma_t\rangle,$$7 with an XGBoost ensemble, chooses rough Bergomi if $$\frac{dS_t}{S_t}=\Sigma_t\, dB_t,\qquad B_t=\rho W_t+\sqrt{1-\rho^2}\,W_t^\perp,\qquad \Sigma_t=\langle \xi_t,\sigma_t\rangle,$$8 and Heston otherwise, lifts simulated volatility paths to degree-three signatures, and accelerates Gaussian signature-kernel evaluations with Random Fourier Features. Reported case studies on S&P 500 equity-index options include AAPL, for which $$\frac{dS_t}{S_t}=\Sigma_t\, dB_t,\qquad B_t=\rho W_t+\sqrt{1-\rho^2}\,W_t^\perp,\qquad \Sigma_t=\langle \xi_t,\sigma_t\rangle,$$9 and the framework selects Heston, and META, for which $\xi_t$0 and it selects rough Bergomi. In that study the Deep Kernel / RFF variant gives the tightest bounds and the smallest duality gaps while adding less than 20% runtime (Shah, 10 Aug 2025).

For frictional hedging, signatures are used not merely as features but as the state variables of a control problem. In a Bachelier setting with permanent impact

$\xi_t$1

and temporary impact

$\xi_t$2

the optimal trading speed is derived in linear feedback form in the time-augmented signature of the control variables, with coefficients characterized by an infinite-dimensional Riccati-type system. The paper’s qualitative conclusion is that market impact smooths optimal trading strategies, so low-truncated signature approximations become highly accurate and robust in frictional markets, contrary to the frictionless case (Jaber et al., 28 Nov 2025).

The same distinction between direct feature learning and model-based volatility learning appears in hedging studies. One paper compares shallow linear signature hedging with a calibrated signature volatility model and reports that learning the dynamics of volatility through a signature volatility model, calibrated on the expected signature of the volatility, yields more accurate and stable hedging across different payoffs and volatility dynamics. In its deep-learning comparison, feedforward neural networks with signature features outperform LSTMs in most cases with orders of magnitude less training compute (Jaber et al., 3 Aug 2025).

At the structural level, hedging can be decomposed in the price filtration by

$\xi_t$3

If the price-filtration completeness depth $\xi_t$4 is finite, the residual $\xi_t$5 is expanded through signature coordinates beyond that depth, and its norm is bounded by a weight-dependent projection error. This identifies higher signature levels as the unhedgeable tail of the model (Xodarev, 16 May 2026).

6. Diagnostic uses of “signature” and related empirical volatility constructions

The term “signature” is also used in the literature in ways that are distinct from rough-path signatures. One such example is the roughness signature function, a two-frequency power-variation diagnostic designed to distinguish rough continuous processes, pure-jump processes, and mixed models. In the empirical application to three S&P 500 volatility measures, realized volatility and the option-extracted volatility estimator show signs of roughness, with the option-extracted estimator appearing smoother than realized volatility, while the VIX appears to be driven by a continuous martingale with jumps (Christensen, 2024).

Another distinct usage is the volatility signature plot in limit order book modeling. In the Extended State-dependent Hawkes Process, the upward slope of the volatility signature plot is reproduced through transient local super-criticality in disequilibrium states triggered primarily by Marketable Limit Orders. The paper argues that physical gating of inadmissible transitions is necessary for stable simulation and accurate reproduction of macro-level volatility (Kimura, 27 Apr 2026).

A further empirical use appears in cryptocurrency volatility. There, the “signature of maturity” refers to the evolution of yearly inequality measures of daily high–low fluctuations, summarized by the Gini index, the Kolkata index, and the $\xi_t$6 factor. The reported result is that cryptocurrencies began with much higher fluctuation inequality than national currencies, but over time their fluctuation behavior tends toward the range characteristic of national currencies rather than stocks (Ghosh et al., 2024).

These strands are related by vocabulary rather than by a common mathematical construction. Signature volatility models in the strict sense are models where volatility itself is parameterized by path signatures; roughness signature functions, volatility signature plots, and maturity signatures are empirical diagnostics or phenomenological summaries. A common misconception is therefore to treat all uses of “signature” in volatility research as instances of the same methodology. The supplied literature does not support that identification.

In contemporary quantitative finance, signature volatility models occupy a specific position: they are pathwise, algebraic representations of volatility that combine universality of path-function approximation with tractable pricing, calibration, and hedging machinery. Their distinguishing feature is not merely non-Markovian flexibility, but the fact that such flexibility is expressed through linear functionals of iterated integrals, so that transform methods, regression schemes, and structural results on martingality, arbitrage, completeness, and Greeks remain available within a single framework.