Signature-Linear Volatility and the Survival of the Riccati Structure

Abi Jaber and Gérard build a class of volatility models in which the instantaneous variance σ_t² is expressed as a linear functional of the time-extended signature of a driving Brownian motion — σ_t² = ⟨ℓ, 𝕊(B̃)_{0,t}⟩ for B̃_t = (t, B_t) and a coefficient ℓ in the dual of the tensor algebra T((R²)) — and prove that the characteristic functional of the log-price and integrated variance retains an exponential-affine form analogous to the Heston characteristic function, with the scalar Riccati ODE of classical affine models replaced by a tensor-algebra-valued Riccati equation. The universality theorem underlying this class, which asserts that every continuous linear functional on the space of continuous paths of bounded p-variation can be expressed as an inner product with the signature, guarantees that the signature-linear family contains all path-dependent volatility specifications of practical interest as special cases — including Stein–Stein, Bergomi, and Heston in the Markovian limit, and rough and path-dependent variants in the infinite-dimensional regime — while maintaining a single unified Fourier pricing framework across all these models. The key technical contribution is the identification of the tensor-algebra Riccati equation as the correct generalization of the classical Heston Riccati ODE to path-dependent volatility. In the Heston model, the log of the characteristic function satisfies dH/dt = F(H, u) with F quadratic in H and the solution given by an analytic formula; in the signature-linear model, H(t) is valued in the tensor algebra T((R²)) and satisfies the same quadratic-in-H structure, now interpreted in the tensor product sense, with the solution no longer analytic but computable by numerical integration of the tensor-valued ODE. The connection to polynomial diffusions — where the expected signature E[𝕊(B̃)_{0,t}] satisfies a linear ODE on the tensor algebra, expressible as matrix exponentiation for any finite truncation level N — provides an additional computational pathway that is complementary to the Fourier method and more efficient for computing quadratic hedging ratios. The practical consequence for a derivatives desk is that path-dependence and Fourier tractability are not in opposition: the same FFT infrastructure used for Heston pricing can be adapted to signature-linear models by replacing the scalar Riccati solve with a tensor-valued solve, maintaining compatibility with the full infrastructure of fast calibration, smile interpolation, and greeks computation that the classical affine framework provides.

Let B̃_t = (t, B_t) ∈ R² be the time-extended driving path with B a standard Brownian motion; the signature is 𝕊(B̃)_{0,t} = (1, ∫_0^t dB̃_s, ∫_0^t ∫_0^{s_2} dB̃_{s_1} ⊗ dB̃_{s_2}, ...) ∈ T((R²)) = ⊕_{n≥0} (R²)^{⊗n}, with iterated integrals in the Stratonovich sense. The signature-linear instantaneous variance is σ_t² = ⟨ℓ, 𝕊(B̃)_{0,t}⟩ with ℓ ∈ T((R²))* = ⊕_{n≥0} ((R²)^{⊗n})* a sequence of tensors (ℓ_n)_{n≥0} with ℓ_n ∈ (R²)^{⊗n*}. The characteristic functional Φ(u,v) = E[exp(iu log S_T + iv ∫_0^T σ_s² ds)] takes the form exp(⟨H(T), 1⟩) where H : [0,T] → T((R²)) solves the tensor-algebra Riccati equation dH/dt = F_{u,v}(H) with F_{u,v}(H) = (iu − u²/2)ℓ_0 + (iv)ℓ + (iv + ρ iu) ℓ ⊗_s H + (1/2) H ⊗_s H + ..., where ⊗_s denotes the symmetrized tensor product and ρ is the correlation between B and the log-price driver. The truncated equation at level N retains only tensor components up to degree N, yielding a finite-dimensional ODE system of size Σ_{k=0}^N 2^k = 2^{N+1} − 1 for the N-truncated coefficient vector.

The survival of the Riccati structure under promotion from scalar to tensor-algebra coefficients is not obvious and relies on the Chen identity 𝕊(B̃)_{0,T} = 𝕊(B̃)_{0,t} ⊗_s 𝕊(B̃)_{t,T}, which asserts that the signature over a concatenated path is the shuffle product of the signatures over the two sub-paths. This identity is the path-space analogue of the semigroup property of Markov processes and is the structural fact that makes the tower property of conditional expectations compatible with the signature representation, enabling the exponential-affine form of the characteristic functional to propagate from one maturity to the next. Without the Chen identity there would be no tensor-Riccati equation and Fourier pricing would require full Monte Carlo simulation. The truncation at level N is a necessary approximation since the full tensor algebra is infinite-dimensional. The approximation error introduced by level-N truncation can be bounded using the factorial decay of signature norms: for a path of bounded variation, ||𝕊(B̃)^n_{0,t}|| ≤ ||B̃||_{1-var}^n / n!, so the contribution of degree-n terms to the characteristic functional decays factorially in n and the truncation error at level N is of order 1/N! × ||B̃||_{1-var}^{N+1}. For rough paths (H < 1/2), the appropriate norm is the p-variation norm with p = 1/H > 2, and the factorial decay applies to the Lyons extension theorem bounds, with the constants depending on H in a way that makes the truncation less accurate for very rough paths at moderate N. Comparison with the rough Bergomi model — which is the canonical non-affine rough volatility model for which Fourier pricing is unavailable and Monte Carlo is required — reveals the scope of the signature-linear extension: Bergomi's instantaneous variance σ_t² = σ_0² exp(ν W^H_t − ν²t^{2H}/2) is not linear in the signature of W^H (the exponential introduces nonlinearity), and so the Bergomi model lies outside the signature-linear class and cannot be priced by the tensor-Riccati formula; but any polynomial function of W^H_t up to degree N is within the class at truncation level N, so that the polynomial approximations to the Bergomi exponential studied in the polynomial volatility literature are special cases of the signature-linear framework at finite N.

For a desk calibrating to SPX implied volatility surfaces on a daily basis, the signature-linear framework offers a significant operational advantage over rough Bergomi: the Fourier pricing formula enables fast calibration via FFT on the order of seconds per parameter iteration, compared to Monte Carlo simulation for Bergomi which requires tens of minutes per iteration for adequate precision and therefore makes real-time recalibration impractical. The desk can maintain a calibrated signature-linear model with a fixed truncation level N = 4 or 5 — capturing Heston, path-dependent, and first-order rough effects — and recalibrate it intraday at each surface update, using the resulting ℓ* coefficients as the model state from which all Greeks, greeks-of-Greeks, and path-dependent exposures are derived analytically via automatic differentiation through the tensor-Riccati solve. The unified model family property — that Heston, Bergomi-polynomial, and rough-vol-linear models are all instances of the same tensor-Riccati framework at different choices of ℓ — enables a principled model comparison via likelihood ratio tests or information criteria computed on the same calibration dataset, providing a statistically grounded procedure for model selection that does not require ad hoc comparison of calibration errors across models fitted by different methods.

The non-negativity constraint σ_t² = ⟨ℓ, 𝕊(B̃)_{0,t}⟩ ≥ 0 for all paths is the binding constraint on the coefficient ℓ that has no simple characterization in terms of the tensor algebra structure: ensuring that the inner product with ℓ is non-negative for all realizations of the signature requires ℓ to belong to the positive cone of the tensor algebra dual, a closed convex cone with empty interior in any finite truncation, making constrained optimization over ℓ subject to non-negativity a semi-infinite programming problem with constraints that are checked on a discrete set of simulated paths rather than verified analytically. Violations of non-negativity during calibration produce complex-valued characteristic functions that make Fourier inversion ill-defined and must be detected and resolved by projection back onto the positive cone, introducing a repair step that is not covered by the convergence theory. The curse of dimensionality in the tensor-algebra coefficients is most severe in the cross-asset extension: for d driving factors (d = 2 for a correlated price-variance system, larger for multi-asset or stochastic interest rate models), the dimension of the level-N truncated tensor algebra grows as d^N, making N = 5 with d = 3 require a coefficient vector of size 3^5 + ... = 364, and N = 6 with d = 3 gives 1093 parameters; identifying these parameters from a fixed set of option prices at a single calibration date is heavily underdetermined and requires strong regularization that biases the calibration toward a low-complexity model regardless of what the data support.