Joint SPX/VIX Calibration as Linear Optimization over Signature Coefficients

Cuchiero, Gazzani, Möller, and Svaluto-Ferro construct signature market models in which the SPX log-price is expressed as a polynomial functional of the signature of a primary diffusion process X, and demonstrate that within this class the joint calibration to SPX vanilla options and VIX options and futures — historically the hardest simultaneous fitting problem in derivatives modeling — can be cast as optimization over the signature coefficient vector ℓ, with the VIX² available in closed form as a signature functional of the conditional expected signature. The approach transforms the joint calibration problem from a nonlinear search over the parameters of a bespoke SDE — which requires a separate parameterization for the VIX dynamics beyond what the SPX dynamics specify — into a linear-algebraic problem in the signature coefficient space, where both SPX option prices and VIX option prices are computed as linear functions of ℓ (after appropriate linearization) and the optimization is therefore convex or nearly convex. The central computational device is the polynomial diffusion embedding: the expected signature E[𝕊(X)_{s,t} | F_s] satisfies the linear ODE d/dT E[𝕊(X)_{s,s+T}] = A · E[𝕊(X)_{s,s+T}] where A is the generator matrix of the associated polynomial diffusion on the truncated tensor algebra, enabling efficient computation of the VIX² = (1/Δ) E_t[∫_T^{T+Δ} σ_u² du] = ⟨ℓ^VIX, E[𝕊(X)_{T,T+Δ} | F_T]⟩ as a matrix-exponential times the current signature state. The VIX is therefore an affine-polynomial function of the state 𝕊(X)_{0,T} at the VIX fixing date T, guaranteeing that VIX options can be priced by simulation of the polynomial diffusion on the truncated tensor algebra without auxiliary SDE parameters. The universality of the signature basis ensures that the model class is rich enough to approximate any consistent joint SPX/VIX dynamics to arbitrary accuracy: the universal approximation theorem for signatures asserts that every continuous path functional can be approximated by a linear functional on the signature, and the polynomial diffusion structure on the truncated tensor algebra provides the computational machinery to evaluate these functionals and their conditional expectations efficiently. The result is a model class that achieves the long-sought joint calibration goal through linear algebra rather than through ad hoc parameter engineering.

The primary diffusion X = (X^1, ..., X^d) : [0,T_max] → R^d satisfies a polynomial SDE dX_t = b(X_t) dt + σ(X_t) dW_t where b and σ are polynomial functions of X of degree at most 1 and 1/2 respectively (linear drift, square-root diffusion); the SPX log-price is log(S_t/S_0) = ⟨ℓ^S, 𝕊(X)_{0,t}⟩ for coefficient ℓ^S ∈ T((R^d))^*. The VIX²_T is VIX²_T = ⟨ℓ^VIX, m(Δ; 𝕊(X)_{0,T})⟩ where ℓ^VIX encodes the 30-day variance swap rate and m(τ; y) = E[𝕊(X)_{T,T+τ} | 𝕊(X)_{0,T} = y] satisfies the ODE dm/dτ = A_N m(τ), m(0) = y, with A_N the N × N generator matrix of the polynomial diffusion on the level-N truncated tensor algebra. Option prices are C(K,T) = E[(S_T − K)_+] computed via characteristic function Fourier inversion using the tensor-Riccati formula for log S_T. The calibration minimizes the weighted sum J(ℓ^S, ℓ^VIX) = Σ_{i,j} w_{ij}(C_model(K_i, T_j) − C_market(K_i, T_j))² + Σ_{p,q} u_{pq}(C^VIX_model(K_p, T_q) − C^VIX_market(K_p, T_q))² over (ℓ^S, ℓ^VIX) subject to martingale constraints encoded as ⟨ℓ^{drift}, E[𝕊(X)_{0,T_j}]⟩ = 0 for each T_j.

The joint calibration problem has historically been approached by adding VIX-specific parameters on top of a base SPX model — for example, adding a separate jump component to explain the VIX smile while keeping the SPX dynamics unchanged — a procedure that produces inconsistent joint models where the parameters driving SPX smiles and VIX smiles can be independently tuned without enforcing the theoretical relationship VIX²_T = E_T[(1/Δ)∫_T^{T+Δ} σ²_u du]. The signature framework enforces consistency by construction: both SPX and VIX are functionals of the same signature state 𝕊(X)_{0,t}, so any calibration that fits both smiles simultaneously is automatically consistent with the theoretical VIX definition, and the only remaining freedom is in the choice of (ℓ^S, ℓ^VIX) subject to this consistency constraint. The linear-algebraic structure of the calibration — which becomes exact when the option pricing formulas are linearized in ℓ about a reference coefficient vector — enables the use of convex optimization solvers with polynomial-time convergence guarantees rather than the non-convex gradient descent typically required for model calibration. In practice the linearization error introduces a non-convex correction that must be handled by successive linearization (Gauss–Newton or sequential linear programming), but the initialization from the linear solution is close to the global optimum and the convergence is fast. The Tikhonov regularization term λ_reg ||ℓ||² added to J prevents overfitting to market data and provides smoothness of the calibrated coefficients across maturities, with λ_reg selected by cross-validation on held-out strikes. The polynomial diffusion generator A_N has a specific sparsity structure: because the generator of the polynomial diffusion on the tensor algebra maps degree-n tensors to degree-n tensors (via the drift and diffusion terms), plus degree-n+2 tensors (via the quadratic Itô correction), A_N is block upper bidiagonal in the graded structure, which enables efficient matrix-exponential computation via Padé approximation with forward differences rather than full dense exponentiation. For d = 2 and N = 6, A_N is a 63 × 63 sparse matrix whose exponentiation at each maturity T requires O(N^3) = O(2^{3N/d}) operations, a manageable cost for the precision needed in calibration.

For a desk running a joint SPX/VIX book — positions in SPX variance swaps, VIX futures, VIX calls, and SPX OTM puts — the signature calibration provides a single model state (ℓ^S, ℓ^VIX) from which all Greeks across the combined book can be computed consistently via automatic differentiation through the matrix-exponential formula for expected signatures and the Fourier formula for SPX option prices. The model vega of a VIX call and the model vega of an SPX variance swap are derivatives of the same functional of ℓ, so cross-instrument hedging ratios — how many units of one product to hold against another — are directly available from the calibrated model without requiring a separate correlation parameter. The VIX-in-closed-form property provides a real-time VIX forecast that is consistent with the calibrated SPX surface: VIX_T_model = √{⟨ℓ^VIX, m(Δ; 𝕊(X)_{0,T})⟩} is computable at each moment from the current signature state and the generator matrix A_N, and comparing this to the observable VIX spot level gives an instantaneous model-versus-market VIX diagnostic. A persistent positive gap (model VIX below market VIX) indicates that the model is underestimating variance uncertainty relative to the market, possibly due to inadequate calibration to the VIX smile wings, and should trigger a recalibration or a model-error reserve against VIX-related positions.

The specification of the primary diffusion X is the fundamental modeling choice on which the expressivity of the signature market model depends: for d = 1 and a univariate OU process, the model collapses to a standard mean-reverting variance process with no rough vol features, while for larger d and richer polynomial SDE specifications the model gains expressivity at the cost of a larger parameter space. The choice of d is constrained by the available calibration instruments — with M instruments and N_sig signature coefficients at level N, the calibration is underdetermined for N_sig > M — and in practice d = 2 or 3 with N = 4 or 5 is the maximum expressivity achievable with a typical SPX/VIX option surface. Any rough-vol features that would require H < 1/2 in the Hurst index sense cannot be captured by a polynomial diffusion X, since polynomial diffusions have H = 1/2 local paths, so the signature model with polynomial-diffusion driver is a smooth-vol approximation to rough-vol dynamics. The martingale constraint E[S_T | F_0] = S_0 e^{−rT} encoded as a linear constraint on ℓ^S is approximate after linearization: the exact martingale constraint is nonlinear in ℓ through the exponential S_t = S_0 exp(⟨ℓ^S, 𝕊(X)_{0,t}⟩), and the linearized version is accurate only near the calibrated ℓ^S value. During recalibration when ℓ^S changes substantially across days — for example during a large market move that shifts the skew materially — the linear martingale constraint may not be tight enough, producing calibrations where E_model[S_T] departs from the forward price by an amount that grows quadratically in the coefficient displacement, and requiring a post-calibration adjustment step that recenters ℓ^S to enforce the exact martingale condition.