Joint SPX/VIX Calibration as Linear Optimization over Signature Coefficients

The paper builds signature-based market models in which the SPX dynamics are polynomial in the signature of a primary process, and shows that the notoriously hard joint SPX/VIX smile calibration can be cast as optimization over signature coefficients, with the VIX computed in closed form as a signature functional. We read it as a constructive route to the joint-calibration 'holy grail' that stays linear-algebraic rather than relying on a bespoke SDE.

S_t is expressed via ⟨ℓ, 𝕊(X)_{0,t}⟩, with VIX² obtained as a conditional signature functional; calibration minimizes smile error over the linear coefficients ℓ subject to martingale constraints.

The appeal is that expressivity comes from the signature basis while the calibration objective stays tractable. It reframes joint calibration from 'find the right model' to 'find the right linear functional on a universal basis'.

Offers the desk a single, auditable representation in which SPX and VIX information are calibrated jointly rather than reconciled after the fact — relevant whenever a memo touches index volatility and its volatility-of-volatility.

Signature truncation and the dimensionality of the coefficient vector govern both expressivity and overfit risk; out-of-sample stability of the calibrated functional is the open question.