An Entropic, Dispersion-Constrained Solution to the Joint SPX/VIX Puzzle

The paper solves the long-standing joint SPX/VIX calibration puzzle exactly by formulating a dispersion-constrained martingale Schrödinger problem: among all models calibrated to SPX smiles at two maturities and the VIX smile, find the one of minimal relative entropy to a reference, subject to martingality and a dispersion (VIX-defining) constraint. We read it as the definitive entropic formulation of joint calibration.

Minimize H(Q | Q_ref) over martingale couplings Q on (SPX_{T1}, VIX_{T1}, SPX_{T2}) calibrated to the three smiles, subject to the dispersion constraint E[ (1/Δ) ∫ d⟨log S⟩ | F_{T1} ] = VIX_{T1}²; solved by a Sinkhorn-type fixed point.

The achievement is exactness: a nonparametric model fitting SPX and VIX smiles simultaneously without approximation, with the Schrödinger / entropic structure supplying both existence and a numerical scheme. It is the benchmark against which parametric joint models are judged.

Provides the desk a reference object for what exact joint SPX/VIX consistency looks like; parametric models in use can be assessed by their distance from this entropic optimum.

The exact solution is discrete-time and nonparametric, so it informs rather than replaces a dynamic model; computational cost and the choice of reference model shape the result.