Two scalars carry the entire SPX calibration: a mean-reversion speed and a correlation. They fit an implied volatility surface of 2455 quotes across six maturities from May to October 2026. Every other model parameter has already been assigned to the VIX side. If the separation survives across dates, ownership of the calibration shifts within a trading desk.
We could not run any of this ourselves. The framework requires VIX futures, VIX index options and SPX index option surfaces, and we have none of the three. Nothing below is an independent test. SPY options or a volatility ETF would not serve as substitutes because the coupling function follows directly from the VIX index definition.
Start with the VIX
Zaugg and Grzelak reverse the usual construction. Conventional joint models specify the SPX instantaneous variance first and derive the VIX from it, creating the familiar calibration tension. Their model instead gives the VIX explicit dynamics through a mean-reverting local-volatility diffusion, dX = k(theta_t - X)dt + sigma_LV(t,X) X dW.
The piecewise-constant long-run mean theta_t is bootstrapped from the futures strip. A mean-reversion-adjusted Dupire formula, following Drimus and Farkas, supplies the local volatility surface. Only then is the SPX instantaneous variance recovered as a latent function of the VIX level, v_t = psi_t(X_t). The coupling function psi maps the VIX to SPX variance while enforcing the 30-day index definition as an identity.
Most of the technical work lies in that identity. Consistency requires Theta X_t^2 to equal the conditional expectation of integrated variance over the next 30 days, where Theta = 30/365. Feynman-Kac then yields the backward relation psi_t(x) = phi_t(x) - Theta(2x mu(t,x) + sigma^2(t,x)). On any 30-day window, psi determines its value on every earlier window. The modeller specifies it once on a terminal interval, then repeatedly solves a PDE to propagate it back to today. For polynomial VIX processes, an (m+1)x(m+1) linear system produces the terminal curve. Other processes, including their local-volatility VIX, require numerical conditional moments followed by a polynomial projection.
The entire test comes from one end-of-day snapshot: 27 April 2026, with VIX spot at 18.02 and SPX spot at 7152.72. Thetadata supplies the option quotes, while the futures settlements come from CBOE. There are six VIX expiries and 234 quotes, alongside six SPX expiries and 2455 quotes. With k fixed at 12, the bootstrapped theta values (23.33, 22.28, 22.68, 22.5, 23, 23.37) reproduce the six futures forwards, exactly as the bootstrap requires. The resulting terminal curve is 1.16x^2 - 4.19x - 101.37. Setting rho = -0.95 fits the SPX skew, and all four stages finish in under about three minutes.
The strongest result is k invariance
The decomposition has practical force. Changing k leaves every model-produced VIX option price unchanged because sigma_LV is recalibrated to preserve each fixed-time marginal of X_t. The SPX surface moves instead. VIX smiles determine the marginals, while k determines the persistence of volatility shocks. That persistence appears in the SPX skew. The two markets answer two different questions without competing over the same coefficient.
The terminal-interval reduction also does real work. In the GBM example, exact consistency has a simple cost: with sigma = 0.7, psi equals c x^2, where c is about 0.98 throughout the whole interval. It amounts to a scaling correction.
Positivity remains an after-the-fact check. The workflow tests psi on [t_0, T_F) and applies a floor where needed. The authors acknowledge the missing guarantee directly: "To our knowledge, there is no general condition to ensure positivity." We solved for the positive root of their fitted quadratic and found that psi reaches zero at a VIX near 11.3, versus a spot of 18.02. Whether the floor binds depends on the propagated psi over [0, 0.5], which the paper does not report. The authors also describe the consistent deformation on the terminal interval as non-unique. The model is therefore under-determined at the point where the modeller makes the choice.
Designed to fit
The VIX option calibration supplies no independent evidence. As the paper says, the calibrated local volatility model "replicates the smoothed VIX options market by design." Its evaluation bands are wide as well. Across the six expiries, maximum relative implied-vol bid-ask spreads are 18.92%, 20.80%, 16.81%, 19.87%, 21.90% and 19.18%.
The SPX side carries the actual test. Its spreads range from 6.06% to 8.82%, and the paper claims that two numbers cover six maturities.
Figures compare the fit with those bid-ask bands. We did not find an RMSE, an average error in vol points, or another error metric in the paper. The authors themselves describe only one experiment. The abstract calls it "a numerical experiment," while the conclusion says the model was applied to "market data from April 2026."
The closing sentence on the SPX fit goes further. It says the model price is "largely in line with the market bid-ask prices observed at the time, proofing the VDV model as valid joint SPX/VIX framework." A validated framework is being claimed from one snapshot. One date. Without repeated fits, the paper tells us nothing about whether rho remains at -0.95 or k remains at 12 tomorrow.
And -0.95 lies close to the wall. The authors identify the main miss themselves: the largest discrepancy appears in out-of-the-money SPX calls. They attribute it to constant correlation, which cannot reproduce the asymmetry of the SPX-VIX relation. Their defence is reasonable. Linear rho is a modelling choice within the framework, rather than a flaw in the VIX-first ordering, and state-dependent correlation could enter without altering the coupling machinery. Yet the second of only two SPX parameters is already at -0.95. A steeper day leaves very little room.
A troublesome constant
One detail links the closed-form check with the market calibration. When Heston is embedded in the framework (kappa = 2, theta = 0.04, xi = 0.5), the projection recovers a_2 = 1.085647 against the exact 1.085647. It also gives a_1 = -4.11e-8 against an exact zero. The quadratic term is essentially perfect.
The constant behaves differently. The projected a_0 is -0.003426, compared with an exact -0.007137. In unscaled variance units, the difference is 0.0037, a factor of two. After rescaling the market fit, its a_0 = -101.37 becomes about -0.0101. Carrying the benchmark's 0.0037 error across makes it roughly a third of the market fit's constant term. This arithmetic is ours, produced on a different process, and the paper does not measure the projection error in its own market-data exercise.
What could settle it?
Repeated calibration over a few months of dates would show whether k and rho move smoothly and whether the terminal curve stays positive across the simulated VIX range. A hedging experiment would also matter because the paper motivates the framework partly through participants using SPX options to hedge VIX exposure and vice versa. We did not find either test in the paper. The work remains a pricing and calibration exercise, which is how the authors present it.
The architecture is the part worth carrying forward today: three markets, three parameter blocks, plus a stated invariance identifying which market moves each block. That offers more than another flexible surface, and it runs in three minutes. Its evidence still comes from one Monday in April.