This paper studies how to fit option prices for the S&P 500 index (SPX) and the VIX index together across many monthly expiries. The authors show that a widely used shortcut, called “Markovian stitching,” forces each month’s fit to be independent of earlier history once you know the current SPX level. That shortcut cannot be checked with the usual monthly option quotes. As a result, different full‑path models can match every monthly price but still disagree on the prices of claims that span multiple months.

At a theoretical level the authors prove two things. First, any full‑path model that exactly matches the given SPX and VIX marginals can be transformed into a “block‑preserving SPX‑Markovization” that leaves each monthly joint law unchanged. In plain language: you can always make a stitched, Markovian version of a valid global model without breaking the monthly fits. Second, stitched models can still be a strict subset of all globally feasible models because the Markovization erases dependence on earlier history beyond the current SPX value. That erased dependence can matter for multi‑period payoffs.

To decide what completion to choose in practice, the paper notes that the usual choice is the minimum‑information completion under a Kullback‑Leibler (KL) relative‑entropy criterion. That choice picks the stitched law. If one wants non‑Markov memory, then extra information is needed: either cross‑period targets, a history‑dependent prior, or a different objective. The authors also formalize the constraints used in calibration: (C1) match each month’s SPX and VIX marginal distributions, (C2) enforce the martingale condition that expected future SPX equals today’s level (under the risk‑neutral measure), and (C3) enforce a dispersion constraint tied to the VIX log‑contract.

On the numerical side they introduce an augmented‑Bregman mirror‑descent algorithm to reconcile finite discretizations with the three constraint families. The scheme keeps the fit to observable option quotes while penalizing violations of the martingale and dispersion conditions. In test problems the method exposes conditional-row discrepancies and keeps prescribed marginals about 25 times tighter than a cyclic row projection baseline. A finite‑state example confirms that the block‑preserving Markovization leaves monthly fits unchanged but can cause material cross‑period price changes. On smoothed market SPX and VIX surfaces their numerical sweep reports that the worst fitted smile error stays below about 0.70 volatility points while conditional diagnostics improve as penalty budget increases.