Dynamic h(L)GJMM: conditional volatility surfaces, martingale couplings, and filtration-dependent option pricing

A joint mixture model calibrated to a vanilla surface by one linear programme, Kienitz (2026a)defines a price martingale with exactly the fitted marginals whose augmented state — price and latentcell — is Markov, and run forward from any state it generates a whole conditional implied-volatilitysurface in closed form. We ask whether such a perfectly arbitrage-consistent martingale producesplausible conditional surfaces, how its coupling — which the vanilla surface leaves free — should beselected, and how the information available to the holder of an early-exercise claim enters its price.The conditional surfaces studied are conditional on the augmented state, the price together with thelatent mixture cell observed at the quoted dates. Conditioning on prices alone averages them over theposterior of the cell, which the prices do not reveal.We consider volatility surfaces fitted by the above method. Then, the conditional surfaces are finitemixtures, free of arbitrage with forward equal to the current price. The calibrated vertex producesdownward-opening (concave) conditional smiles to a non-negligible extend of the states, especiallyon short maturities. They are not arbitrage, and three proved facts explain these shapes. First, thelevel-variance of a period is fixed by the marginals. Second, the jump window caps the kurtosis of alevel jump, and, third, vertices of the programme are sparse.Recalibrating with a wider jump window and then selecting the coupling at the new fitted weightsby a Kullback–Leibler projection reduces the concave smiles from about 30% to below 5% in ourexperiments. The method raises the long-horizon rates slightly. Only the final fixed-weight selectionstep leaves every vanilla price of the re-calibrated model unchanged. A laziness constraint on thecoupling, linear and imposed in the calibration, brings the six-month and one-year frown probabilitiesof a reduced set-up to below 2% at a cost of a fraction of a basis point for the fit. Replacing the linearcalibration programme by an entropic calibration of the whole joint law, a strictly convex programmewith a unique solution, avoids vertices in the weights and in the coupling, and fits better. It bringsthe frown probabilities to about 1% without further constraints. Against Heston and Bates surfaces,whose conditional smiles never open downwards, the orignially calibrated vertex frowns in up to 40%of the states, the entropic calibration with a Laplace reference in below 3% , and a lattice image of thetruth shows that most of the remaining forward-smile error is due to the lattice.We compare the construction with the SANOS-based generative and risk-neutral market modelsDYSANOS and GROMM, and find concave simulated smiles in an independent implementation ofGROMM. Static and dynamic no-arbitrage consistency therefore do not determine the conditionalsmile shapes. Suppressing frowns is a declared shape preference used to select among arbitrage-freecouplings, not an arbitrage correction or an identification result.For Bermudan claims we distinguish the augmented-information value, the price-observable valueand the value of the spot-Markov projection, which is a different model. We establish a relationbetween the information availability for the augmented case by independent policy simulation. Athree-level example with upward and downward le