Peacocks by Linear Programming joint mixture models and explicit martingale transport

Fitting each expiry of an implied-volatility surface and constructing a martingale coupling of thefitted marginals afterwards solves two problems in the wrong order: the fitted marginals need notadmit a coupling of the intended kind, and the vanilla surface does not determine which couplingprices path-dependent contracts.We instead calibrate the joint law applying finite mixture models of Gaussian, resp. LogNormal typeinstead. A mixture component, or cell, carries a component forward (the level) and a componenttotal variance. On a fixed dictionary of cells the vanilla fit, the normalisation and a conditional-meancondition on the cell transitions are linear in the weights and the transition masses, so the jointcalibration is one linear programme.Every feasible point defines an explicit price martingale with exactly the fitted marginals. Its aug-mented state, the price together with the mixture cell, is Markov, while the price alone need not be.Feasibility is therefore a constructive sufficient guarantee that the marginals increase in convex order.The latter is not necessary and an explicit two-component marginals in convex order that no admissiblepair attains on any dictionary containing their cells is constructed. Due to modelling choice, thetransition kernel, every conditional law and the extremal couplings for a given payoff within the classare explicit.The vanilla surface leaves the coupling, and with it every path-dependent price, underdetermined, andthe construction makes this measurable rather than hidden. For forward-start options, locally cappedcliquets, Bermudan and American options we report class ranges and model-free martingale transportranges separately. Class ranges are narrower because of the modelling restriction. They are ranges ofthe model class, not tighter no-arbitrage bounds. The non-identification goes beyond the vanillas: aone-parameter memory dial preserves every marginal and every joint law of two consecutive quoteddates. Yet it can be applied to move Bermudan put prices within the price interval, e.g. to achive aclose match to a local-volatility value on the same vanillas.On a listed single-stock American chain the class reprices the European call quotes inside the spread.With the discount curve fitted to the same American puts, its early-exercise premium fits the putcross-section in sample better than a specified flat-volatility benchmark. The example establishesfeasibility and comparative in-sample fit, not out-of-sample performance, and the coupling againremains unidentified.