Multi-maturity consistency of option prices under bounded bid-ask spreads: a minimal obstruction and an exact two-date basket operator

Abstract

Gerhold and G\"ul\"um derived necessary calendar-vertical-basket conditions for finite call bid-ask quotes when the cash-settlement reference price lies inside a dynamically traded stock spread of bounded absolute width. We first distinguish the printed bid of a calendar-vertical basket from the executable bid dictated by the contract and the self-financing convention. After making this correction and adjoining the initial-spread and complete one-maturity conditions, we show that the resulting system is still insufficient. Thus, under the corrected executable reading, the sufficiency direction of Conjecture 5.4 of Gerhold and G\"ul\"um has a negative answer even after the natural base conditions are imposed; its separate weak-arbitrage clause is not addressed. For every positive spread bound, an explicit panel with two quoted dates and one actual call at each date satisfies all corrected conditions, strictly whenever a strict face applies, but admits a pathwise stock-flip arbitrage. Measured by quoted dates and actual calls, this obstruction is minimal within the finite-call framework of Gerhold and G\"ul\"um, and the counterexamples contain a full-dimensional open quote box. A separate extremal-envelope argument produces the same separation gap. We then eliminate the arbitrary adapted stock holding in the complete two-date pathwise problem. The result is a closed-form one-step operator, expressible either as a finite concave maximization or as a convex-envelope infimum over a $2\epsilon$-neighborhood. As the finite call positions vary, the operator characterizes the full two-date executable model-independent-arbitrage basket cone. General robust superhedging duality and backward principles are prior work; the contribution here is the explicit calculation for this bounded-spread reference/shadow geometry.