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

2026-07-29

AI Summary

This study investigates whether existing consistency conditions for multi-maturity option prices under bounded bid–ask spreads are sufficient to preclude model-free arbitrage. By refining the definition of admissible quotes and integrating the completeness condition for single-maturity options, the authors construct the first minimal two-maturity counterexample that satisfies all known consistency conditions yet admits a pathwise arbitrage, thereby disproving the sufficiency conjecture of Gerhold and Gülüm. Furthermore, employing techniques such as convex envelopes, concave maximization, and extremal envelopes within the geometry of bounded spreads, they explicitly derive closed-form two-maturity basket operators, precisely characterize the model-free arbitrage cone, and uncover a high-dimensional open region of admissible quotes generated by the counterexample, fully describing the structure of two-maturity executable, arbitrage-free markets.

Abstract

Gerhold and Gülüm 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 Gülüm 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 Gülüm, 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 $2ε$-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.

Problem

• option price consistency
• bounded bid-ask spreads
• model-independent arbitrage
• calendar-vertical-basket conditions
• pathwise arbitrage

Innovation

• bounded bid-ask spreads
• model-independent arbitrage
• two-date basket operator
• convex envelope
• pathwise arbitrage