Multiple-Maturity Consistency Under Bid–Ask Spreads

Definitions and mathematical statement

Let $\mathcal D$ denote the given option-price data, and let $\mathcal T$ denote their set of maturities, with $|\mathcal T|>1$. Let $\varepsilon\in[0,\infty)$ be a proposed upper bound on the discounted bid--ask spread of the underlying asset.

The requested problem is to give explicit necessary and sufficient conditions, in terms of $\mathcal D$ and $\varepsilon$, for $\varepsilon$-consistency, and to determine the smallest admissible value of $\varepsilon$.

To express the optimization target without assuming attainment, write \[ \mathcal E(\mathcal D) :=\{\varepsilon\in[0,\infty):\mathcal D\text{ is }\varepsilon\text{-consistent in the intended source sense}\}, \qquad \varepsilon_{\inf}(\mathcal D):=\inf\mathcal E(\mathcal D), \] with $\inf\varnothing=+\infty$. The requested smallest admissible bound exists precisely when $\mathcal E(\mathcal D)$ has a least element; in that case it equals $\varepsilon_{\inf}(\mathcal D)$. Whether this infimum is admissible is part of making the requested minimum precise.

The phrase "$\varepsilon$-consistent in the intended source sense" is an unresolved predicate here. The supplied record does not define it or specify the option contracts, the interpretation of their quoted prices, the discounting convention, the admissible underlying-price processes, or the probabilistic and pricing requirements. Thus the displayed set records the intended target but does not supply a complete mathematical definition of consistency. In particular, the record does not specify when and in what probabilistic sense the discounted spread bound must hold. These hypotheses must be recovered from the cited source before a fully quantified model-specific statement can be given.

The supplied dated assessment distinguishes this multiple-maturity problem from the one-maturity case: it reports necessary and sufficient conditions for one maturity, and necessary conditions together with equivalent semiexplicit criteria for multiple maturities, while reporting no comparably explicit necessary-and-sufficient characterization for the latter. Accordingly, the intended target appears to be a more explicit characterization, rather than merely an equivalent criterion. This interpretation and the reported literature status require review.

Short English statement

For option prices at more than one maturity, characterize when they are consistent with a prescribed bound $\varepsilon$ on the underlying asset's discounted bid--ask spread, and determine the least feasible bound, including whether it is attained. The supplied record leaves the technical meaning of consistency unspecified. Its dated curation labels the problem partially solved and identifies a more explicit multiple-maturity characterization as the remaining target.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003427$, source problem code catalogue entry; registry locations: data/unsolvedmath/problems.json and the supplementary catalog/problems.json. Title: "Multiple-Maturity Consistency Under Bid--Ask Spreads." These identifiers refer to the supplied record; no independent inspection of the registry is asserted.
• Supplied report citation: Mathematics of Quantitative Finance (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/13, pp. 683--769. https://doi.org/10.4171/owr/2017/13. The record attributes the problem to Oberwolfach workshop open-problem discussions, lists proposed year 2017, and supplies no individual proposer. The citation's 2018 title date and the record's 2017 year are retained without reconciliation.
• Supplied literature reference: "Gerhold--Gulum, consistency under bid-ask spreads." https://pmc.ncbi.nlm.nih.gov/articles/PMC7155110/.
• Supplied literature reference: "Oberwolfach quantitative-finance report." https://doi.org/10.4171/owr/2017/13.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially_solved. It reports necessary and sufficient conditions for the one-maturity problem, and necessary conditions and equivalent semiexplicit criteria for the multiple-maturity problem, but no comparably explicit necessary-and-sufficient characterization. This is supplied dated curation derived from the OWR record, not an independently verified assessment; its time-sensitive status and cited claims require independent verification.
• The supplied statement status is corrected_verified; its accompanying note says that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. This supplied label does not establish independent mathematical verification of this draft.