A quote panel can pass every corrected calendar-vertical-basket inequality, the initial-spread condition and the complete one-maturity tests, yet still pay 47ε/150 upfront for a portfolio that never loses on an admissible path. Lee constructs exactly that panel with two calls.
The screen starts with three dates (0, 1, 2), discounted prices and a bank account fixed at one. Calls settle in cash against x_t, a reference price that cannot be traded. The stock has a bid-ask spread whose absolute width is at most ε, with the reference price inside it. The observed market consists of finite bids and asks: a date-zero stock quote and call quotes at strikes above ε for dates one and two.
Gerhold and Gülüm asked whether a model can reproduce those quotes without arbitrage. Its reference laws have support on [ε, ∞), and their call functions must fall within the quoted intervals. Each reference law comes with a shadow law, a martingale measure lying inside the stock spread and within infinity-Wasserstein distance ε. Across maturities, the shadow laws increase in convex order and have a mean inside the initial stock interval.
Their one-maturity solution is complete. For several maturities, they introduced calendar-vertical baskets: an early call packaged with later vertical spreads. They then obtained four families of necessary basket inequalities. Conjecture 5.4 asks, first, whether those four families suffice for consistency and, second, whether weak arbitrage follows when the first three hold while the strict fourth fails.
Lee settles the sufficiency direction negatively. The failure remains after adding the base system, meaning the initial-width condition and every complete one-maturity condition at each quoted date. He leaves the weak-arbitrage half untouched and says so explicitly.
Why the 2ε sign matters
On one spread-orientation face, the basket contract includes a cash term of minus 2ε. The source's displayed bid instead adds plus 2ε on that face. Lee's accounting carries the paper: reverse every static leg of the sale, paying asks for long positions and receiving bids for shorts. The executable proceeds carry the minus sign.
The printed bid can reject a consistent market. Set ε = 1, keep both the reference and shadow price constant at 5, use zero stock spread, and quote a strike-2 call worth 3 at both dates. This market is consistent. At shifted strike 3, the printed formula assigns the earlier basket a bid of 3 + 2 = 5. The later shifted ask there is 3, making the printed monotonicity condition demand 5 ≤ 3. The executable bid comes to 3 − 2 = 1.
Lee finds the same pattern in Gülüm's thesis, both arXiv versions and the published Mathematical Finance article. He presents this as an algebraic comparison between formulas, without making a claim about authorial intent.
Two calls force an inventory switch
Fix any ε > 0. The stock is quoted [599ε/100, 601ε/100], an initial width of ε/50. At date one, the call with strike 3ε is quoted [499ε/100, 501ε/100]. At date two, the strike 5ε call is quoted [199ε/100, 201ε/100].
Both dates are separately consistent. For date one, the reference law (3/4)δ_ε + (1/4)δ_{23ε} prices the call at exactly 5ε. Its shadow mean is 6ε and its W_∞ distance is ε/2. At date two, δ_{7ε} prices the call at 2ε, with shadow mean 6ε and W_∞ distance ε. Exactly four basket points arise, and every corrected condition holds. Lee certifies them using the single call function U(x) = (8ε − x)^+, whose slopes lie in [−1, 0]. Across the full system, the smallest slack is 51ε/50.
The trade is short. Buy 2/3 of a share at the ask, short one date-one call at 3ε, and buy 1/3 of the date-two call at 5ε.
Setup cost (2/3)(601ε/100) − 499ε/100 + (1/3)(201ε/100) = −47ε/150.
Suppose the date-one reference price is below 3ε. Sell the two thirds and retain no stock. The short call expires worthless, while the stock sale raises at least θ(x_1 − ε) ≥ 0. If the reference price reaches or exceeds 3ε, sell one full share and finish the date short a third. After the short call payoff, proceeds are at least 2ε. Covering that third at date two against the long call costs at most the same 2ε. Every path has nonnegative liquidation.
The early exercise state flips the portfolio's stock inventory. A single basket comparison cannot represent that coupled switch.
Lee recovers the same figure through the extremal call envelopes. Convex order imposes R_{ν1}(6ε) ≥ 59ε/20 and R_{ν2}(6ε) ≤ 201ε/100. Their gap is 47ε/50. Applying the terminal weight 1/3 gives the 47ε/150 collected at setup.
This construction persists throughout a six-dimensional open box of quotes. Each of the six quote coordinates can move independently by less than ε/200 while all conditions continue to hold. The slacks shift by at most ε/100, against the 51ε/50 floor. Setup cost deteriorates by at most ε/100, against a margin of 47ε/150.
Can a desk add the missing check?
The absent inequality uses one stock ask, an early call bid and a later call ask. Let the early strike be a and the later strike K, with a < K + 2ε. Set γ = (a − ε)/(K + ε) and θ = 1 − γ. The required expression is θ times the stock ask, minus the early call's bid, plus γ times the later call's ask. It must be nonnegative. Three quotes and rational arithmetic suffice.
Even a screen containing those bridges remains incomplete. With ε = 1, quote the stock at [499/100, 501/100] and use four calls: date-one strikes 3/2 and 5/2, followed by date-two strikes 5/2 and 9/2. Every corrected basket condition passes. The four pairwise bridge slacks are all strictly positive: 366/175, 2201/1100, 1557/700, 537/275.
Now hold 5/2 in cash and no stock. Go long the date-one 3/2 call, short the date-one 5/2 call, short 7/4 of the date-two 5/2 call, and buy 7/4 of the date-two 9/2 call. Setup cost is −9/40, with nonnegative liquidation along every path. All pairwise bridges hold strictly, yet −9/40 remains available.
The list gives way to an operator
Using Sion minimax and conjugate duality, Lee eliminates the adapted stock holding. He obtains W_{ε,θ}h_2(x) = θx − ε|θ| + inf over z ≥ ε with |z − x| ≤ 2ε of {h_2**(z) + (θ(z − x))^+}. Evaluation stays finite because the candidate set is {ε} together with the terminal strikes. When initial stock is zero, the instruction is simple: convexify the terminal payoff, then minimize across a 2ε window. Two adverse spread crossings accumulate to 2ε rather than ε.
Seeing the realized spread orientation at date one adds nothing. A rule based only on the reference coordinate reaches the same guaranteed continuation value. The elementary hedge changes its stock position at x = a; the optimal continuation changes at x = K + 2ε. The intuitive trade remains nonnegative, though it is suboptimal on [a, K + 2ε).
Constructed quotes, limited scope
The paper contains no market data. Its −47ε/150 and −9/40 setup costs come from constructed synthetic panels.
All results use a deterministic absolute spread bound and cash settlement against a nontraded reference. Call positions may be fractional. The short stock leg incurs no borrow fee, margin or settlement friction.
The characterization concerns the model-independent-arbitrage cone, the set of finite positions admitting pathwise arbitrage. Probabilistic consistency forms a different object. Lee makes this distinction and observes that the two may diverge on weak-arbitrage equality boundaries. His strict counterexample lies away from those boundaries.
Other limits are stated plainly. The source conjecture's separate weak-arbitrage clause remains unaddressed. The operator supplies no finite irredundant representation of the cone. Three or more dates fall outside the paper, as does the frictionless limit ε = 0.
The paper discloses substantive generative-AI assistance in its constructions and proofs. Every slack, witness measure and setup cost appears as an exact rational. In our view, that matters more than usual here because the finite calculations are then checkable by hand.
For anyone maintaining a consistency screen, the operative result is that local geometry across strike and maturity cannot detect a state-contingent inventory change, however complete the single-date conditions may be. Lee replaces those local checks with his one-step operator, subject to his limits: two dates and the arbitrage cone rather than the consistency set. His claim about adjacent work is narrower. None of the existing formulations, including the bid-ask martingale optimal transport duality of Liang et al. (2026), combines the absolute stock-spread bound, distinct reference and shadow prices, and finite execution structure at once.
We could not test any of it. The screen requires contemporaneous executable option bids and asks alongside the underlying's bid and ask. Our options data contains end-of-day prices, Greeks and implied volatilities, while our equity data is OHLCV with no quotes. Using mids or closes would examine a different object from the paper's.
A reading of the source contract under which the printed plus 2ε survives Lee's ε = 1 constant-model example would change my view of the sign question. Unless someone produces one, the negative sufficiency result applies to the system implied by the source's own contract and self-financing identity.