A flat book should still carry a skew: δ = (w/2)·log(q/(1−q)). Cotton derives the level from customer direction and local market width, with no free parameter beyond w. The dealer shades her mid by half the local market width times the log-odds of customer direction. He makes no empirical claims here, and says so plainly. The magnitudes are falsifiable in principle, though every number below remains an untested prediction. Corollary 2 supplies the first test I would run: width responds to imbalance only at second order.

Cotton's stationary dealer

Inventory enters this model only when the dealer answers enquiries. Opportunities arrive through a Poisson process with mean inter-arrival time τ. Each sealed-bid enquiry comes from a seller with probability q and a buyer with probability 1−q, always for size s. Several dealers quote and the best price wins. Relative to a commonly discerned fair price, the best competing quote has constant hazard h. Its exponential mean is w = 1/h, which Cotton calls the market width.

Every won trade pays a fixed ϵ in adverse selection. Carrying x costs c(x) per unit time, and the analysis stays in ergodic steady state. Cotton deliberately excludes arrival clustering and a fluctuating q, deferring both to joint work with Papanicolaou. The stationary core retains exact symmetry, which is the object isolated here.

An indifference liquidation cost ν(x) characterises the policy. It is the dealer-to-dealer haircut at which she would clear her inventory, and only its first two differences enter. Slope S(x) gives the skew. Convexity C(x) gives the discretionary share of width. Two exact identities follow: half-width(x) = w + ϵ + C(x), while mid(x) = fair value − S(x). Reversing the width identity means a dealer's discretionary width exactly discloses the second difference of her inventory cost. Cotton says he has not found that result stated elsewhere. He credits the skew half to Ho and Stoll, where he says it is implicit, and to everything written since.

The proof applies one algebraic identity to the steady-state consistency equation:

q·e^(−hS) + (1−q)·e^(hS) = 2√(q(1−q))·cosh(h(S−δ)).

This converts an imbalanced mixture of exponentials into a balanced mixture translated by δ and rescaled. The balanced solution therefore handles the imbalanced problem after three adjustments. Skew moves by δ. Non-discretionary width increases by γ = (1/h)·log(1/(2√(q(1−q)))). Cost of carry is multiplied by M(q) = 1/(2√(q(1−q))) ≥ 1. Cotton describes the practical result as one balanced solve serving every imbalance.

For 60/40 flow, δ = 0.203w, γ = 0.020w, M = 1.021. At 80/20, δ = 0.693w, γ = 0.223w, M = 1.25. In the mathematics, order imbalance becomes a tax on inventory that is computable from the tape and unbounded as q approaches 0 or 1.

One by-product is immediately usable even if the broader model does not persuade you. With constant hazard, the optimal response lands exactly w beyond its strike. Expected net gain on a won trade is consequently s·w at every inventory level. Expected profit per enquiry varies only through fill ratio, and at markup m that ratio is e^(−m/w), log-linear with slope −1/w. With zero adverse selection and zero marginal inventory cost, an optimally quoting dealer wins e^(−1) ≈ 37% of her enquiries. Persistent deviations measure her effective strike.

Why lean with an empty book?

The symmetry gives Cotton's answer. At zero inventory, the balanced solution has zero shifted skew, leaving the raw skew exactly equal to δ. My own gloss is simpler: the side won less often is worth less per enquiry. Cotton handles priority carefully. Bergault and Guéant had observed numerically that flow-aware market makers skew at flat inventory, while Butz and Oomen documented the practice among electronic FX dealers. Corollary 1 supplies a closed form for behaviour already visible on the tape.

The closed form contributes a level. With 60/40 flow, the prescription is a fifth of a width. A desk can argue about that number.

Corollary 2 carries the empirical burden

Corollary 2 gives δ ≈ 2w(q−1/2) and γ ≈ 2w(q−1/2)². Cotton treats this asymmetry as the theoretical case for skewing into flow before widening. The derivation fixes ϵ as a single constant per won trade, independent of direction, size and q. One-sided flow may instead reflect informed sellers, making ϵ a function of q. Since ϵ enters through Δ = w + ϵ, any movement in ϵ enters at first order. The claimed ordering then collapses because its conditioned constant has moved. Post-trade markouts by direction bucket would test the issue.

Then there is w.

Cotton replaces global exponentiality with a local condition: only the hazard at the quotes actually made enters. In the numerical check accompanying the paper, the zero-inventory skew error is about a quarter of the hazard's relative variation across visited strikes, and it shrinks linearly with that variation. This is an honest, useful relaxation. It also makes w self-referential. The estimate comes from regressing log fill ratio on the dealer's own markups over the strike range visited by the current policy. Adding δ changes the operating point, where the hazard need not equal the measured value. Remark 2 instructs the reader to use reciprocal hazard at the operating point. I did not find the feedback from δ into measured w addressed anywhere in the paper.

The paper places CWLS, the constant-width linear-skew benchmark descending from Avellaneda and Stoikov, with precision. CWLS is exactly optimal only under balanced flow when c(x) ∝ cosh(2hC0x/s) − 1. That cosh form remains quadratic up to a relative error of (hS_δ(x))²/12. Under a genuinely quadratic cost, the inventory-side CWLS error grows as the square of the skew-to-width ratio, and Cotton gives no constant for it. With 60/40 flow, the omitted intercept δ is 0.203w. Cotton identifies this flow term as the benchmark's missing piece. His objection to the Avellaneda and Stoikov line is direct: constant width follows from dropping terms, while those same terms are required to say anything about width.

Anyone treating carry as a funding fee rather than a risk charge should also notice one feature. A linear component in c(x) creates a square-root kink in optimal skew at zero inventory.

We could not test any of this. The mechanism requires RFQ records, while we have OHLCV bars. Those bars contain no customer direction counts for estimating q and no win-or-lose outcome for each quote. Cotton arrives at the same constraint from another direction. Illiquidity produces short time series and unreliable backtests, so he proposes decision-level scoring instead. His four-cell table charges s·w whenever the model would have traded and the trader did not.

The setup echoes a note of ours on Wasserstein allocation. The mathematics did exactly what it claimed, while the practical edge depended on a single input selected from the same data. Here that input is w. Cotton proposes the relevant test himself: sort instruments cross-sectionally by flow imbalance. Mids should then move at first order in imbalance, while quotes widen only at second, as Corollary 2 predicts.

An RFQ study showing first-order mid shifts with widths staying put at second order would change my view of the width claim. If widths instead move at first order in |q−1/2|, the explanation is ϵ(q), and the corollary's ordering comes from holding adverse selection constant.