On a Simple Relationship Between Order Imbalance, Skew and Width in Over-The-Counter Trading

Peter Cotton

REVIEW 3 major objections 4 minor 21 references

On a Simple Relationship Between Order Imbalance, Skew and Width in Over-The-Counter Trading

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read One symmetry absorbs all order imbalance in OTC quoting

desk verdict A genuinely useful symmetry for RFQ market making, with an exactness claim that needs a boundary condition patch before it is true as stated. read the letter →

arxiv 2608.07690 v1 pith:3CW4MKQP submitted 2026-08-07 q-fin.TR q-fin.MFq-fin.ST

Peter Cotton This is my paper · ORCID

classification q-fin.TRq-fin.MFq-fin.ST MSC 91G1593E20

keywords marketmakingorderimbalanceskewbid-askspreadrequestforquoteinventorycostover-the-countertradingsteady-statesymmetry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Market makers in over-the-counter markets face one-sided customer flow: sellers arrive more often than buyers, or the reverse. This paper claims that imbalance does not add a new problem. The steady-state optimal quoting policy with imbalance $q$ is exactly the balanced-flow policy after three mechanical adjustments: the skew is translated by a computed amount, the quote width is widened, and the cost of carrying inventory is multiplied by $M(q)=1/(2\sqrt{q(1-q)})$. No free parameter enters beyond the observable market width $w$, so one numerical solution of the balanced problem serves every imbalance. This explains why a dealer with a flat book should still lean quotes into the flow, and why practitioners skew before they widen.

What carries the argument

The engine is the elementary identity $$q $e^{{-hS}}$+(1-q)$e^{{hS}}$=2\sqrt{q(1-q)}\$\cosh$\bigl(h(S-\delta)\bigr),\qquad \delta=\frac{1}{2h}\log\frac{q}{1-q},$$ which rewrites an imbalanced mixture of two exponentials as a balanced $\cosh$, translated by $\delta$ and rescaled by $2\sqrt{q(1-q)}$. Substituted into the steady-state consistency equation that equates the cost of carrying inventory with the option value of the next trading opportunity, it shows the imbalanced problem is the balanced problem with carrying cost multiplied by $M(q)=e^{h\gamma}$. The economics comes from identifying $S$ as the slope of the inventory cost (the skew), $C$ as its convexity (the discretionary width), and $w=1/h$ as the reciprocal hazard of the best competing quote.

What would settle it

Run the steady-state analysis with a non-exponential win curve, for instance a Weibull survival function with shape parameter different from 1, and compare the imbalanced solution to the balanced solution adjusted by $(\delta,\gamma,M(q))$; if they differ, exactness fails. A simpler empirical version is to estimate from a request-for-quote archive the local hazard of the best competing quote over the strikes a dealer actually visits; if it is not close to constant, the predicted zero-inventory skew $\delta=(w/2)\log(q/(1-q))$ should be off by roughly the hazard's relative variation.

Watch

Extended reading notes

Core claim

The paper's central discovery is Theorem 1: the imbalanced market making problem compresses onto the perfectly balanced one. With arrival imbalance $q$ and market width $w$, the imbalanced steady state is the balanced state with carrying cost $M(q)c(\cdot)$, skew shifted by $\delta=(w/2)\log(q/(1-q))$, and non-discretionary width $\Delta$ widened to $\Delta+\gamma$, where $M(q)=1/(2\sqrt{q(1-q)})$ and $\gamma=w\log(1/(2\sqrt{q(1-q)}))$. Three corollaries follow: a zero-inventory dealer should quote mid shifted by $\delta$; skew responds to imbalance at first order while width responds only at second order; and one-sided flow acts like an additional cost of carrying inventory. The popular constant-width, linear-skew policy is exactly optimal only in one corner, balanced flow with a $\cosh$-shaped carrying cost, and the symmetry supplies the flow term that this benchmark omits.

Load-bearing premise

The exact equivalence assumes the best competing quote has a locally constant hazard rate, an exponential distribution, at the prices the dealer actually quotes; if that hazard varies materially over the strikes she visits, the translation, widening, and cost multiplier are only approximate.

Editorial extensions

If this is right

A dealer with zero inventory should quote a midpoint shifted by $\delta=(w/2)\log(q/(1-q))$ away from fair value, with the sign set by the net buying or selling pressure.
Skew responds to imbalance at first order while quote width responds only at second order, so mild one-sidedness moves the mid almost linearly and barely changes the spread.
One-sided flow acts like an additional cost of carrying inventory, so even a product with zero holding cost effectively taxes a dealer's book when arrivals are imbalanced.
A single balanced steady-state solve, computed once, yields the imbalanced solution for every $q$ through the same three-part correction.
The constant-width, linear-skew heuristic is exact only for balanced flow with a cosh-shaped carrying cost; outside that corner, a quadratic inventory-cost ansatz is internally inconsistent.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the local-exponential assumption holds only approximately in a real request-for-quote market, the same $(\delta,\gamma,M(q))$ correction should still be the leading-order adjustment, with the error controlled by the hazard's relative variation across the strikes a dealer visits; quote-archive data could measure that directly.
The identity (4) is a general two-state symmetrization, so the same compression may carry over to other asymmetric two-sided models, such as limit order book systems with unequal buy and sell arrival rates, wherever an analogous consistency equation appears.
A cross-sectional prediction follows from the theorem: across OTC instruments, zero-inventory skew should scale with $w\log(q/(1-q))$ and the width correction with $w\log\cosh(\delta/w)$, giving a no-free-parameter empirical check.
The decomposition reframes constant-width, linear-skew heuristics as missing not merely inventory-cost curvature but the entire flow term, so flow-aware heuristics should add a flow intercept before adding any nonlinearity.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper finds a real and useful symmetry in a standard request-for-quote market-making model: the steady state with imbalanced arrival flow compresses onto the balanced problem via a skew shift, a width widening, and a cost multiplier. That packaging is new, and it turns a practitioner regularity (flow-shading at flat inventory) into an explicit formula, delta = (w/2) log(q/(1-q)). The corollaries are clean: skew is first-order in imbalance, width second-order, and the constant-width/linear-skew heuristic reappears as a small-skew special case. The paper is honest about the algebraic lineage (birth-death symmetrization, refs [19-21]) and the literature review is careful about what is and is not new. I found no self-citation or curve-fitting issues; the predictions are parameter-free given an observable width, which is exactly the right kind of claim for a theory note.

The main soft spot is real and needs fixing. Eq. (3) replaces the enquiry value sup_m (m-K)e^{-hm} with the unconstrained exponential form, which is only valid when the optimal markup is nonnegative. The paper explicitly says "away from the region where the max(.,0) binds" in Section 2.2 but never carries that qualification into Theorem 1. The stress-test is right that this is not a measure-zero nuisance: for a convex carrying cost, the strike K drifts below -1/h at large inventory, those states carry positive stationary mass, and the translation S to S - delta shifts the boundary so the compression does not commute with the max operator. The theorem needs an explicit interiority assumption (or a proof that the max never binds on the support) before the "exactly" claim is true as written. This is fixable, but it is a genuine gap in the central result, not a stylistic nit.

Two smaller issues: Eq. (3) is stated without derivation (the reader says it follows from a standard average-reward HJB setup, and I agree, but a sketch would help), and the numerical verification script referenced in Remark 2 is not included, so that part is not independently checkable. Neither is serious.

Who is this for? Anyone modeling OTC dealer markets or designing RFQ algorithms. The symmetry is a useful benchmark even if the boundary patch changes the theorem's fine print. The paper deserves a serious referee; my own verdict would be conditional revision, not reject. I would bring it to a reading group and cite it if I wrote on this topic.

Referee Report

3 major / 4 minor

Summary. The paper studies a steady-state market-making model in which trade opportunities arrive as a Poisson process and each arriving customer is a seller with probability q and a buyer with probability 1−q. The dealer quotes a markdown or markup against a common fair price, and wins the trade if her quote beats the best competing response; the best competitor's displacement is assumed to be exponentially distributed with mean width w. The dealer's policy is characterized by an indifference liquidation cost ν(x), and the central result (Theorem 1) is an algebraic identity: the steady-state consistency equation for the imbalanced problem is equivalent to that of a balanced (q=1/2) problem after translating the skew by δ=(1/2h)log(q/(1−q)), widening the non-discretionary width Δ by γ=(1/h)log(1/(2√(q(1−q)))), and multiplying the carrying cost by M(q)=1/(2√(q(1−q))). Several corollaries follow: a flat book should still skew toward the flow, skew responds at first order in imbalance while width responds at second order, and the constant-width linear-skew heuristic is identified as the small-skew solution for a specific cost structure. The paper also sketches practical uses for RFQ markets, including filling-ratio diagnostics and transfer learning for reinforcement learning.

Significance. If the exactness claim holds, the paper provides a clean and useful symmetry result: one balanced solve supplies solutions for every imbalance level via explicit parameter adjustments that involve no fitted constants. The elementary identity (4) is verifiable, and the paper is honest about the historical provenance of the symmetrization and about the local nature of the exponential assumption. The result is potentially significant for OTC/RFQ market making, where one-sided flow is common, and the corollaries offer falsifiable cross-sectional predictions (first-order skew shift, second-order width response) that could be tested on RFQ archives. The main weaknesses are the compressed derivation of the consistency equation and an unaddressed boundary issue with the nonnegativity constraint on markups, both of which affect the exactness claim.

major comments (3)

[§2.2, Eq. (1); §2.3, Eq. (3); Theorem 1] The proof of Theorem 1 uses the unconstrained exponential form of the enquiry value G(K) in Eq. (3), but Eq. (1) contains a max(0,·) operator. When K < -1/h, the optimal markup is m=0 and G(K) = -K, so Eq. (3) is not valid at those states. The paper does not state an interiority condition in Theorem 1 and does not prove that the constraint never binds on the support of the ergodic inventory distribution. For a convex carrying cost, K(x) can be negative for large |x|, and such states can carry positive stationary mass. Moreover, the skew translation S → S - δ shifts the location of the boundary K = -1/h, so the max operator does not commute with the compression. The theorem's 'same solution' claim is therefore unproven as stated; either add an explicit assumption that the max never binds on the visited inventory states, or extend the derivation to the constrained region.
[§2.3, Eq. (3)] Eq. (3) is the load-bearing consistency equation, but the paper states it with only the sentence 'Comparing the two yields a functional equation for ν.' Given that Theorem 1 and all corollaries rest on this equation, the manuscript should provide a full derivation from the dealer's optimality (e.g., the average-reward Hamilton–Jacobi–Bellman equation or an indifference argument), or at least a precise statement of the normalization and boundary conditions. Without this, a reader cannot verify the exactness of the transformation claimed in Theorem 1.
[Remark 2] The paper claims in Remark 2 that the exponential assumption is needed only locally, with an error of the order of the hazard's relative variation, verified numerically in an external script. This is a substantive weakening of Theorem 1's hypothesis, but the error estimate is not derived and the numerical check is not described. If the theorem is to be advertised as exact, the conditions for exactness (global constant hazard, plus the interiority condition above) should be stated as assumptions; the local version should be clearly labeled as an approximation with a documented error analysis.

minor comments (4)

[§2, first paragraph] 'All trades are of sizes' appears to be a typo; it should read 'of size s'.
[§2.2, Eq. (1)] The notation '1/h(m↑(x;s))' is easily misread; clarify that h is evaluated at the markup, e.g., write m = max(K + 1/h(m), 0).
[Abstract and keywords] 'Keywords:market making' is missing a space after the colon; the abstract would also benefit from defining q and w explicitly (the body defines them, but the abstract uses these symbols without introduction).
[Corollary 1] The proof of Corollary 1 uses Sδ(0)=0 'by symmetry'; this should be justified (e.g., by uniqueness of the balanced solution or by an explicit parity argument) rather than asserted.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 follows by substitution of an algebraic identity into the model's own consistency equation, with no fitted constants and no load-bearing self-citation.

full rationale

The paper's central claim, Theorem 1, is derived by substituting the elementary identity (4) into the consistency equation (3); no parameter is fitted to the predicted skew or width, and the translation delta, widening gamma, and cost multiplier M(q) all emerge from the algebra of the exponential mixture rather than from calibration. The exponential win-curve assumption is an explicit model input, not a hidden restatement of the conclusion; Corollaries 1-3 are immediate consequences of the resulting closed-form expressions. There are no load-bearing self-citations: the companion paper by the author is mentioned only as an extension, and the birth-death symmetrization is credited to external classical references. The one genuine weakness is a rigor gap, not circularity: Eq. (3) uses the unconstrained supremum h^{-1} e^{-1-hK}, whereas Eq. (1) permits zero markup when K < -1/h, and the paper does not state an explicit interiority condition ruling out binding of the max(0,.) operator on the support of the stationary inventory distribution. That omission affects the exactness of Theorem 1 as stated, but it does not make the theorem's conclusion an input to its own derivation. Thus the derivation is self-contained and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted constants and no new physical or model entities. The central claim depends on a small set of modeling assumptions (Poisson stationary arrivals, local exponential win curve, interior optimal quotes) and standard calculus. The observable market width w and imbalance q are inputs, not free parameters.

assumptions (5)

domain assumption The market maker's problem is in ergodic steady state and is characterized by an indifference liquidation cost nu(x) satisfying the average-reward HJB consistency equation (3).
Invoked in Section 2.3 without a full derivation; all subsequent results operate on this equation.
domain assumption Best competing response is exponentially distributed with mean w = 1/h, at least locally at the quotes actually made.
Introduced in Section 2 immediately before Eq. (1); Remark 2 weakens to local constancy but the exact theorem still depends on it.
domain assumption Arrivals are Poisson with a fixed imbalance q, each trade is size s, and adverse selection epsilon is a constant.
Model setup in Section 2; the symmetry is exact only in this stationary, constant-size, constant-adverse-selection setting.
ad hoc to paper Optimal markups stay in the interior region where the max(0) constraint in Eq. (1) is not binding, so the exponential G(K) formula applies at all visited inventories.
The consistency equation (3) and Theorem 1 use the unconstrained solution; the paper does not prove this region is visited, and the theorem statement omits the caveat.
standard math The envelope theorem and the exponential survival functional form are used to derive Eq. (3) and Remark 2.
Standard calculus; not disputed.

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Cite this review

Pith. "Pith review of On a Simple Relationship Between Order Imbalance, Skew and Width in Over-The-Counter Trading." pith.science (2026). https://pith.science/paper/3CW4MKQP

@misc{pith2026260807690,
  author       = {Pith},
  title        = {Pith review of: On a Simple Relationship Between Order Imbalance, Skew and Width in Over-The-Counter Trading},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3CW4MKQP}},
  note         = {Machine review of arXiv:2608.07690}
}

read the original abstract

We consider a market maker who can only obtain and dispose of inventory by responding to a sequence of sealed-bid enquiries, and whose customers arrive with imbalanced intent: sellers more often than buyers, or the reverse. Under the assumption that the best competing response is exponentially distributed around a commonly discerned fair price, we observe a symmetry in the steady state solution that compresses the imbalanced problem onto the perfectly balanced one. Order imbalance is absorbed, exactly, by a translation of the market maker's skew, a widening of her quotes, and a multiplication of her effective cost of carry. The adjustment is simple even though the solution it adjusts is not, and it involves no free parameter beyond the observable market width. The exponential assumption is needed only locally, at the quotes actually made, and the width that enters is the locally observed one. Among the consequences: a market maker with zero inventory should still skew; skew responds to imbalance at first order whereas width responds only at second order; and the popular "constant width, linear skew" heuristic is recovered as the small-skew solution in the special case of balanced flow and quadratic holding cost.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Garman, M. B. (1976). Market microstructure.Journal of Financial Economics3(3), 257–275

work page 1976

[2] [2]

Amihud, Y., Mendelson, H. (1980). Dealership market: market-making with inventory. Journal of Financial Economics8(1), 31–53

work page 1980

[3] [3]

Ho, T., Stoll, H. R. (1981). Optimal dealer pricing under transactions and return uncer- tainty.Journal of Financial Economics9(1), 47–73

work page 1981

[4] [4]

Avellaneda, M., Stoikov, S. (2008). High-frequency trading in a limit order book.Quan- titative Finance8(3), 217–224

work page 2008

[5] [5]

Gu´ eant, O., Lehalle, C.-A., Fernandez-Tapia, J. (2012). Dealing with the inventory risk: a solution to the market making problem.Mathematics and Financial Economics7(4), 477–507

work page 2012

[6] [6]

(2016).The Financial Mathematics of Market Liquidity: From Optimal Exe- cution to Market Making

Gu´ eant, O. (2016).The Financial Mathematics of Market Liquidity: From Optimal Exe- cution to Market Making. Chapman & Hall/CRC

work page 2016

[7] [7]

Gu´ eant, O. (2017). Optimal market making.Applied Mathematical Finance24(2), 112– 154. 8

work page 2017

[8] [8]

Cartea, ´A., Jaimungal, S., Ricci, J. (2014). Buy low, sell high: a high frequency trading perspective.SIAM Journal on Financial Mathematics5(1), 415–444

work page 2014