inventory

Order Imbalance, Skew and Width

Control theory applied to inventory and pricing in over-the-counter markets.

A dealer in an over-the-counter market trades one way: customers request quotes, several dealers respond, the best price wins. When the customers are imbalanced — sellers more often than buyers, or the reverse — every dealer knows to shade her quotes with the flow. The optimal market making literature, built on symmetric arrivals, has never said by how much.

It turns out the answer is a symmetry. In the steady state, the market making problem with arrival imbalance $q$ compresses exactly onto the perfectly balanced problem: the imbalance is absorbed by a translation of the skew, a widening of the quotes, and a multiplication of the effective cost of carry — three constants involving nothing but the imbalance and the observable market width $w$.

Imbalanced flow sellers with probability q the problem you face Balanced flow q = ½, carry × M(q) the problem you solve the symmetry one balanced solve, every q skew shifts by δ first order in q − ½ quotes widen by γ second order carry × M(q) flow taxes inventory

Three explicit corrections: $\delta = \tfrac{w}{2}\log\tfrac{q}{1-q}$, a widening $\gamma$, and the carry multiplier $M(q) = 1/(2\sqrt{q(1-q)})$.

Three consequences follow. A dealer with a flat book should still skew, by $\delta$ exactly. Skew responds to imbalance at first order while width responds only at second — which is why practitioners skew before they widen. And the constant-width, linear-skew benchmark of Avellaneda–Stoikov reappears as a corner case, with the symmetry supplying its missing column: what to do about flow. For the derivation see the paper; the theorem is verified numerically to machine precision by an accompanying script.

Demos

The claims run live in your browser: the symmetry demo solves the imbalanced and balanced problems independently and lands one on the other to within $10^{-13}$; the market making simulation races the optimal dealer against flow-blind benchmarks on a common enquiry stream; and the learning demo shows the symmetry cutting the sample complexity of reinforcement learning and of its inverse. The demos page lists them.

The wider program

Storage economics stabilizes inventory from below by the stockout constraint and from above by the cost of carry. For value-dense, non-degrading goods the carry can be tens of basis points a year — effectively absent. What then keeps optimal inventory bounded? The working hypothesis of this repository: nothing physical does. The stabilizer is microstructural — the dealer market’s bid–offer is the endogenous replacement for the missing carrying cost. The working notes develop the formulation.

Cite

Cotton, P. (2026). “On a Simple Relationship Between Order Imbalance, Skew and Width in Over-The-Counter Trading.” Working paper; work completed around 2015, first written up 2022. 
@unpublished{cotton2026skew,
  author = {Cotton, Peter},
  title  = {On a Simple Relationship Between Order Imbalance,
            Skew and Width in Over-The-Counter Trading},
  note   = {Working paper; work completed around 2015,
            first written up 2022},
  year   = {2026},
  url    = {https://inventory.microprediction.org}
}

Bibliography

Organized by the role each work plays relative to the imbalance equivalence paper. Every entry also appears on the literature map.

The core

Dealer-inventory classics

The economics of OTC immediacy

The optimal market making line

RFQ markets and empirics

The probability side

The storage program

Working on a related paper? Open an issue on the inventory repo and we'll add it.