inventory

Control theory applied to inventory and pricing. A research site — papers, interactive demonstrations, and literature maps — extending Cotton & Papanicolaou, Trading Illiquid Goods, toward a theory of storable goods with very low carrying costs.

Papers

The flagship result, from On a Simple Relationship Between Order Imbalance, Skew and Width in Over-The-Counter Trading (Cotton; work completed around 2015). A market maker trades only by responding to sealed-bid enquiries, and her customers arrive with imbalanced intent: sellers with probability q, buyers with probability 1−q.

The imbalanced problem is the balanced one in disguise. Order imbalance is absorbed, exactly, by three adjustments tied to the observable market width w: the skew translates by δ = (w/2) log(q/(1−q)), the quotes widen by γ, and the cost of carry multiplies by M(q) = 1/(2√(q(1−q))). Consequences: a dealer with a flat book should still skew; skew responds to imbalance at first order while width responds at second; and one-sided flow taxes inventory exactly like a carrying cost.

The paper locates the constant-width linear-skew benchmark of Avellaneda–Stoikov as a corner case and supplies its missing column: what to do about flow. The exponential assumption on the best competing quote is needed only locally, at the quotes actually made (numerical certificate). The companion Cotton–Papanicolaou paper and the program's working notes are on the papers page.

Interactive demonstrations

The imbalance symmetry — both problems solved numerically and independently in your browser; the corrected balanced solution lands on the imbalanced one to within 10⁻¹³. The dots on the line are the theorem.
Optimal and sub-optimal market makers — three dealers face the same enquiry stream and the same competing quotes: the optimal policy, constant-width linear-skew with no flow term, and no skew at all. The flow term is worth a few percent; no skew at all is ruinous.

Literature

Literature map — the paper and its neighbours; dashed edges are gaps it fills or predictions not yet tested.
Annotated bibliography — organized by the role each work plays.

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 cost of carry can be a few 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.

Toward a mathematical formulation — storage relation as a term-structure constraint, transport of the coverage-forecast curve, the explosiveness boundary, filtering.
The bid–offer as endogenous cost of carry — explosive inventory in low-carry goods; the oscillator; hypotheses and scoreboard.
Weymar (1965) — distillation of the grounding thesis; program-level maps: storage theory · control & OR · instability.

Cite

@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://github.com/microprediction/inventory}
}

Comments — especially from optimal-control readers — are welcome; the notes are written to be attacked. Repository on GitHub.