inventory (view as web page)

Control theory applied to inventory and pricing. A research repository — working notes, preprints and literature maps — extending Cotton & Papanicolaou, Trading Illiquid Goods, toward a theory of storable goods with very low carrying costs.

@unpublished{cotton2026skew,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}
}

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 — and the resulting closed loop behaves as an oscillator with amplitude-dependent damping, parking itself at the edge of stability.

Two documents ground the program:

• Weymar's 1965 MIT thesis (distilled in literature/weymar1965.md): the spot price of a storable good is a boundary-value problem — the storage relation supplies the slopes of the expected-price curve as a function of the expected coverage path, and a long-run anchor expectation supplies the level. His behavioural argument for discarding explosive inventory paths leans on carrying costs being material; at tens of basis points it barely binds.
• Cotton & Papanicolaou, Trading Illiquid Goods (literature/trading_illiquid.pdf, unpublished): dealer market making in sealed-bid auctions as stochastic control. The optimal policy is characterized by an inventory indifference cost ν(x) whose slope is the quoted skew and whose convexity is the discretionary width — so the quote surface is the first two derivatives of a storage value function.

Joining the two: the quadratic holding-cost term that entered the dealer model as a convenience is, for low-carry goods, the load-bearing physical object, and several quantities usually modelled independently — inventory, price level, volatility, quoted width and skew — turn out to be linked by layered constraints, in loose analogy with how HJM ties drifts to volatilities (an analogy, not a construction: there is no traded curve here).

This repository is for theory — papers, preprints, notes written to gather feedback.

1. Formulate the constrained joint dynamics of inventory, price, volatility and dealer width/skew (notes/formulation.md), including the filtering problem that arises when flow forecasts are observable but the stock is latent.
2. Characterize the explosive-inventory boundary at low carrying cost — cheap-control singular limits, turnpike loss, and the Van der Pol reading of the commodity cycle (notes/spread_as_endogenous_carry.tex).
3. Publish the core results (work completed around 2015) as a clean solo paper — papers/skew_width_imbalance/skew_width_imbalance.tex — relating optimal dealer skew and width to flow imbalance (the exact point of departure from Avellaneda & Stoikov: skew responds to imbalance, even at zero inventory). Then revise Cotton & Papanicolaou, Trading Illiquid Goods, as the extension, citing it.

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

flowchart LR
    W[Weymar 1965<br/>storage relation, coverage path] --> F[notes/formulation.md<br/>layered constraints, filtering]
    CP[Cotton & Papanicolaou<br/>skew = slope, width = convexity of ν] --> F
    CP --> S[notes/spread_as_endogenous_carry.tex<br/>bid-offer as endogenous carry]
    F --> S
    S --> O[oscillator, limit cycle,<br/>marginal stability]
    LT[literature/map_theory.md] -.-> F
    LC[literature/map_control.md] -.-> F
    LI[literature/map_instability_theory.md] -.-> O

• papers/skew_width_imbalance/skew_width_imbalance.tex — On a Simple Relationship Between Order Imbalance, Skew and Width in Over-The-Counter Trading (work completed around 2015): the sealed-bid dealer model, the imbalance equivalence theorem (skew translated, width widened, carry multiplied by 1/(2√(p(1−p)))), zero-inventory skew, and CWLS as a corner case
  • papers/skew_width_imbalance/verify_local_exponentiality.py — numerical certificate for the theorem on the grid, and the check that the result survives, with local width, when the win curve is only locally exponential
  • papers/skew_width_imbalance/novelty_imbalance.md — verified prior-art review and positioning; papers/skew_width_imbalance/data_scout_imbalance.md — public datasets for the empirical section
• papers/exponential_normal_form/exponential_normal_form.tex — Exponential Rigidity and a Log-Value Normal Form for Imbalanced Market Making: for arbitrary win curves, imbalance is an exact translation in the coordinates of the effective log-value Φ = −log G; rigid strike translations balance the problem exactly for the affine-exponential family κ + Be^(−hK), while rigidity of the submitted quotes, or a finite-mean tail, singles out the exponential (two rigidity theorems and a quote-rigidity corollary); the failure to lift the pointwise gauge to an inventory potential is a curvature functional whose value at the affine point is the widening γ; a one-operator perturbation hierarchy plus envelope transfer covers Weibull, Gompertz, gamma, mixtures and empirical win curves; a distribution-free parity theorem separates what imbalance does by arithmetic (skew odd, width even) from what exponentiality adds (the coefficients)
  • papers/exponential_normal_form/verify_normal_form.py — numerical certificate: exact claims at machine precision, expansion remainders at their claimed orders
• papers/normal_form_rl/architecture.md — normal-form RL: exact exponential controller + perturbative tangent controller + small learned residual; the critic in the normal coordinates where imbalance vanishes for every win curve; response library, sensitivity-penalized robustness, minimax regret, the anchor as a Bellman control variate
  • papers/normal_form_rl/verify_tangent_rates.py — rate certificate: on the bounded lattice the consistency solution is exactly average-reward optimal; regret of the exponential controller is O(ε²) and of the tangent controller O(ε⁴)
• docs/ — the web site: paper PDF, interactive demos (symmetry, simulation; solver in docs/mm_core.js), an interactive literature map, and an annotated bibliography
• papers/skew_width_imbalance/sifin_cover_letter.tex — cover letter for the SIAM J. Financial Mathematics short-communication submission
• notes/formulation.md — the mathematical formulation: storage relation as a term-structure constraint, transport of the coverage-forecast curve, volatility and width constraints, an MFG aggregation conjecture, the explosiveness boundary, filtering
• notes/spread_as_endogenous_carry.tex (+ PDF) — working note: the bid–offer as endogenous cost of carry; the imbalance multiplier; constant-width linear-skew as the fingerprint of the explosive-capable regime; the oscillator; hypotheses and scoreboard
• notes/parked.md — ideas deliberately deferred
• literature/ — the grounding documents and literature maps
  • weymar1965.md — distillation of Weymar's thesis
  • trading_illiquid.pdf — the Cotton & Papanicolaou deck
  • map_theory.md — supply of storage, competitive storage, convenience-yield and HJM-style models, empirics, instability
  • map_control.md — the OR/control line: (s,S), servo/bullwhip, joint pricing-inventory control, warehouse problem, market making, MFG, cheap control
  • map_instability_theory.md — limit cycles in economics, stability-is-destabilizing, self-organized criticality, the averaging toolkit