Refill in this model reaches the quote later than depletion and with less force. Angstmann, Diana and Gebbie place new market-maker orders beyond the visible quote, at a placement edge. Supply must diffuse inward from there, losing volume to cancellation before it reaches the touch. The displayed quote is what remains after that journey.
Why keep both sides of the book?
The authors build on the reaction-diffusion latent order book they cite from Donier and coauthors. Bid and ask densities remain separate fields. Each diffuses to represent aggregate order revision; each faces cancellation at rate ν. On contact, the sides annihilate with reaction coefficient κ. The model runs in operational time u, with calendar time supplied by an unspecified clock U(t).
When both sides have diffusivity D and cancellation rate ν, imbalance φ = ρ_B − ρ_A follows a closed linear equation for prescribed sources. Its zero locates the price. Spread geometry, though, resides in the total density. If the model keeps only imbalance, linearising around the price produces a threshold spread s = 2ε_φ/|∂_xφ|. That gradient sets linear impact too. As the authors put it, such a model "provides an implied threshold spread but no independently varying placement scale."
Their added scale is the withheld interval (q_b, q_a), where the market-making sector places no new orders. Its width is Σ. Standing density enters from the edges and decays over the cancellation length ℓ = √(D/ν). A quote forms where one side's density reaches its execution threshold ε_i. With edge density ρ^q, the quote lies d = ℓ ln(ρ^q/ε) inside the placement edge. The resulting spread is s ≃ Σ − d_A − d_B. This requires Σ > d_A + d*_B; "the formula should not be continued to negatives," the authors warn.
Placement responds to pending inventory through Σ = Σ_0 + χ_s Q_Σ. Q_Σ counts total volume awaiting paired opposite-side replenishment. An ask fill starts a round trip completed by a bid-side purchase. Because the driver is gross pending volume, Q_Σ can remain positive when signed inventory nets to zero. Immediate completion gives Q_Σ = 0 and returns Σ to Σ_0. The authors confine this closure to the inventory channel, while acknowledging that adverse selection can contribute.
The delayed refill
A market order takes volume from the quote p_i. Replenishment for side i starts at placement edge q_i, a distance d away; an ask execution pairs with refill on the bid side. At the quote, the density response follows a killed heat kernel. Its peak lag without cancellation is d²/(2D). Cancellation brings the peak forward to (√(1+4νd²/D) − 1)/(4ν).
The loss persists when responses are integrated over all lags. Refill placed at the edge (d = d*) changes quote density by ε/ρ^q times the change from equal local depletion. For a general distance, the factor is e^(−d/ℓ). The integrated quote displacement per unit of refill is 1/(2νρ^q). Hold ρ^q, D and ν fixed, and ε disappears from that expression, even though quote location, lag and response shape still depend on it.
The authors distinguish density response from first-passage arrival. Both give integrated survival e^(−d/ℓ), but with cancellation off, first-passage arrival peaks at d²/(6D), a third of the density-kernel lag d²/(2D). Their lag distributions should remain separate. The authors also define quote capacity as volume removed per unit of quote displacement over a window Δu. In short windows it is ε√(πνΔu), with a next-order factor of 1 + νΔu/3. Capacity tends to zero as the window shrinks. Any estimated coefficient of volume per unit quote move therefore depends on the window; the model has no non-zero instantaneous one.
What can one-minute bars show?
We could not test this. The quantities above require side-specific standing depth at each price level, along with distinct records of placements, cancellations and executions. One-minute OHLCV bars lack even the quote, much less ρ^q or the withheld interval.
Four inputs before a spread level
The paper text gives no numerical output; its numerical experiments are in a linked repository. Σ_0 and χ_s are unestimated, as is the completion law. The authors leave ε_i to the execution or observation rule, since the equations do not choose it. A spread level thus requires Σ_0, χ_s, the completion law and ε_i. The model does supply signed comparative statics. At fixed placement, ∂s/∂ln ε_i = ℓ. At fixed Σ, D, ν and ε, reducing edge density ρ^q widens the spread, the authors' account of a shallower book quoting wider. A change in liquidity may also alter placement, diffusion and cancellation, they caution. Nobody here has measured ℓ, which governs that slope.
The paper does not write out midpoint impact per executed share. It reaches coupled Volterra equations for midpoint and spread instead. Other participants' supply within the withheld interval enters as a perturbation. At fixed placement, its stationary effect on density is bounded by ‖s_other‖∞/ν. The authors limit this treatment to small induced density and quote displacements on the execution and spread scales. The fixed-placement bound cannot tell us where the placement edge finally lies.
A clock the trader still needs
The authors concede the operational-time limit: "finite spread and price impact arise from distinct response sectors, while their calendar-time appearance requires a separate observation clock." Their distinction works within the model. Placement determines Σ; depletion and refill kernels govern quote motion. Those responses still share density fields and remain coupled, as the authors acknowledge. A trading lag has to be priced in seconds. Without U(t), d²/(2D) is expressed in units nobody can schedule against.
The asymmetry is the reason to read this paper. Depletion strikes the quote at full weight, while equal-sized refill arrives later and contributes only ε/ρ^q. Resilience estimates that net the flows at the touch combine kernels the model separates. The authors make the same argument for net current J^− − J^+. An event-level book-data fit showing refill attenuation scaling with ε/ρ^q would turn that account into something a trader could use.