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The term structure of the flow skew

The finite-horizon behavior of the imbalance equivalence. Companion to On a Simple Relationship Between Order Imbalance, Skew and Width in Over-The-Counter Trading; a note on the finite-horizon case is in preparation.

The steady-state theory says one-sided flow displaces a dealer's quoted mid by the constant $\delta = \tfrac{1}{2k}\log(A^b/A^a)$. At finite horizon that is not what happens. In the model of Guéant, Lehalle and Fernandez-Tapia with asymmetric arrival scales, the problem is exactly conjugate to a balanced one, but the imbalance lands in the terminal condition: the dealer's flow is balanced around the shifted price $S - \delta$ while her book is marked at $S$. That mismatch makes the flow displacement a term structure: zero at maturity, converging to $-\delta$ at the spectral-gap rate as the horizon grows.

Look for: the displacement curve leaving zero at maturity and climbing to the dashed $-\delta$ line at the spectral rate; the spread's finite-horizon transient; and the violet line — a dealer who marks her terminal book at the flow-adjusted price $S - \delta$ — flat at $-\delta$ for every horizon. One accounting convention turns the term structure back into the constant.

4.0 0.10
Steady-state shift −δ
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the long-horizon limit
Spectral gap
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convergence rate of the term structure
Half-life of the transient
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ln 2 / gap, in model time
Mid displacement against horizon $\tau = T - t$ (inventory 0)
standard marking (book at S) flow-adjusted marking (book at S − δ) −δ asymptote
At $\tau = 0$ the terminal condition cancels the flow skew entirely: both depths equal $c_0$ whatever the imbalance. The climb to $-\delta$ is governed by the spectral gap of the symmetrized system; the violet dealer, marking at the price her flow is balanced around, has no transient to climb.
Spread difference against horizon (inventory 0)
standard marking flow-adjusted marking (identically zero)
The spread transient is zero at maturity, zero in the long-horizon limit, and nonzero in between, with a sign that depends on the parameters. It is second order in the imbalance at zero inventory; away from zero inventory it is first order.

Engine: the paper's exact spectral representation, computed in your browser (fh_core.js): one symmetric eigendecomposition serves every horizon and both markings. Model of Guéant, Lehalle and Fernandez-Tapia (2012) with asymmetric scales; $\sigma = \sqrt{2}$, $k = 1$.