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The imbalance symmetry

Companion demo to On a Simple Relationship Between Order Imbalance, Skew and Width in Over-The-Counter Trading.

A dealer faces sealed-bid enquiries that are sellers with probability $q$ and buyers with probability $1-q$. The theorem states that the imbalanced problem is the balanced one in disguise: solve the balanced problem with carrying cost multiplied by $M(q) = 1/(2\sqrt{q(1-q)})$ and tilt the indifference cost by $\delta = \tfrac{w}{2}\log\tfrac{q}{1-q}$ per unit. (The widening $\gamma$ is the same factor as $M(q)$ read in the overhead frame, not a further correction.) Below, both problems are solved numerically and independently.

The dots landing on the line. That coincidence is the theorem, and the “max deviation” readout is its size — solver precision, not approximation.

0.65 0.010
Skew shift δ
–
first order in q − ½
Width change at x = 0
–
second order in q − ½
Carry multiplier M(q)
–
multiplies the carrying cost
Max deviation, dots vs line
–
numerical solver agreement
Skew S(x) against inventory
imbalanced problem, solved directly balanced solve at cost M(q) c, shifted by δ
The skew no longer vanishes at zero inventory: S(0) = δ. A dealer with a flat book still shades her quotes toward the flow.
Skew responds at first order, width at second (per unit of market width)
δ/w, the skew shift width change at x = 0, per unit of w
Near balance the skew shift is linear in the imbalance while the width change is quadratic — why practitioners skew before they widen. The width change is the imbalanced dealer's width at a flat book minus the balanced dealer's at the same carrying cost; it comes entirely from the multiplied carry, so it depends on the carry slider. The vertical hairline marks the current slider value of q.

Model and solver: exponential best-competitor with width w = 1, grid x = −10…10, Levenberg–Marquardt on the steady-state consistency equation (mm_core.js).