Your signal says fair is 3 cents above the mid in a 4 cent wide market and decays with a four second half life. Crossing pays 2 cents. Resting on the bid fills 35% of the time, and conditional on filling the signal has decayed to 1 cent and the fill carries 1.5 cents of adverse selection. Do you cross or rest?

Your signal says fair is 3 cents above the mid in a 4 cent wide market and decays with a four second half life. Crossing pays 2 cents. Resting on the bid fills 35% of the time, and conditional on filling the signal has decayed to 1 cent and the fill carries 1.5 cents of adverse selection. Do you cross or rest?

Approach: Value each choice as expected edge per share, using the signal conditional on the fill for the passive order rather than the signal you hold at the moment of the decision.

Cross, worth a certain 1 cent per share against 0.525 cents in expectation for resting. Crossing captures the 3 cent signal and pays the 2 cent half spread, leaving 1 cent with no uncertainty. Resting earns the 2 cent half spread plus the 1 cent of signal that survives to the fill, less 1.5 cents of adverse selection, so a fill is worth 1.5 cents, and at a 35% fill rate the expected edge is 0.525 cents. The comparison turns entirely on the half life: a signal that decays in four seconds is mostly gone by the time a passive order fills, and the fills that do arrive are selected against. A signal with a four hour half life reverses the answer, because the passive order keeps its edge and the half spread comes free.

Follow-up: At what fill probability are crossing and resting equal, holding the decayed signal and the adverse selection fixed?

Key concepts: signal decay, adverse selection, passive fill, expected edge.