Tuning your skew to stop the bleeding exactly

Question

Your fill intensity on each side falls off exponentially with distance from mid: a quote posted $\delta$ cents from mid fills at rate $\lambda(\delta) = A\,e^{-k\delta}$ orders per minute, with $A = 10$ and $k = 0.5$ per cent. Flat, you'd post both quotes $4$ cents from mid.

You're now long, and you want your quotes to net a drain of exactly $2$ orders per minute (sell rate minus buy rate).

How should you skew, where do you place bid and ask, to hit that target?

Accepted Answer

Set up the two fill rates. Skew shifts the whole quote pair down by some amount \Delta: the ask (sell side, reduces inventory) moves to 4 - \Delta cents from mid, filling faster; the bid (buy side, adds inventory) moves to 4 + \Delta cents from mid, filling slower. \lambda{	ext{sell}} = 10\,e^{-0.5(4-\Delta)}, \qquad \lambda{	ext{buy}} = 10\,e^{-0.5(4+\Delta)}. Net drain (target = 2). We want sells to exceed buys by 2 per minute: \lambda{	ext{sell}} - \lambda{	ext{buy}} = 10\,e^{-2}\big(e^{0.5\Delta} - e^{-0.5\Delta}\big) = 2. Since 10\,e^{-2} = 10 	imes 0.1353 = 1.353, and e^{x} - e^{-x…

Tuning your skew to stop the bleeding exactly

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