Skewing quotes to bleed off a long position

Question

You're long 3{,}000 shares of a stock worth 10.00. Flat, you'd quote 9.98 / 10.02, and at those prices you estimate each side fills a 1{,}000-share order with probability 0.30 per minute. You skew both quotes down 2 cents to 9.96 / 10.00. Your fill model: moving a quote 2 cents toward the mid raises that side's fill probability to 0.50; moving it 2 cents away lowers it to 0.15. What is your expected inventory change per minute before and after the skew?

Accepted Answer

Track expected inventory change as (expected shares bought) minus (expected shares sold) per minute, on 1{,}000-share clips. Before the skew (symmetric 9.98/10.02). Both sides fill with probability 0.30: E[\Delta q] = \underbrace{0.30 	imes 1000}{	ext{buy on bid}} - \underbrace{0.30 	imes 1000}{	ext{sell on ask}} = 300 - 300 = 0. Symmetric quotes drift your inventory nowhere, bad, because you're already long 3{,}000 and want to shrink. After the skew (down 2 cents, 9.96/10.00). Your ask moved toward the mid, so its fill probability rises to 0.50; your bid moved away from the mid, so its f…

Skewing quotes to bleed off a long position

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