How far to skew to double your sell rate

Question

Your ask currently sits 3 cents above mid and fills with probability 0.20 per interval. Fill probability decays exponentially with distance from mid: p(\delta) = A\,e^{-k\delta}, and for this stock k = \ln 2 \approx 0.693 per cent. You're long and want to sell twice as often. How many cents closer to the mid should you move your ask to double the fill probability?

Accepted Answer

Set up the doubling condition. Moving the ask from \delta0 = 3 cents to a new distance \delta1 changes the fill probability by the ratio of exponentials: \frac{p(\delta1)}{p(\delta0)} = \frac{A e^{-k\delta1}}{A e^{-k\delta0}} = e^{-k(\delta1 - \delta0)}. We want this ratio to equal 2: e^{-k(\delta1 - \delta0)} = 2 \;\Longrightarrow\; -k(\delta1 - \delta0) = \ln 2 \;\Longrightarrow\; \delta0 - \delta1 = \frac{\ln 2}{k}. Plug in k = \ln 2. The \ln 2 cancels beautifully: \delta0 - \delta1 = \frac{\ln 2}{\ln 2} = 1 	ext{ cent.} So move your ask 1 cent closer to the mid, from 3 cents above to 2 c…

How far to skew to double your sell rate

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