break-even-half-gap.tex

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\begin{document}

In a buy\textrightarrow sell sequence with a positive gap, we end up with more of the quote asset and less of the base asset. Setting our loss ratio equal to our profit ratio, we get

\[
\frac{F_B}{L} = \frac{p_sL - p_bL - F_Q}{p_bL}
\]

where $ F_B $ and $ F_Q $ are the fees accumulated for the base and quote assets respectively, $ p_b $ and $ p_s $ are the buy and sell prices, and $ L $ is the market's lot size.

Rearranging, we get 

\[
p_bF_B = p_sL - p_bL - F_Q
\]

Noting that for mid-gap price, $ r $, and half-gap width  $ g $, 

\[
p_b = r - g
\]
\[
p_s = r + g
\]

substituting,

\[
(r - g)F_B = (r + g)L - (r - g)L - F_Q
\]

and solving for $ g $ yields our break-even half-gap.


\[
g = \frac{rF_B + F_Q}{F_B + 2L}
\]

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