decred.org/dcrdex@v1.0.5/docs/images/break-even-half-gap.tex (about)

     1  \documentclass[]{article}
     2  \begin{document}
     3  
     4  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
     5  
     6  \[
     7  \frac{F_B}{L} = \frac{p_sL - p_bL - F_Q}{p_bL}
     8  \]
     9  
    10  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.
    11  
    12  Rearranging, we get 
    13  
    14  \[
    15  p_bF_B = p_sL - p_bL - F_Q
    16  \]
    17  
    18  Noting that for mid-gap price, $ r $, and half-gap width  $ g $, 
    19  
    20  \[
    21  p_b = r - g
    22  \]
    23  \[
    24  p_s = r + g
    25  \]
    26  
    27  substituting,
    28  
    29  \[
    30  (r - g)F_B = (r + g)L - (r - g)L - F_Q
    31  \]
    32  
    33  and solving for $ g $ yields our break-even half-gap.
    34  
    35  
    36  \[
    37  g = \frac{rF_B + F_Q}{F_B + 2L}
    38  \]
    39  
    40  \end{document}