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I was wondering if anyone could help me understand Figure 2 Rama Cont's Price Impact paper? It is on arxiv as well.

In Figure 2 (screen from arxiv version), they demonstrate how to derive change in midprice and I do not understand why the result is $-\delta$? (-1 tick)? enter image description here

$\delta \frac{L^{b}-M^{s}}{D}=\delta \frac{7-15}{5}=-\delta 1.6$ I understand the market order depletes one full tick, but they define $D$ as market depth and even tells us it is 5, but clearly they are using 8.

Could anyone point me in the right direction, please?

Thanks!

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The way I understand this:

  1. $15$ shares are market sold in figure 1, this corresponds with $15 / 5 = 3$ ticks
  2. $7$ limit bid orders are added after step 1. This fully replenishes one level and partially replenishes a second one.

After this the best bid is $0 - 3 + 2 = 1$ tick lower than at the start.

I'm not sure why you say they are using 8, this seems to be the delta in shares offered, calculated as $-15 + 7 = 8$.

It seems they use $\lceil x \rceil$ to denote rounding of $x$ towards 0. Rounding towards zero is necessary as the bid or ask level only changes if a level is completely filled or emptied.

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    $\begingroup$ Thanks a lot, Bob! I did not realize it was a rounding symbol; I thought it was just squared brackets. $\endgroup$
    – Jan Sila
    Apr 2, 2023 at 21:00
  • $\begingroup$ You're welcome, I must say the typesetting on arxiv isn't great making it easy to miss. $\endgroup$
    – Bob Jansen
    Apr 3, 2023 at 17:42

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