# Order Flow Imbalance calculation

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)?

$$\delta \frac{L^{b}-M^{s}}{D}=\delta \frac{7-15}{5}=-\delta 1.6$$ 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!

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.

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