Comparing Order Books Across Venues After Trading Fees

# Translating Order books accounting for fees


# Translating Order books accounting for fees












I am trying to understand how fee structure plays into how I should best execute a trade.

Say there are two exchanges with the following order book:

Exchange A:

Bid Qty | Bid Price | Ask Price | Ask Qty

```
60   |   7189.5 | 7190.0   |90
4    |   7189.0 | 7190.5   |90
1    |   7188.5 | 7191.0   |90
4    |   7188.0 | 7191.5   |90
12   |   7187.5 | 7192.0   |90
```

Exchange B:

Bid Qty | Bid Price | Ask Price | Ask Qty

```
106  |   7197.5 | 7197.9   |186
405  |   7196.8 | 7198.1   |190
2    |   7196.4 | 7198.7   |100
2    |   7185.3 | 7199.2   |9
15   |   71838  | 7199.5   |19
```

Lets now say that exchange A has a maker rebate of 2.5BPS (ie -0.025%) and a taker fee of 7.5BPS (+0.075%)

Exchange B has a maker fee of 2BPS (ie +0.025%) and a taker fee of 5.0BPS (+0.05%)

Is there any way to translate them to in a way normalise the order books for fees to see where its best to execute a trade all else being equal.

I might have missed a bunch of info that is necessary. Happy to add to this once they come to light.

## Answer by lehalle (score 1)

https://quant.stackexchange.com/a/49888

When fees are not symmetric, to take fees into account on orderbook needs to know if you want to provide or consume liquidity: you have in fact two different views (ie two ranking) on the same orderbook:

Say you are a buyer, and do the calculation for the first limit only. $P^B(i)$ and $P^A(i)$ are respectively the prices at the bid and ask on venue $i$, where marker fees (ie for liquidity providers) are $f^m(i)$ and taker fees (for liquidity consumers / removers) $f^t(i)$. Remember that for rebates, you have to put a minus sign in the fees.

- if you you want to provide liquidity, you have to compare $$P^B(i)\cdot(1+f^m(i))\stackrel{?}{\lt} P^B(j)\cdot(1+f^m(j)),$$

- if you want to consume liquidity the game is taking place at the ask: compare $$P^A(i)\cdot(1+f^t(i))\stackrel{?}{\lt} P^A(j)\cdot(1+f^t(j))$$ (you add fees to the price because you are buying, hence you will spend more money; for a seller you will get less money, so you have to put a minus sign in front of the fees).

But the reasoning cannot stop at this point:

- For liquidity consumption, it is simple: just go to the cheapest venue.

- But for liquidity provision, you need to account for the probability to obtain a transaction.

Say for instance that the best bid of venue $i$, once fees are taken into account, is cheapest than the the best bid of venue $j$, even that it is equal to the second bid of venue $j$. How can you be sure that consumers (ie seller) will not go to venue $j$ first? since the fees are not the same, the order can be inverted for them: it may be more attractive for consumers on venue $j$... In such a case you will wait on venue $i$, expecting a better net price, but nobody will ever come to trade there with you...

To avoid this configuration, the price improvement to wait on venue $i$, ie $$\Delta P^m(i|j):=P^B(j)\cdot(1+f^m(j)) - P^B(i)\cdot(1+f^m(i))$$should have the same sign as the price improvement to come and consume on this venue, ie $$\Delta P^t(i|j):=P^B(i)\cdot(1-f^t(i))- P^B(j)\cdot(1-f^t(j)).$$

This is somehow a theoretical equilibrium, thus you can have a look at the rate at which market orders are consuming your side, at your price limit to assess the probability to obtain e transaction. You can do this on a long term historical basis (estimating the Markov chain of rates given the state of the orderbook on several days), or in real-time.

For details have a look at Market Microstructure in Practice (1st or 2nd edition).