Today, I want to write about order book matching algorithms. These algorithms are scratching the surface of HFT and normally you do not need to know much about them except if you trade fully automated intraday or if you want to incorporate a more realistic market simulation in your backtester. Nevertheless I think they provide an interesting glimpse into the mechanics of an exchange.

## What is an orderbook?

First i want to explain what an orderbook is. When the limit orders for a specific asset arrive at an exchange, they are collected in a list. This list is termed the orderbook. There is one orderbook per asset. This is divied into the sell orders an the buy orders. It looks roughly as follows:

ID Time Size Price   Price Size Time ID
4 8:00:04 250 100   101 750 8:00:01 1
6 8:00:10 500 100   101 500 8:00:05 5
2 8:00:01 750 97   101 750 8:00:30 8
7 8:00:10 150 96   102 250 8:00:02 3

Of course it is in a database inside a computer, so the above is only a representation. On the left hand side I have grouped the buy orders, whereas on the right side i have the sell orders. Here the price of the buy orders is lower than the price of the sell orders. There are special cases where this is not the case, but for clarity we can disregard them. The size is the amount of the asset which should be traded. Time marks the time when the order arrived at the exchange. The field id is an identifier for the order. There are also other fields like the participants, but the ones I have written above are the most important ones. Note that orders are sorted with respect to price and when they have the same price they are sorted wrt. their arrival time at the exchange. When a market order is routed to the exchange, the exchange has to decide which orders are filled. This decision algorithm is called an order book matching algorithm.

## Price-Time-Priority/FIFO

The most simplest order book matching algorithm is a price-time-priority algorithm. That means that the matching priority is first price and then time. The participants are rewarded for offering the best price and coming early. Let us assume we have the same order book as above and a buy order for 1000 assets comes in. The relevant parts from above is the sell side.

Sell
Price Size Time ID
101 750 8:00:01 1
101 500 8:00:05 5
101 750 8:00:30 8

When assuming price-time-priority, the orders are simply filled from above. So the first order of 750 assets is filled and the remaining 250 assets are filled from order id 5. So the order book after the buy order looks like the following.

Sell
Price Size Time ID
101 250 8:00:05 5
101 750 8:00:30 8

## Price Pro-rata

Price Pro-rata matching is often found in futures market and means that priority is first given to the price and then the orders are filled proportionally to their size. Assume again the order book from above. The motivation is, that order id 8 has a large size and should be preferred. We buy again 1000 assets. The total of assets with the best price of 101 is 2000, so each of the limit orders 1, 5, 8 is half filled. The order book after the execution of the buy order looks as follows:

Sell
Price Size Time ID
101 375 8:00:01 1
101 250 8:00:05 5
101 375 8:00:30 8

## Conclusion

There are many more order book matching algorithms, so I only covered the most important and widespread ones. They have different features like being good for small liquidity, but unfortunately I did not find a comprehensive reference.

The order book matching algorithm depends on the exchange, so when in doubt, consult the documentation of your exchange. Especially if you wonder why you are not filled when you joined too late at a price level.

18 October 2016

#### Category

Quantitative Finance