equities - Price functions based on order book events

Assume some equity traded on a given exchange based on an electronic limit open-order book $B$ that makes sequential updates as a function of time $t$. What are "natural" or common price functions $P: B \rightarrow \mathbb{R}_{\ge0}$?

Two natural price functions are

1. The average of the best bid and best offer


2. The price of the most recent transaction

A disadvantage of the first price function is that it doesn't take into account the whole depth of the book. A disadvantage of the second price function is that it only updates when a transaction occurs.

Are there more sophisticated price functions that take into account the whole depth of the book, and change for every update to the order book?

Answer

I recommend reading Cao, Hansch, and Wang (2004) "The Informational Content of an Open Limit Order Book". They present a simple model for an order-book price called the weighted price ($\mbox{WP}$):

$$\mbox{WP}^{n_1 - n_2} = \frac{\sum_{j=n_1}^{n_2} (Q_j^d P_j^d + Q_j^s P_j^s)}{(Q_j^d + Q_j^s)}$$

Where:

• $n$ is the order book level


• $Q_j$ is the size at level $j$


• $P_j$ is the price at level $j$


• $d$ is the "demand" side and $s$ is the "supply" side