The order book is a list of all current buy or sell orders for a given financial security. The rise of electronic stock exchanges introduced a debate about the relevance of the information it encapsulates of the activity of traders. Here, we approach this topic from a theoretical perspective, estimating the amount of mutual information between order book layers, i.e., different buy/sell layers, which are aggregated by buy/sell orders. We show that (i) layers are not independent (in the sense that the mutual information is statistically larger than zero), (ii) the mutual information between layers is small (compared to the joint entropy), and (iii) the mutual information between layers increases when comparing the uppermost layers to the deepest layers analyzed (i.e., further away from the market price). Our findings, and our method for estimating mutual information, are relevant to developing trading strategies that attempt to utilize the information content of the limit order book.
Keywords: deep layers of order book; entropy; entropy estimation; limit order book; mutual information; mutual information estimation; price and volume; recursive copula.