Limit Order Book and its modelling in terms of Gibbs Grand-Canonical Ensemble

Abstract

In the domain of the so called Econophysics some attempts already have been made for applying the theory of Thermodynamics and Statistical Mechanics to economics and financial markets. In this paper a similar approach is made from a different perspective, trying to model the limit order book and price formation process of a given stock by the Grand-Canonical Gibbs Ensemble for the bid and ask processes. As a consequence we can define in a meaningful way expressions for the temperatures of the ensembles of bid orders and of ask orders, which are a function of maximum bid, minimum ask and closure prices of the stock as well as of the exchanged volume of shares. It is demonstrated that the difference between the ask and bid orders temperatures can be related to the VAO (Volume Accumulation Oscillator) indicator, empirically defined in Technical Analysis of stock markets. Furthermore the distributions for bid and ask orders derived by the theory can be subject to well defined validations against real data, giving a falsifiable character to the model.