One popular approach to model the limit order books dynamics of the best bid and ask at level-1 is to use the reduced-form diffusion approximations. It is well known that the biggest contributing factor to the price movement is the imbalance of the best bid and ask. We investigate the data of the level-1 limit order books of a basket of stocks and study the numerical evidence of drift, correlation, volatility and their dependence on the imbalance. Based on the numerical discoveries, we develop a non-parametric discrete model for the dynamics of the best bid and ask, which can be approximated by a reduced-form model that incorporates the empirical data of correlation and volatilities with analytical tractability that can fit the empirical data of the probability of price movement.
A Reduced-Form Model for Level-1 Limit Order Books