  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 nonparametric discrete model for the
dynamics of the best bid and ask, which can be approximated by a reduced-form
model with analytical tractability that can fit the empirical data of
correlation, volatilities and probability of price movement simultaneously.
