From orders to prices: A stochastic description of the limit order book to forecast intraday returns.

We propose a microscopic model to describe the dynamics of the fundamental events in the limit order book (LOB): order arrivals and cancellations. It is based on an operator algebra for individual orders and describes their effect on the LOB. The model inputs are arrival and cancellation rate distributions that emerge from individual behavior of traders, and we show how prices and liquidity arise from the LOB dynamics. In a simulation study we illustrate how the model works and highlight its sensitivity with respect to assumptions regarding the collective behavior of market participants. Empirically, we test the model on a LOB snapshot of XETRA, estimate several linearized model specifications, and conduct in- and out-of-sample forecasts.The in-sample results based on contemporaneous information suggest that our model describes returns very well, resulting in an adjusted $R^2$ of roughly 80%. In the more realistic setting where only past information enters the model, we observe an adjusted $R^2$ around 15%. The direction of the next return can be predicted (out-of-sample) with an accuracy above 75% for time horizons below 10 minutes. On average, we obtain an RMSPE that is 10 times lower than values documented in the literature.