The What, When and Where of Limit Order Books

We model the limit order book (LOB) as a continuous Markov process and develop an algebra to describe its dynamics based on the fundamental events of the book: order arrivals and cancellations. We show how all observables (prices, returns, and liquidity measures) are governed by the same variables which also drive arrival and cancellation rates. The sensitivity of our model is evaluated in a simulation study and an empirical analysis. We estimate several linearized model specifications based on the theoretical description of the LOB and conduct in- and out-of-sample forecasts on several frequencies. The in-sample results based on contemporaneous information suggest that our model describes up to 90% of the variation of close-to-close returns, the adjusted $R^2$ still ranges at around 80%. In the more realistic setting where only past information enters the model, we still observe an adjusted $R^2$ in the range of 15%. The direction of the next return can be predicted, out-of-sample, with an accuracy of over 75% for short time horizons below 10 minutes. Out-of-sample, on average, we obtain $R^2$ values for the Mincer-Zarnowitz regression of around 2-3% and an $RMSPE$ that is 10 times lower than values documented in the literature.