Before revision:  We also characterize the dynamics of the  demand curve and of the clearing price in the equivalent martingale measure. We find that the volatility of the clearing price is inversely proportional to the sum of buy and sell order flow density (evaluated at the clearing price ), which confirms the intuition that volatility is inversely proportional to volume. We also demonstrate that our approach is implementable. We use real order book data and simulate option prices under a particularly simple parameterization of our model.
After revision:  We also characterize the dynamics of the demand curve as quantity as a function of price (as opposed to price as a function of quantity). We calibrate our model to real order book data and simulate option prices under a particularly simple parameterization of our model.