  This paper develops a model of liquidity provision in financial markets by
adapting the Madhavan, Richardson, and Roomans (1997) price formation model to
realistic order books with quote discretization and liquidity rebates. We
postulate that liquidity providers observe a fundamental price which is
continuous, efficient, and can assume values outside the interval spanned by
the best quotes. We confirm the predictions of our price formation model with
extensive empirical tests on large high-frequency datasets of 100 liquid Nasdaq
stocks. Finally we use the model to propose an estimator of the fundamental
price based on the rebate adjusted volume imbalance at the best quotes and we
empirically show that it outperforms other simpler estimators.
