We show that if a contact open book \((\Sigma,h)\) on a \((2n+1)\)-manifold \(M\) (\(n\geq1\)) is induced by a Lefschetz fibration \(\pi:W \to D^2\), then there is a 1-1 correspondence between positive stabilizations of \((\Sigma,h)\) and positive stabilizations of \(\pi\). We also show that any exact open book, an open book induced by a compatible exact Lefschetz fibration, carries a contact structure. Moreover, we prove that there is a 1-1 correspondence (similar to the one above) between convex stabilizations of an exact open book and convex stabilizations of the corresponding compatible exact Lefschetz fibration. We also show that convex stabilization of compatible exact Lefschetz fibrations produces symplectomorphic completions.