Families with fiber isomorphic to $\mathbb{P}^1$ and the relative Proj construction

Question

Let $\pi: X \to S$ be a flat, smooth morphism of schemes of relative dimension $1$, such that each fiber is isomorphic to $\mathbb{P}_k^1$.

Is $X$ necessarily isomorphic $\operatorname{Proj}(\mathcal{F})$ for some some quasi-coherent sheaf $\mathcal{F}$ of $\mathcal{O}_S$-modules (which is also a sheaf of graded $\mathcal{O}_S$-algebras)?