Why are quadratic normal volatility models analytically tractable?

We discuss the class of quadratic normal volatility (QNV) models, which have drawn much attention in the financial industry due to their analytic tractability and flexibility. We characterize these models as those that can be obtained from stopped Brownian motion by a simple transformation and a change of measure that depends only on the terminal value of the stopped Brownian motion. This explains the existence of explicit analytic formulas for option prices within QNV models in the academic literature. Furthermore, via a different transformation, we connect a certain class of QNV models to the dynamics of geometric Brownian motion and discuss changes of nuḿeraires if the nuḿeraire is modeled as a QNV process.