The present study investigates the price (co)volatility of four dairy commodities -- skim milk powder, whole milk powder, butter and cheddar cheese -- in three major dairy markets. It uses a multivariate factor stochastic volatility model for estimating the time-varying covariance and correlation matrices by imposing a low-dimensional latent dynamic factor structure. The empirical results support four factors representing the European Union and Oceania dairy sectors as well as the milk powder markets. Factor volatilities and marginal posterior volatilities of each dairy commodity increase after the 2006/07 global (food) crisis, which also coincides with the free trade agreements enacted from 2007 onwards and EU and US liberalization policy changes. The model-implied correlation matrices show increasing dependence during the second half of 2006, throughout the first half of 2007, as well as during 2008 and 2014, which can be attributed to various regional agricultural dairy policies. Furthermore, in-sample value at risk measures (VaRs and CoVaRs) are provided for each dairy commodity under consideration.