Food price volatility has re‐emerged as an important topic of political discussion since the food price crisis of 2007–2008. Different volatility drivers have been identified for different markets in the theoretical and empirical literature. However, there is no comprehensive analysis that considers a large number of potential drivers and investigates their joint effects in a dynamic model of interrelated markets. Our study provides such a volatility analysis for the oilseeds and vegetable oils markets. We use a common GARCH approach and a VAR model to identify volatility drivers and spillover effects. Our results show that exchange rate volatility is very important. However, the hotly debated financialisation of commodity markets is not found to be volatility increasing in our monthly data. Impulse response functions show strong spillover effects. Because many volatility drivers found to be important in other markets have no significant effect in our study, our results suggest that volatility drivers are market specific. This implies that any volatility‐reducing policies need to be designed for the market in question.