The research question we consider is whether incremental complexity in option pricing models is justified by incremental model performance. We apply the model confidence set as a formal model comparison approach in appraising stochastic volatility jump‐diffusion option pricing models, spanning affine and nonaffine specifications. Jumps in price with stochastic (constant) arrival intensity produce superior (inferior) outcomes. A parsimonious negative exponential price jump distribution outperforms the popular normal distribution. Jumps in volatility synchronized or not) worsen model performance. A parsimonious nonlinear hyperbolic drift extension of the Heston model performs particularly well. Nonlinear CEV models generally do not produce appreciable model performance.