Out-of-sample density forecasts with affine jump diffusion models

•We conduct density forecast evaluations of the affine jump diffusion models.•We use the S&P 500 stock index and its options contracts.•Our results support the time-varying jump risk premia models.•The options’ information improves density forecasting ability.•Beta transformation is used for density parameter updating. We conduct out-of-sample density forecast evaluations of the affine jump diffusion models for the S&P 500 stock index and its options’ contracts. We also examine the time-series consistency between the model-implied spot volatilities using options & returns and only returns. In particular, we focus on the role of the time-varying jump risk premia. Particle filters are used to estimate the model-implied spot volatilities. We also propose the beta transformation approach for recursive parameter updating. Our empirical analysis shows that the inconsistencies between options & returns and only returns are resolved by the introduction of the time-varying jump risk premia. For density forecasts, the time-varying jump risk premia models dominate the other models in terms of likelihood criteria. We also find that for medium-term horizons, the beta transformation can weaken the systematic effect of misspecified AJD models using options & returns.