Score-Driven Modeling with Jumps: An Application to S&P500 Returns and Options

We introduce a novel score-driven model with two sources of shock, allowing for both time-varying volatility and jumps. A theoretical investigation is performed which yields sufficient conditions to ensure stationarity and ergodicity. We extend the model to consider a time-varying jump intensity. Both an in-sample and an out-of-sample analysis based on the S&P500 time series show that the proposed methodology provides excellent agreement with observed returns, outperforming more standard Generalized Autoregressive Conditional Heteroskedasticity (GARCH) specifications with jumps. Finally, we apply our models to option pricing via risk neutralization. Results show this novel approach produces reliable implied volatility surfaces. Supplementary Materials including proofs, the derivation of the conditional Fisher information, and two figures showing additional empirical results are available online.