This paper extends the class of stochastic volatility diffusions for asset returns to include Poisson jumps of time-varying intensity. It finds that any reasonably descriptive continuous-time model for equity-index returns must allow for discrete jumps and stochastic volatility with a pronounced negative relationship between return and volatility innovations. The dominant empirical characteristics of the return process appear to be priced by the option market. The paper explores alternative representations for daily equity-index return dynamics within a general jump-diffusion setting. It considers specifications both inside and outside the popular class of affine models that generally provide tractable pricing and estimation procedures. The paper identifies the features of the return dynamics that account for the inadequate performance of classical models. It also explores the relationship between the estimated specification and the associated derivatives prices. The paper finds that a combination of fairly standard and parsimonious representations of stochastic volatility and jumps accommodates the dominant features of the S&P500 equity-index returns and offers an attractive alternative to complex four-factor pure diffusion specifications. The paper also finds empirical support for the affine jump-diffusion model of Bates and Bakshi, Cao and Chen. The paper concludes that there is a general correspondence between the dominant features of equity-index returns and option prices. The paper also finds that the estimated parameters for the model that are unaffected by the adjustment for volatility and jump risks are generally similar to those obtained in previous work exploiting only equity options. The paper also finds that the jump component and the asymmetric return-volatility relationship induce a smirk in the typical implied volatility pattern which resembles that extracted from options data. The paper also finds that relatively small premia for the uncertainty associated with volatility and jumps are sufficient to replicate most of the salient features of the term structure of implied volatility. The paper concludes that a large number of characteristics of the stock price process, which seem to be implied or priced by the derivatives contracts, are independently identified as highly significant components of the underlying dynamics uncovered in the empirical analysis of the S&P500 returns. The paper also concludes that the results indicate a general correspondence between the dominant features of the equity-index returns and option prices. The paper also concludes that the need for a general, yet efficient framework for inference leads to the adoption of a variant of the simulated method of moments (SMM) technique. The paper also concludes that the results indicate that both stochastic volatility and discrete jump components are critical ingredients of the data generating mechanism. The paper also concludes that a pronounced negative correlation between return and volatility innovations is necessary to capture the skewness in S&P500 returns. The paper also concludes that a relatively low-frequency jump component accounts for the fat tails of the returns distribution. The paper also concludes that jumps occur on average 3-4 times a year. The paper also concludes that the discontinuities are relatively small, with most of the jumps lying within the ±3% range. The paper also concludes that all variants of the model without a negative return-volatility relation orThis paper extends the class of stochastic volatility diffusions for asset returns to include Poisson jumps of time-varying intensity. It finds that any reasonably descriptive continuous-time model for equity-index returns must allow for discrete jumps and stochastic volatility with a pronounced negative relationship between return and volatility innovations. The dominant empirical characteristics of the return process appear to be priced by the option market. The paper explores alternative representations for daily equity-index return dynamics within a general jump-diffusion setting. It considers specifications both inside and outside the popular class of affine models that generally provide tractable pricing and estimation procedures. The paper identifies the features of the return dynamics that account for the inadequate performance of classical models. It also explores the relationship between the estimated specification and the associated derivatives prices. The paper finds that a combination of fairly standard and parsimonious representations of stochastic volatility and jumps accommodates the dominant features of the S&P500 equity-index returns and offers an attractive alternative to complex four-factor pure diffusion specifications. The paper also finds empirical support for the affine jump-diffusion model of Bates and Bakshi, Cao and Chen. The paper concludes that there is a general correspondence between the dominant features of equity-index returns and option prices. The paper also finds that the estimated parameters for the model that are unaffected by the adjustment for volatility and jump risks are generally similar to those obtained in previous work exploiting only equity options. The paper also finds that the jump component and the asymmetric return-volatility relationship induce a smirk in the typical implied volatility pattern which resembles that extracted from options data. The paper also finds that relatively small premia for the uncertainty associated with volatility and jumps are sufficient to replicate most of the salient features of the term structure of implied volatility. The paper concludes that a large number of characteristics of the stock price process, which seem to be implied or priced by the derivatives contracts, are independently identified as highly significant components of the underlying dynamics uncovered in the empirical analysis of the S&P500 returns. The paper also concludes that the results indicate a general correspondence between the dominant features of the equity-index returns and option prices. The paper also concludes that the need for a general, yet efficient framework for inference leads to the adoption of a variant of the simulated method of moments (SMM) technique. The paper also concludes that the results indicate that both stochastic volatility and discrete jump components are critical ingredients of the data generating mechanism. The paper also concludes that a pronounced negative correlation between return and volatility innovations is necessary to capture the skewness in S&P500 returns. The paper also concludes that a relatively low-frequency jump component accounts for the fat tails of the returns distribution. The paper also concludes that jumps occur on average 3-4 times a year. The paper also concludes that the discontinuities are relatively small, with most of the jumps lying within the ±3% range. The paper also concludes that all variants of the model without a negative return-volatility relation or