This paper examines a class of continuous-time models with stochastic volatility that incorporate jumps in returns and volatility. We develop a likelihood-based es- timation strategy and provide estimates of model parameters, spot volatility, jump times and jump sizes using S&P 500 and Nasdaq 100 index returns. Estimates of jump times, jump sizes and volatility are particularly useful for identifying the effects of these factors during periods of market stress, such as those in 1987, 1997 and 1998.

This paper studies the empirical performance of jump-diffusion models that allow for stochastic volatility and correlated jumps affecting both prices and volatility. The results show that the models in question provide reasonable fit to both option prices and returns data in the in-sample estimation period. This contrasts previous findings where stochastic volatility paths are found to be too smooth relative to the option implied dynamics. While the models perform well during the high volatility estimation period, they tend to overprice long dated contracts out-of-sample. This evidence points towards a too simplistic specification of the mean dynamics of volatility.

Previous studies document that volatility risk is priced in the cross-section of stock returns. Driven by evidence from option pricing literature that stock prices exhibit both stochastic volatility and jumps, I test whether market jump risk is priced differently from volatility risk. In addition to earlier findings which document a negative price for volatility risk, I find that jump risk is priced separately, and is negative. Furthermore, I document significant differences between volatility and jump risk factor loadings of value vs. growth, and small vs. big portfolios. Due to differences in their volatility and jump risk loadings, investors require an additional return of 0.86 % per month on a portfolio, which longs stocks in the smallest size decile and shorts stocks in the biggest size decile. Similarly, a portfolio which longs value stocks and shorts growth stocks will on average require an additional return of 0.59 % per month.