This paper examines continuous-time stochastic volatility models incorporating jumps in returns and volatility. We develop a likelihood-based estimation strategy and provide estimates of 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. Using formal and informal diagnostics, we ¢nd strong evidence for jumps in volatility and jumps in returns. Finally, we study how these factors and estimation risk impact option pricing.

Particle learning (PL) provides state filtering, sequential parameter learning and smoothing in a general class of state space models. Our approach extends existing particle methods by incorporating the estimation of static parameters via a fully-adapted filter that utilizes conditional sufficient statistics for parameters and/or states as particles. State smoothing in the presence of parameter uncertainty is also solved as a by-product of PL. In a number of examples, we show that PL outperforms existing particle filtering alternatives and proves to be a competitor to MCMC.Comment: Published in at http://dx.doi.org/10.1214/10-STS325 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

This paper examines model specification issues and estimates diffusive and jump risk premia using S&P futures option prices from 1987 to 2003. We first develop a time series test to detect the presence of jumps in volatility, and find strong evidence in support of their presence. Next, using the cross section of option prices, we find strong evidence for jumps in prices and modest evidence for jumps in volatility based on model fit. The evidence points toward economically and statistically significant jump risk premia, which are important for understanding option returns.THERE ARE TWO CENTRAL, RELATED, issues in empirical option pricing. The first issue is model specification, which comprises identifying and modeling the factors that jointly determine returns and option prices. Recent empirical work on index options identifies factors such as stochastic volatility, jumps in prices, and jumps in volatility. The second issue is quantifying the risk premia associated with the jump and diffusive factors using a model that passes reasonable specification hurdles.The results in the literature regarding these issues are mixed. For example, tests using option data disagree over the importance of jumps in prices:

This paper analyzes the role of jumps in continuous-time short rate models. I first develop a test to detect jump-induced misspecification and, using Treasury bill rates, find evidence for the presence of jumps. Second, I specify and estimate a nonparametric jump-diffusion model. Results indicate that jumps play an important statistical role. Estimates of jump times and sizes indicate that unexpected news about the macroeconomy generates the jumps. Finally, I investigate the pricing implications of jumps. Jumps generally have a minor impact on yields, but they are important for pricing interest rate options.

This paper examines specification issues and estimates diffusive and jump risk premia using S&P futures option prices from 1987 to 2003. We first develop a test to detect the presence of jumps in volatility, and find strong evidence supporting their presence. Based on the cross-sectional fit of option prices, we find strong evidence for jumps in prices and modest evidence for jumps in volatility. We are not able to identify a statistically significant diffusive volatility risk premium. We do find modest but statistically and economically significant jump risk premia.