Volatility in Discrete and Continuous Time Models: A Survey with New Evidence on Large and Small Jumps

The topic of volatility measurement and estimation is central to nancial and more generally time series econo- metrics. In this paper, we begin by surveying models of volatility, both discrete and continuous, and then we summarize some selected empirical ndings from the literature. In particular, in the rst sections of this paper, we discuss important developments in volatility models, with focus on time varying and stochastic volatility as well as nonparametric volatility estimation. The models discussed share the common feature that volatilities are unobserved, and belong to the class of missing variables. We then provide empirical evidence on "small" and "large" jumps from the perspective of their contribution to overall realized variation, using high frequency price return data on 25 stocks in the DOW 30. Our "small" and "large" jump variations are constructed at three truncation levels, using extant methodology of Barndor¤-Nielsen and Shephard (2006), Andersen, Bollerslev and Diebold (2007) and Aït-Sahalia and Jacod (2009a,b,c). Evidence of jumps is found in around 22.8% of the days during the 1993-2000 period, much higher than the corresponding gure of 9.4% during the 2001-2008 period. While the overall role of jumps is lessening, the role of large jumps has not decreased, and indeed, the relative role of large jumps, as a proportion of overall jumps has actually increased in the 2000s.