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The paper examines volatility activity and its asymmetry and undertakes further specification analysis of volatility models based on it. We develop new nonparametric statistics using highfrequency option-based VIX data to test for asymmetry in volatility jumps. We also develop methods to estimate and evaluate, using price data alone, a general encompassing model for volatility dynamics where volatility activity is unrestricted. The nonparametric application to VIX data, along with model estimation for S&P Index returns, suggests that volatility moves are best captured by infinite variation pure-jump martingale with symmetric jump compensator around zero. The latter provides a parsimonious generalization of the jump-diffusions commonly used for volatility modeling.

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