The paper undertakes a non-parametric analysis of the very high frequency movements in stock market volatility using very finely sampled data on the S&P VIX index compiled by the CBOE. The data suggest that stock market volatility is best described as a pure jump process without a continuous component. The finding stands in contrast to nonparametric results, reported here and elsewhere, that the stock price itself is not a pure jump process but rather contains a continuous martingale component. The jumps in stock volatility are found to be so active that this discredits many recently proposed stochastic volatility models, including the classic affine model with compound Poisson jumps that is widely used in financial modeling and practice. Additional empirical work presents strong evidence for many common jumps, or co-jumps, in both the stock price and stock volatility.
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