Financial time series exhibit periods where the volatility is consistently low that alternate with periods of consistently high volatility. This feature is referred to as volatility clustering. The variation of volatility can be attributed to nonhomogeneous information arrival, which can be proxied by trading volume [see Mandelbrot and Taylor (1967, Operations Research), and Karpoff (1987, Journal of Financial and Quantitative Analysis), inter alia]. Loosely speaking, the argument is that trading does not take place in a uniform fashion across time: new information will result in a more dense trading pattern with higher trading volumes, which in turn result in higher volatilities.

Garch and stochastic volatility models have been used in the literature to mimic volatility clustering [see Engle (1982, Econometrica), and Ghysels, Harvey and Renault (1996) and the references therein. Nevertheless, although persistent, volatility should exhibit some degree of mean reversion. It is counterintuitive to think of the volatility reaching 1000% with any meaningful probability, even for very long horizons. Empirically, the volatility persistence is in many cases estimated to be equal to one, which consists a puzzle. A number of alternatives have been proposed, including models with structural breaks, or models where volatility is fractionally integrated.

New families of models have been recently introduced, where prices are explicitely modelled in business rather than calendar time. The resulting calendar time processes are in many cases Levy. The variance gamma model is a standard example of this approach.

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