Stochastic volatility models and the family of GARCH models have been designed to capture well-known stylized facts of the volatility in financial data:

• volatility is stochastic,
• volatility has large fluctuations,
• volatility has jumps,
• volatility often exhibits long range dependence,
• volatility clusters on a high level, i.e. it shows persistence over long time intervals.

We investigate discrete and continuous time financial models with stochastic volatility with respect to these stylized facts. Our focus is on a new continuous time GARCH model, driven by a Lévy process, which captures many of the above features and allows for modelling high density data.