Volatility Models: from GARCH to Multi-Horizon Cascades

We overview different methods of modeling stock prices and exchange rates volatility, focusing on their ability to reproduce the empirical properties in the corresponding time series. The properties of volatility change across the time scales of observations. Adequacy of volatility models for describe price dynamics at several time horizons simultaneously Special attention is a central topic of this study. We propose a detailed survey of recent volatility models, accounting for multiple horizons. These models are based on different and sometimes competing theoretical concepts, belonging either to GARCH or stochastic family of models and often borrowing methodological tools from statistical physics. We compare their properties and comment on their practical usefulness and perspectives.