Volatility Models: from GARCH to Multi-Horizon Cascades

We overview different methods of modeling volatility of stock prices and exchange rates, focusing on their ability to reproduce the empirical properties in the corresponding time series. The properties of price fluctuations vary across the time scales of observation. The adequacy of different models for describing price dynamics at several time horizons simultaneously is the 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. They belong either to GARCH or stochastic volatility model families and often borrow methodological tools from statistical physics. We compare their properties and comment on their pratical usefulness and perspectives.