**CHAPTER 8**

**Volatility**

**T**his chapter starts with the basic definitions of volatility and practical methods of its forecasting. I go on to provide an overview of the heteroskedastic volatility models, which allow for modeling well-documented volatility clustering. Then, I turn to the realized volatility that has been drawing strong attention in current econometric research. Finally, I outline the application of volatility measures in market risk management.

**BASIC NOTIONS**

Volatility is a generic notion for measuring price variability. This concept is very important in risk measurement and is widely used in defining trading strategies. Standard deviation of returns is usually used for quantifying volatility. For a data sample with *N* returns *r*_{i} at *i* = 1,2, …, *N* and the average value , or *realized volatility* (called also *historical volatility*), is defined as

Usually, returns are calculated on a homogeneous time grid with spacing Δ*t*. For financial reporting, volatility is often calculated as the annualized percentage, that is, σ(*T/Δt*)^{1/2}100 percent, where *T* is the annual period in units of Δ*t*. For example, in the case of daily returns, Δ*t* = 1, and *T* = 252 (which is the yearly number of working days). More complicated expressions for calculating volatility on inhomogeneous time grids are offered ...