The analysis of volatility in financial markets has become a first rank issue in modern
financial theory and practice: Whether in risk management, portfolio hedging, or option
pricing, we need to have a precise notion of the market's expectation of volatility. Much
research has been done on the analysis of realized historic volatilities,
Roll (1977) and references therein. However, since it seems unsettling to draw
conclusions from past to expected market behavior, the focus shifted to implied
volatilities, Dumas, Fleming and Whaley (1998). To derive implied volatilities the Black and
Scholes (BS) formula is solved for the constant volatility parameter 
using
observed option prices. This is a more natural approach as the option value is decisively
determined by the market's assessment of current and future volatility. Hence implied
volatility may be used as an indicator for market expectations over the remaining
lifetime of the option.
It is well known that the volatilities implied by observed market
prices exhibit a pattern that is far different from the flat
constant one used in the BS formula. Instead of finding a
constant volatility across strikes, implied volatility appears to
be non flat, a stylized fact which has been called
''smile''effect. In this chapter we illustrate how implied
volatilites can be analyzed. We focus first on a static and
visual investigation of implied volatilities, then we concentrate
on a dynamic analysis with two variants of principal components
and interpret the results in the context of risk management.