The analysis of implied volatilities

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 a 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 volatilities 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.