The presentation of this chapter closely follows the work of Fengler et al. (2002). The principal components analysis is applied to the changes of implied volatilities for fixed ranges of days to maturity by Skiadopoulos et al. (1998) who find two principal components can already sufficiently explain the dynamics of smiles. The conditions to ensure the absence of arbitrage in the volatility models are derived by Schönbucher (1998). Furthermore, Härdle and Hafner (2000) show that the prices of out-of-the-money options strongly depend on volatility features such as asymmetry.
Härdle et al. (2000) develop an adaptive method of estimation which does not use any information about the time homogeneity of the observed process. It can be used to estimate the principal components. For the effect of the implied volatilities changes on the dynamic hedging of exotic and complex options we refer to Taleb (1997).