Principal Components Analysis in implied volatility modeling (Analiza skladowych glownych w modelowaniu implikowanej zmiennosci)

We analyze the implied volatility surface structure of ODAX options as traded on DTB (currently Eurex). We apply PCA to cross sections of the implied volatility surface taken along the same moneyness (m) or the same time to maturity (T). For data from the period October 3rd 1997 – November 30th 1998 a substantial reduction of the dimensionality of the problem was achieved. It turned out that (i) for the m-sections the first two principal components contained 99.997% of information, and (ii) for the T-sections the first three principal components contained 99.892% of information. Moreover, the obtained principal components were very similar for different m's or T's allowing us to assume that that the space spanned by the eigenvectors is identical across several groups (of m's or T's), i.e. leading us to the so called Common PCA.