The Dynamics of Smiles

This empirical study is motivated by the models of Dupire (1992) and Derman and Kani (1997). We investigate the number and shape of shocks that move implied volatility smiles and subsequently we look at the correlation of changes in volatility with changes in the underlying asset. We achieve this by applying Principal Components Analysis on the changes in implied volatilities over time, for fixed ranges of days to maturity and in two different metrics: the strike and the moneyness metric. In contrast to earlier papers in the interest rate literature, we decide on how many principal components explain the implied volatilities dynamics, not by using rules of thumb, but by using Velicer's non-parametric criterion. Subsequently we use a ''Procrustes'' type rotation in order to interpret the retained components. The retained rotated principal components are used for the calculation of the correlation coefficients. We find similar results in both metrics regarding the number and shape of shocks. Two principal components explain the dynamics of smiles. After the rotation the first one is interpreted as shift and the second one has a Z-shape. The correlations for the first principal component depend metric, while for the second are positive under both metrics