Principal Component Analysis of Implies Volatility Smiles and Skews
Abstract: Several principal component models of volatility smiles and skews have been based on daily changes in implied volatilities, by strike and/or by moneyness. Derman and Kamal (1997) analyze S&P500 and Nikkei 225 index options where the daily change in the volatility surface is specified by delta and maturity. Skiadopoulos, Hodges and Clewlow (1998) apply PCA to first differences of implied volatilities for fixed maturity buckets, across both strike and moneyness metrics. And Fengler et. al. (2000) employ a common PCA that allows options on equities in the DAX of different maturities to be analyzed simultaneously. There is an important difference between the research just cited and the approach taken in this paper. Instead of applying PCA to daily changes in implied volatilities, a PCA is applied to daily changes in the deviations of fixed strike volatilities from at-the-money volatility. The advantages of this approach are both empirical and theoretical: daily variations in fixed strike deviations from ATM volatility are much less noisy than the daily changes in fixed strike or fixed delta volatilities; and the models of the skew in equity markets that were introduced by Derman (1999) can be expressed in a form where fixed strike volatility deviations from ATM volatility always have the same relationship with the underlying index. The model presented here extends Derman's models to allow non-parallel shifts in the skew as the index moves. It uses PCA to actually quantify the sensitivities of implied volatilities to changes in the underlying price. It has applications to all types of implied volatility surfaces, including currency option smiles and swaption skews. But the present paper focuses on its application to the skew in the FTSE 100 between 4th January 1998 and 31st March 1999. It is found that the sensitivity of a fixed strike volatility to movements in the index changes according to market conditions and that the range of the skew (the difference between low strike volatility and high strike volatility) will normally fluctuate over time. However in jumpy markets the range of the skew is quite static and shifts in fixed strike volatilities are more likely to be parallel, as predicted by Derman's models.