  Principal component analysis (PCA) is a useful tool when trying to construct
factor models from historical asset returns. For the implied volatilities of
U.S. equities there is a PCA-based model with a principal eigenportfolio whose
return time series lies close to that of an overarching market factor. The
authors show that this market factor is the index resulting from the daily
compounding of a weighted average of implied-volatility returns, with weights
based on the options' open interest (OI) and Vega. The authors also analyze the
singular vectors derived from the tensor structure of the implied volatilities
of S&P500 constituents, and find evidence indicating that some type of OI and
Vega-weighted index should be one of at least two significant factors in this
market.
