Tensor PCA for Implied Volatility Surfaces
Principal component analysis (PCA) is a useful tool when trying to uncover factor models from historical asset returns. For the implied volatilities of U.S. equities, there is a PCA-based model with a so-called principal eigenportfolio whose returns time series lies close to that of an overarching market factor. Specifically, this market factor is a new volatility index that we have constructed to be a weighted average of implied-volatility returns with weights based on the options' vega and open interest (OI). This OI-weighted index is one among several possible new indices that can be constructed by collecting implied volatilities from options on many individual equities. We analyze the singular values from the tensor structure of implied volatilities from the S&P500 constituents, and find evidence indicating there to be at least two significant factors in this market, with increased significance in a second factor in the months leading up to a volatility event.