Principal Component Analysis
Principal component analysis (PCA) may be performed on a time series of historical term structure data in an attempt to identify the dominant factors driving its evolution. PCA produces factors maximising successive contributions to overall variance. Hence these factors attempt to explain the diagonal of the covariance matrix. The resulting factors are surrogate volatility factors derived from an empirical analysis of term structure data.
Principal component analysis provides a direct indication of the number and general shape of factors driving the term structure movements. A historical estimate of the magnitude of the volatility functions is also obtained as part of the analysis. These driving factors are both econometrically and financially justifiable, but like all historical calibration methodologies, will not exactly recover market prices of traded derivative instruments.
Many analyses have used spot interest rates as a description of the term structure; here we consider a finite set of forward rates of predetermined tenor, that span the whole term structure. Within the HJM framework each instantaneous forward rate is a stochastic variable in its own right, displaying some degree of correlation with other forward rates. To model a realistic evolution of the term structure one needs to determine a set of forward rate variances (volatilities) as well as covariances (correlations) between the forward rates.