Mortgages, loans, financial matters

The HJM framework has the advantage that it allows seamless extension from one to several sources of uncertainty. The choice of number of volatility factors to use is driven by a number of often conflicting considerations. Additional sources of uncertainty introduce additional degrees of freedom to the evolution of the term structure which allows decorrelation (decreasing correlation between forward rates with larger differences in their maturities) of rates to be properly accounted for. However, additional factors increase the numerical complexity, slowing down computation time.

Even when calibrating via PCA, where this choice is driven by the number of factors explaining a sufficiently large percentage of total volatility, there are several points to consider:

PCA tells us how many factors are significant in explaining the movement of the term structure within the historical data set. We wish to use these factors as predictors of future movements of the term structure. While adding additional factors will improve our ability to explain the historical term structure movements, this may not be the case for explaining future movements.

The historical factors and their specific loadings (magnitude for various maturities along the term structure) may not be ideal predictors of future volatility factors. Additional factors may not be stable through time adding noise, rather than improving the structural relationships within the volatility predictions.

PCA attempts to maximise the factors’ contribution to the diagonal elements of the covariance matrix, that is the variances. The resulting factors may explain the off-diagonal elements (covariances) with significant error. This may affect valuation of correlation dependent options such as swaptions and spread options.

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.