Market Models Table 6.3a: Eigenvalues of correlation matrix

Component Eigenvalue Cumulative R2

component explains 71.29% of the total variation. The weights on the first principal component in Table 6.3b show that it captures a shift in the same direction, of approximately the same magnitude, for both curves.

The second eigenvalue explains a further 24.7% of the variation in the system. The factor weights in Table 6.3b show that it represents an upwards shift and tilt up at the short end in the UK curve contemporaneous with a downwards shift and tilt down at the short end in the US curve.

The last two principal components capture relatively minor movements due to simultaneous tilts in both curves. Together they explain only an additional 3% of the variation. The third principal component represents a similar tilt in both curves, up at the short maturities and down at the long maturities. The fourth and least significant component represents a tilt up in the short and

down in the long of the UK, but down at the short and up in the long in the US.

In the US-UK example the first two principal components already explain nearly 96% of the variation in the system between January 1992 and March 1995. Thus a very useful reduction in dimensionality, from 21 to 2, has been achieved with a high degree of accuracy.

The results shown here are not specific to these countries, or to a system with just two yield curves. In fact it is often necessary to analyse portfolios based on a number of different yield curves, within and/or across countries. In such cases PCA enables dimensions to be very substantially reduced, but to what extent depends very much on the empirical nature of the system. As a very rough guide, note that 2"~ principal components will capture all possible combinations of up/down shifts of n yield curves.7 Thus if variation is limited to this type of movement, one needs only 4 components for three yield curves, 8 components for four yield curves, and so on. Empirically, of course, some of the more important components are likely to include tilts as well as shifts and the shifts will be of differing magnitudes.

It is often necessary to analyse portfolios based on a number of different yield curves, within and/ or across countries. In such cases PCA enables dimensions to be very substantially reduced

6.2.3 Term Structures of Futures Prices

The NYMEX sweet crude oil futures prices from 1 to 12 months for the period from 4 February 1993 to 24 March 1999 are shown in Figure 6.4.8 This was a relatively quiet period in oil prices. Basis risk for these very liquid contracts is minimal, as all the futures prices are tightly pegged to the spot price. Therefore, it should be expected that a PCA applied to this system would reveal only one major source of information.

As usual, the PCA routine first calculates the return correlation matrix and determines its eigenvalues and eigenvectors. The eigenvalue and eigenvalue analysis shown in Tables 6.4a and 6.4b indicate that indeed there is only one major source of variation: 96% of the variation in these prices over the period could be attributed to roughly parallel shifts in all prices. The tilt component adds only another 3% to the explanation of variation, and there is little need to use more than these two components. Of all the systems examined in this section, the term structure of crude oil futures prices has the strongest correlation. This high correlation is reflected in the factor weights on the first principal component, which are not only very similar but also close to 1.

This high correlation is reflected in the factor weights on the first principal component, which are not only very similar but also close to 1

7 There are 2" possible combinations of up-down shift in n yield curves, but the principal component that captures a shift up in all curves is the same as the one that captures a shift down in all curves, and so on. Assuming that the up-down shifts are of similar magnitude, only half of these movements (2"/2 = 2"~{) are needed.

8 Many thanks to Enron for providing these data.

10l-1-.-.--1-r-1

Feb-93 Feb-94 Feb-95 Feb-96 Feb-97 Feb-98 Feb-99

- ml- m2 m3 m6 -m9 -m12-y2 Figure 6.4 NYMEX sweet crude prices.

Table 6.4a: Eigenvalues of correlation matrix

6.3 Modelling Volatility Smiles and Skews

Principal component analysis greatly facilitates the construction of appropriate scenarios for movements in the implied volatility smile surface corresponding to movements in the underlying price. Various attempts to model volatility