As we can see in Figure 5.4, the resulting eigenfunctions are often very rough.
Smoothing them could result in a more natural interpretation of the obtained weight functions. Here we apply a popular approach
known as roughness penalty. The downside of this technique is that we loose orthogonality in the sense.
Assume that the underlying eigenfunctions of the covariance operator have a continuous and square-integrable
second derivative.
Let
be the first derivative operator and define the roughness penalty by
Moreover, suppose that
has square-integrable derivatives up to degree four and that the second and the third
derivatives satisfy one of the following conditions:
Given a eigenfunction with norm
, we can penalize the sample variance of the principal component
by dividing it
by
:
Define
to be a matrix whose elements are
.
Then the generalized eigenequation (5.11) can be transformed to:
Applying Smoothed Functional PCA (SPCA) to the IV-strings, we get
the smooth-eigenfunctions plotted in Figure 5.5. We use
, the aim is to use a rather small degree of smoothing, in order to replace the high frequency fluctuations only.
Some popular methods, like cross-validation, could be employed as well, Ramsay and Silverman (1997).
The interpretation of the weight functions displayed in Figure 5.5 is as follows:
The first weight function (solid blue) represents clearly the level of the volatility - weights are
almost constant and positive. The second weight function
(finely dashed green) changes sign near the at-the-money point, i.e. can be interpreted as the
in-the-money/out-of-the-money identification factor or slope.
The third (dashed cyan) weight function may play the part of
the measure for a deep in-the-money or out-of-the-money factor or curvature.
It can be seen that the weight functions for the 1M (
)
and 2M maturities (
)
have a similar structure. From a practical point of view it can be interesting to try to get
common estimated eigenfunctions (factors in the further analysis) for both groups.
In the next section, we introduce the estimation motivated by the Common Principal Component Model.