5.6 Common Principal Components Model

The Common Principal Components model (CPC) in the multivariate setting can be motivated as the model for similarity of the covariance matrices in the $ k$-sample problem, Flury (1988). Having $ k$ random vectors, $ \mathbf{x}_{(1)},\mathbf{x}_{(2)},\ldots, \mathbf{x}_{(k)} \in \mathbb{R}^p$ the CPC-Model can be written as:

$\displaystyle \mathbf{\Psi}_j \stackrel{\mathrm{def}}{=}\textrm{Cov}(\mathbf{x}_{(j)})=\mathbf{\Gamma} \mathbf{\Lambda_j} \mathbf{\Gamma}^{\top},$

where $ \mathbf{\Gamma}$ is an orthogonal matrix and $ \mathbf{\Lambda}_j=\textrm{diag}(\lambda_{i1},\ldots,\lambda_{ip})$. This means that eigenvectors are the same across samples and just the eigenvalues - variances of the principal component scores (5.4) differ.

Using the normality assumption, the sample covariance matrices $ \mathbf{S}_j, j=1,\ldots, k$, are Wishart-distributed:

$\displaystyle \mathbf{S}_j \sim W_p(n_j,\mathbf{\Psi}_j/n_j),$

and the CPC model can be estimated using maximum likelihood estimation with likelihood-function:

$\displaystyle \textrm{L}(\mathbf{\Psi}_1,\mathbf{\Psi}_2,\ldots, \mathbf{\Psi}_...
...1}\mathbf{S}_j \right)\right\} (\mathrm{det}\mathbf{\mathbf{\Psi}_j})^{-n_j/2}.$

Here $ C$ is a factor that does not depend on the parameters and $ n_j$ is the number of observations in group $ j$. The maximization of this likelihood function is equivalent to:

$\displaystyle ~\prod\limits_{j=1}^k \left\{\frac{\textrm{det}\ \textrm{diag}(\m...
...p}S_j \Gamma})}{\textrm{det}(\mathbf{\Gamma^{\top} S_j \Gamma})}\right\}^{n_j},$ (5.13)

and the maximization of this criterion is performed by the so-called Flury-Gautschi(FG)-algorithm, Flury (1988).

As shown in Section 5.4, using the functional basis expansion, the FPCA and SPCA are basically implemented via the spectral decomposition of the ``weighted" covariance matrix of the coefficients. In view of the minimization property of the FG algorithm, the diagonalization procedure optimizing the criterion (5.13) can be employed. However, the obtained estimates may not be maximum likelihood estimates.

Figure 5.6: First weight functions, $ \alpha =10^{-7}$, solid blue line is the weight function of the 1M maturity group $ (\hat{\gamma}_1^{1M})$, finely dashed green line of the 2M maturity group $ (\hat{\gamma}_1^{2M})$, and dashed black line is the common eigenfunction $ (\widetilde{\gamma}_1^{c})$, estimated from both groups.

\includegraphics[width=1.12\defpicwidth]{FDAharmdi1lame-7.ps}

Figure 5.7: Second eigenfunctions, $ \alpha =10^{-7}$ , solid blue line is the weight function of the 1M maturity group $ (\hat{\gamma}_2^{1M})$, finely dashed green line of the 2M maturity group $ (\hat{\gamma}_2^{2M})$, and dashed black line is the common eigenfunction $ (\widetilde{\gamma}_2^{c})$, estimated from both groups.

\includegraphics[width=1.12\defpicwidth]{FDAharmdi2lame-7.ps}

Using this procedure for the IV-strings of 1M and 2M maturity we get ``common" smoothed eigenfunctions. The first three common eigenfunctions ( $ \widetilde{\gamma}_1^{c}, \widetilde{\gamma}_2^{c}$, $ \widetilde{\gamma}_3^{c}$) are displayed in Figures 5.6-5.8. The solid blue curve represents the estimated eigenfunction for the 1M maturity, the finely dashed green curve for the 2M maturity and the dashed black curve is the common eigenfunction estimated by the FG-algorithm.

Figure 5.8: Third eigenfunctions, $ \alpha =10^{-7}$, solid blue line is the weight function of the 1M maturity group $ (\hat{\gamma}_3^{1M})$, finely dashed green line of the 2M maturity group $ (\hat{\gamma}_3^{2M})$, and dashed black line is the common eigenfunction $ (\widetilde{\gamma}_3^{c})$, estimated from both groups.

\includegraphics[width=1.12\defpicwidth]{FDAharmdi3lame-7.ps}

Assuming that $ \hat{\sigma}_i(\kappa,\tau)$ are centered for $ \tau = 1M$ and $ 2M$ (we subtract the sample mean of corresponding group from the estimated functions), we may use the obtained weight functions in the factor model of the IV dynamics of the form:

$\displaystyle \widetilde{\sigma}_i(\kappa,\tau)=\sum\limits_{j=1}^R \widetilde{...
...) \langle \widetilde{\gamma}^c_j(\kappa), \hat{\sigma}_i(\kappa,\tau) \rangle ,$ (5.14)

where $ \tau \in\{1M,2M\}$ and $ R$ is the number of factors. Thus $ \widetilde{\sigma}_i$ is an alternative estimation of $ \sigma_i$. This factor model can be used for simulation applications like Monte Carlo VaR. Especially the usage of Common Principal Components $ \widetilde{\gamma}^c_j(\kappa)$ reduces the high-dimensional IV-surface problem to a small number of functional factors.

In addition, an econometric approach, successfully employed by Fengler, Härdle, and Mammen (2004) can be employed. It consists of fitting an appropriate model to the time series of the estimated principal component scores, $ \widetilde{f}^c_{ij}(\tau)=\langle \widetilde{\gamma}^c_j(\kappa),\ \hat{\sigma}_i(\kappa,\tau) \rangle$, as displayed in Figure 5.9. Note that $ \hat{\sigma}_i(\kappa,\tau)$ are centered again (sample means are zero). The fitted time series model can be used for forecasting future IV functions.

Figure: Estimated principal component scores, $ \widetilde{f}^c_{i1}(1M)$, $ \widetilde{f}^c_{i2}(1M)$, and $ \widetilde{f}^c_{i3}(1M)$ for 1M maturity - first row, and $ \widetilde{f}^c_{i1}(2M)$, $ \widetilde{f}^c_{i2}(2M)$, and $ \widetilde{f}^c_{i3}(2M)$ for 2M maturity - second row; $ \alpha =10^{-7}$.

\includegraphics[width=1.5\defpicwidth]{FDApc1mpc2m.ps}

There are still some open questions related to this topic. First of all, the practitioner would be interested in a good automated choice of the parameters of our method (dimension of the truncated functional basis $ L$ and smoothing parameter $ \alpha $). The application of the Fourier coefficients in this framework seems to be reasonable for the volatility smiles (U-shaped strings), however for the volatility smirks (typically monotonically decreasing strings) the performance is rather bad. In particular, the variance of our functional objects and the shape of our weight functions at the boundaries is affected. The application of regression splines in this setting seems to be promising, but it increases the number of smoothing parameters by the number and the choice of the knots - problems which are not generally easy to deal with. The next natural question, which is still open concerns the statistical properties of the technique and the testing procedure for the Functional Common PCA model. Finally, using the data for a longer time period one may also analyze the longer maturities like 3 months or 6 months.