20.3 Stability Analysis of the VDAX's Dynamics

In order to sensibly apply the principal factors in measuring the risk of portfolios we have to study their stability over time. When the principal components and the factor loadings change significantly over time, a global analysis would not be suitable to illustrate the future variance of the implied volatility nor to judge the risks of the option portfolios with sufficient accuracy.

Our procedure considers two aspects: first whether the random portion in daily data is possibly significantly higher than in weekly data. A possible cause for this is the non-synchronous trading caused by a frequent realizations of the quotes in the liquid contracts with short time to maturity and sparsely available prices in the long running contracts. In order to distinguish the possible influences of this effect, we run our analysis analogously based on weekly data. By sufficient stability in the principal components the use of daily respectively weekly data should lead to similar results.

For the second aspect we divide our data into two non-overlapping periods of equal length. Each sub-period contains $ m$ = 220 daily observations of the process of the differences. We run for each sub-period a principal component analysis as described above and compare the respective size of the eigenvalues $ \hat{\lambda}_k^i,
k= 1,2,$ in both sub-periods $ i = 1,2$.


Table 19.3: Explained portion of the variance (in percentage) in different sub-periods
Principal
Components: 1 2 3 4 5 6 7 8
Weekly data
18.03.96-19.12.97 73.85 91.59 95.09 97.38 98.80 99.61 99.88 100
Daily data
18.03.96-19.12.97 70.05 83.12 88.69 91.80 94.86 96.97 98.90 100
Sub-period 1
18.03.96-05.02.97 83.36 91.84 94.65 96.39 97.76 98.78 99.52 100
Sub-period 2
05.02.97-19.12.97 68.22 82.21 87.99 91.35 94.64 96.93 98.86 100


As already mentioned the effect of non-synchronous trading that appears in daily data can be eliminated by using weekly data. From Table 19.3 it emerges that the explanatory power of the first principal component is slightly higher in weekly data. This is not surprising given the expected size of the error terms proportion in daily data. Overall the explanatory proportions of the variance have similar values when using weekly data. This supports the stability of the analysis method used here w.r.t. the bias due to non-synchronous trading in daily data.

From Table 19.3 it emerges that the proportion of the variance explained by the first two principal components declines in the second sub-period. Based on this a stability test is necessary: A two sided confidence interval for the difference of the eigenvalues from both sub-periods is

$\displaystyle \ln \hat{\lambda}_k^1-2q_{\alpha}\sqrt{\frac{2}{m-1}} \leq \ln \hat{\lambda}_k^2 \leq \ln \hat{\lambda}_k^1+2q_{\alpha}
\sqrt{\frac{2}{m-1}},$     (20.4)

where $ q_{\alpha}$ represents the $ \alpha$ quantile of a standard normal distribution, see Härdle and Simar (2003). From this it follows that
$\displaystyle \mid \ln \hat{\lambda}_k^1 - \ln \hat{\lambda}_k^2 \mid \geq 2q_{\alpha}
\sqrt{\frac{2}{m-1}}$     (20.5)

is a second test for $ H_0: \lambda_k^1 = \lambda_k^2$. Under the null hypothesis the respective eigenvalues are the same in both periods. The null hypothesis is rejected when the inequality is fulfilled for a corresponding critical value $ q$. This would indicate an instability of the principal components over time.

Critical values for rejecting the null hypothesis are 0.313 (probability of error 10%), 0.373 (probability of error 5%) and 0.490 (probability of error 1%). The differences of the estimated eigenvalues are 0.667 and 1.183. Both differences are significantly larger than zero with an error probability of 1%. These results prove that the determining factors of the volatility dynamics changes over time. By the determination of the risk of option portfolios it therefore appears necessary to use an adaptive method of the principal components. Here the estimation is periodically done over a moving time window and the length of the time window is adaptively set, see Härdle et al. (2000).