Options on the DAX are the most actively traded contracts at the derivatives exchange
EUREX. Contracts of various strikes and maturities constitute a liquid market at any
specific time. This liquidity yields a rich basket of implied volatilities for many pairs
. One subject of our research concerning the dynamics of term structure
movements is implied volatility as measured by the German VDAX subindices available from
Deutsche Börse AG
(
http://deutsche-boerse.com/
)
These indices, representing different option maturities, measure volatility implied in ATM European calls and puts. The VDAX calculations are based on the BS formula. For a detailed discussion on VDAX calculations we refer to Redelberger (1994). Term structures for ATM DAX options can be derived from VDAX subindices for any given trading day since 18 March 1996. On that day, EUREX started trading in long term options. Shapes of the term structure on subsequent trading days are shown in Figure 6.3.
If we compare the volatility structure of 27 October 1997 (blue line) with that of 28 October 1997 (green line), we easily recognize an overnight upward shift in the levels of implied volatilities. Moreover, it displays an inversion as short term volatilities are higher than long term ones. Only a couple of weeks later, on 17 November (cyan line) and 20 November (red line), the term structure had normalized at lower levels and showed its typical shape again. Evidently, during the market tumble in fall 1997, the ATM term structure shifted and changed its shape considerably over time.
As an option approaches its expiry date , time to maturity
is declining with each trading day. Hence, in order to
analyze the dynamic structure of implied volatility surfaces, we
need to calibrate
as time
passes. To accomplish this
calibration we linearly interpolate between neighboring VDAX
subindices. For example, to recover the implied volatility
at a fixed
, we use the subindices at
and
where
, i.e. we
compute
with fixed maturities of
calendar days by
![]() |
(6.5) |
Proceeding this way we obtain time series of fixed maturity. Each time series is a
weighted average of two neighboring maturities and contains
data points of
implied volatilities.
The data set for the analysis of variations of implied volatilities is a collection of term structures as given in Figure 6.3. In order to identify common factors we use Principal Components Analysis (PCA). Changes in the term structure can be decomposed by PCA into a set of orthogonal factors.
Define
as the
matrix of centered first differences of ATM
implied volatilities for subindex
in time
, where in our case
and
. The sample covariance matrix
can be decomposed by
the spectral decomposition into
![]() |
(6.6) |
where is the
matrix of eigenvectors and
the
diagonal matrix of eigenvalues
of
. Time series of principal components
are obtained by
.
A measure of how well the PCs explain variation of the underlying data is given by the
relative proportion of the sum of the first
eigenvalues to the overall sum
of eigenvalues:
![]() |
(6.7) |
The quantlet
XFGiv04.xpl
uses the VDAX data to estimate the proportion of variance
explained by the first
PCs.
As the result shows the first PC captures around 70% of the total data variability. The second PC captures an additional 13%. The third PC explains a considerably smaller amount of total variation. Thus, the two dominant PCs together explain around 83% of the total variance in implied ATM volatilities for DAX options. Taking only the first two factors, i.e. those capturing around 83% in the data, the time series of implied ATM volatilities can therefore be represented by a factor model of reduced dimension:
![]() |
(6.8) |
where
denotes the
th element of
,
is taken from the matrix of principal components
, and
denotes white
noise. The
are in fact the sensitivities of the implied volatility time
series to shocks on the principal components. As is evident from
Figure 6.4, a shock on the first factor tends to affect all maturities in a
similar manner, causing a non-parallel shift of the term structure. A shock in the second
factor has a strong negative impact on the front maturity but a positive impact on the
longer ones, thus causing a change of curvature in the term structure of implied
volatilities.
Implied volatilities calculated for different strikes and maturities constitute a surface. The principle component analysis as outlined above, does not take this structure into account, since only one slice of the surface, the term structure of ATM options are used. In this section we present a technique that allows us to analyze several slices of the surface simultaneously. Since options naturally fall into maturity groups, one could analyze several slices of the surface taken at different maturities. What we propose to do is a principal component analysis of these different groups. Enlarging the basis of analysis will lead to a better understanding of the dynamics of the surface. Moreover, from a statistical point of view, estimating PCs simultaneously in different groups will result in a joint dimension reducing transformation. This multi-group PCA, the so called common principle components analysis (CPCA), yields the joint eigenstructure across groups.
In addition to traditional PCA, the basic assumption of CPCA is that the space spanned by
the eigenvectors is identical across several groups, whereas variances
associated with the components are allowed to vary. This approach permits us to analyze a
variate random vector in
groups, say
maturities of implied volatilities
jointly, Fengler, Härdle and Villa (2001).
More formally, the hypothesis of common principle components can be stated in the following way, Flury (1988):
where the are positive definite
population covariance matrices,
is an orthogonal
transformation matrix and
diag
is the matrix of
eigenvalues. Moreover, assume that all
are distinct.
Let be the (unbiased) sample covariance matrix of an underlying
-variate normal
distribution
with sample size
. Then the distribution of
is
Wishart, Muirhead (1982), p. 86, with
degrees of freedom:
![]() |
The density of the Wishart distribution is given by
![]() |
![]() |
![]() |
|
![]() ![]() |
(6.9) |
![]() ![]() ![]() |
(6.10) |
where is a constant not depending on the parameters
. Maximizing the
likelihood is equivalent to minimizing the function
![]() ![]() |
Assuming that holds, i.e. in replacing
by
, one gets after some manipulations
![]() |
As we know from section 6.3.2, the vectors in
need to be
orthogonal. We achieve orthogonality of the vectors
via the Lagrange method,
i.e. we impose the
constraints
using the Lagrange
multiplyers
and the remaining
constraints
for
using the multiplyer
. This yields
![]() |
Taking partial derivatives with respect to all
and
, it can
be shown (Flury; 1988) that the solution of the CPC model is given by the generalized
system of characteristic equations
This has to be solved using
![]() |
under the constraints
![]() |
Flury (1988) proves existence and uniqueness of the maximum of the likelihood
function, and Flury and Gautschi (1988) provide a numerical algorithm, which has been
implemented in the quantlet
CPC
.
CPC-Analysis
A number of quantlets are designed for an analysis of covariance matrices, amongst them
the
CPC
quantlet:
As input variables we need a
array A, produced from
covariance matrices, and a
vector of weights N.
Weights are the number of observations in each of the
groups.
The quantlet produces the common transformation matrix B, and the
matrix of asymptotic standard errors betaerror. Next,
eigenvalues lambda and corresponding standard errors lamdbaerror
are given in a vector array of
. Estimated population covariances
psi are also provided. As an example we provide the data sets
volsurf01
,
volsurf02
and
volsurf03
that have been used in
Fengler, Härdle and Villa (2001) to estimate common principle components for the implied
volatility surfaces of the DAX 1999. The data has been generated by smoothing a surface
day by day as spelled out in section 6.2.2 on a specified grid. Next, the
estimated grid points have been grouped into maturities of
,
and
months and transformed into a vector of time series of the "smile", i.e. each
element of the vector belongs to a distinct moneyness ranging from
to
.
We plot the first three eigenvectors in a parallel coordinate plot in Figure 6.5. The basic structure of the first three eigenvectors is not altered. We find a shift, a slope and a twist structure. This structure is common to all maturity groups, i.e. when exploiting PCA as a dimension reducing tool, the same transformation applies to each group! However, from comparing the size of eigenvalues among groups, i.e. ZZ.lambda, we find that variability is dropping across groups as we move from the front contracts to long term contracts.
Before drawing conclusions we should convince ourselves that the CPC model is truly a good description of the data. This can be done by using a likelihood ratio test. The likelihood ratio statistic for comparing a restricted (the CPC) model against the unrestricted model (the model where all covariances are treated separately) is given by
The calculations yield
, which corresponds to the
-value
for the
distribution. Hence we cannot reject the CPC
model against the unrelated model, where PCA is applied to each maturity separately.
Using the methods in section 6.3.2, we can estimate the amount of variability
explained by the first
principle components: again a few number of factors, up to
three at the most, is capable of capturing a large amount of total variability present in
the data. Since the model now captures variability both in strike and maturity
dimension, this can be a suitable starting point for a simplified VaR calculation for
delta-gamma neutral option portfolios using Monte Carlo methods, and is hence a valuable
insight for risk management.