DAX options belong to the most frequently
traded derivatives of the German/Swiss derivative market
``EUREX''. On every trading day one can find a significant number
of liquid time series with varying strike prices and maturities
(,
) on the market, which in principle can be used to
calculate implied volatilities. In view of the often limited data
processing capacities, an updated calculation of numerous
volatilities and partial derivatives of an extensive option
portfolio is still not feasible. Even with the appropriate
available information the isolated consideration of each implied
volatility as a separate source of risk is problematic, since it
results in an unstructured or ``uneven'' volatility surface. If
one were to use generated volatilities in calibrating option
prices, respectively risk models, this can lead to serious
specification errors and significantly deteriorate the results of
the corresponding trading and hedging strategies. As a result of
principal component analysis a ``smooth'' volatility surface, in
contrast to the one outlined above, can be generated with a
manageable amount of information. This allows for a better
calibration of the model and a more precise estimate of portfolio
sensitivities.
For our study of the dynamics of implied volatility we use the
volatility index (VDAX) made
available by the German Stock Exchange (Deutsche Börse AG)
respectively the closing prices of the corresponding VDAX
sub-indices. These indices reflect the implied volatility of the
DAX options ``at the money'' for times to maturity from one to
months. The corresponding values are determined by applying
the Black-Scholes formula (6.23) using prices observed on
the market:
Given the observed implied volatilities from varying times to
maturity at a specific time point and from a strike price
, the expectations of the market participants with respect to
the future actual volatility of the underlying financial
instrument can be estimated. In doing so one must remember that
the implied volatility of the BS model does not directly apply to
the actual variance of the price's process. Although the implied
BS volatility reflects a market expectation, the theoretical
relationship between it and the actual volatility can only be
determined using specific
assumptions, see Schönbucher (1998), Härdle and Hafner (2000).
Implied volatility for ATM-DAX options are calculated for various
lengths of maturity by the German Stock Exchange AG. A detailed
description of how the VDAX and its sub-indices are calculated can
be found in Redelberger (1994). Since March 18,
maturities of 1, 2, 3, 6, 9, 12, 18 and 24 months have been
considered in the calculation. On this date the trading of so
called ``Long Term Options'', i.e., trading of options with
maturities of over
months, were added to the EUREX. Using
closing prices the German Stock Exchange AG calculates for every
trading day a total of eight VDAX sub-indices for the maturities mentioned above. These sub-indices
reflect the volatility of the respective DAX option ``at the
money''. The time to maturity structure for DAX options that are
``at the money'' can be determined for every trading day using the
VDAX indices. Figure 19.2 illustrates a typical
development of the structure, which shows strong changes in the
positioning and form of the structure over time.
![]() |
The analysis done here is not only restricted to specific maturities of liquid options, which are represented by the first four VDAX sub-indices. On the contrary, we include all eight sub-indices in the analysis for the following reasons:
First of all a brisk trade of even the ``most distant'' option
contracts (i.e., the contracts with a remaining time of more than
one year) take place on numerous trading days, so that excluding
the pertaining sub-indices from the analysis would result in a
loss of information. VDAX sub-indices for long maturities have
been calculated by the German Stock Exchange since March 18,
. After the date of December 19,
the quality of the
data available to us declined considerably. In addition to the
daily often unchanged prices, the entries corresponding to the
removed sub-indices were usually missing. Given this we have
restricted our analysis to the time period from March 18,
to December 19,
.
Including relatively non-liquid DAX options with long maturities appears to make sense for another reason: For our analysis we require constant option maturities, since the daily shortening of the time to maturity can lead to enormous biases in the analysis results with data that has not been corrected. This especially holds for options with very short time to maturity. Thus we find it utterly necessary to use interpolated volatilities with corresponding constant time to maturities of the underlying option. Referring back to the calculation of the VDAX used by the German Stock Exchange AG we use the following linear interpolation:
For a fixed time to maturity of
calendar days we calculate daily
volatility indices
using the VDAX sub-indices with the next shorter respectively
longer maturity
and
with
![]() |
(20.1) |
This way, we obtain volatility time series each with constant
maturities. Every time series represents a weighted average of two
consecutive VDAX sub-indices and is based on
daily
observations of the implied DAX volatilities ``at the money``.