20.1 Description of the Data

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 ($ K$, $ \tau$) 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 $ 24$ months. The corresponding values are determined by applying the Black-Scholes formula (6.23) using prices observed on the market:

$\displaystyle C(S,\tau)=e^{(b-r)\tau}S\Phi (y+\sigma\sqrt {\tau}) - e^{-r\tau}K\Phi (y),$

where $ \Phi $ is the distribution function of the standard normal distribution and

$\displaystyle y=\frac {\ln\frac SK+(b-\frac {1}{2}\sigma^2)\tau}{\sigma\sqrt {\tau}},
$

The only parameter from the BS formula that cannot be immediately observed on the market is the actual volatility $ \sigma$ of the price process. In principle the volatility of the process can be estimated from historical financial market data, see Section 6.3.4, however, it is commonly known that the assumption of the BS model, that the financial underlying has a geometric brownian motion, is in reality only approximately fulfilled. Alternatively the BS formula is also used in order to calculate the $ \sigma$ value as the implied volatility for a given market price of a specific option. This does not mean that the market participant should accept the assumption of the Black-Scholes method. On the contrary they use the BS formula as a convenient possibility to quote and price options with these parameters.

Given the observed implied volatilities from varying times to maturity $ \tau$ at a specific time point and from a strike price $ K$, 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, $ 1996$ 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 $ 12$ 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.

Fig.: Time to maturity structure of implied DAX volatilities ``at the money'' 34766 SFEVolaTermStructure.xpl
\includegraphics[width=1\defpicwidth]{abbild2.ps}

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, $ 1996$. After the date of December 19, $ 1997$ 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, $ 1996$ to December 19, $ 1997$.

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 $ \tau^*_1 = 30, \tau^*_2 = 60, \tau^*_3 = 90, \tau^*_4 = 180, \tau^*_5 = 270, \tau^*_6 = 360,
\tau^*_7 = 540, \tau^*_8 = 720$ calendar days we calculate daily volatility indices $ \hat{\sigma}_{I,t}(\tau^*_j), j=1, ..., 8,$ using the VDAX sub-indices with the next shorter respectively longer maturity $ \hat{\sigma}_{I,t}(\tau^-_j)$ and $ \hat{\sigma}_{I,t}(\tau^+_j)$ with


$\displaystyle \hat{\sigma}_{I,t}(\tau^*_j) = \hat{\sigma}_{I,t}(\tau^-_j)\left[...
...igma}_{I,t}(\tau^+_j)
\left[\frac{\tau^*_j-\tau^-_j}{\tau^+_j-\tau^-_j}\right].$     (20.1)

This way, we obtain $ 8$ volatility time series each with constant maturities. Every time series represents a weighted average of two consecutive VDAX sub-indices and is based on $ n = 441$ daily observations of the implied DAX volatilities ``at the money``.