The valuation method with GARCH is applied to the German stock index and options data. For the stock index we use the daily closing values of the DAX from January 1, 1988 to March 31, 1992. The closing values are usually set at 13:30 (Frankfurt time). For the options data on this index we have taken the recorded values of the transaction prices from the German derivative exchange (DTB) from January to March 1992. In order to synchronize the observation time periods of the index and options we interpolate between the last option price before 13:30 and the first price after, as long as the difference is no more than two hours.
No evidence for autocorrelated DAX returns was found but the
squared and absolute returns are highly autocorrelated. We
estimate a GARCH(1,1)-M model
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The Quasi-Maximum-Likelihood-Estimator of the TGARCH model
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(15.26) |
After the model was fitted to the data from 1988 to 1991, the next
step in calculating the option price for the observed time period
from January to March 1992 is to use the simulation method
described above and then compare this to the market prices. Here
we will concentrate on call options. Since the DAX option,
which is traded on the DTB, is a European
option, the results for put options can be calculated as usual
from the put-call-parity. We consider nine call options with
maturity dates January 17, March 20, and June 19, 1992. In order
to distinguish the case of in, out and at the money, we have
chosen the strike prices 1550, 1600 and 1650 for the January
option 1600, 1650 and 1700 for the March and June options. We
simulate the price of the January option from January 3rd to the
16th (10 days), for the March option from January 3rd to the 19th
(57 days) and for the June option from January 3rd to the 31st of
March (64 days). The June option with a strike price of 1700 began
on January 16th so that there are no observations for the first 10
trading days. Due to low trading volume not all market prices are
available thus we reduced the number of observations, in Table
14.4, even further.
A remaining question is how to choose the starting value of the volatility process. We set the starting value equal to the running estimator of the volatility (GARCH or TGARCH), in which the volatility process is extrapolated and the parameters are held constant. Alternatively one can use the implied volatility, see Section 6.3.4.
To calculate the Black-Scholes price at time the implied
volatility at time
is used. To obtain a measure of the
quality of the estimate, we define the relative residuals as
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Overall the GARCH as well as the TGARCH options valuation model
performs substantially better than the Black-Scholes model. For
options in and at the money the improvement of the TGARCH forecast
compared to the GARCH model is small. When the option, however, is
out of the money there is a large reduction of the loss criterion.
In the simulation study out of the money options react the most
sensitive to stochastic volatility and the leverage effect. In the
situation with real data this is most obvious for the January-1650
option, where Black-Scholes performs poorly and TGARCH performs
better than GARCH. For the March and June options the difference
is not so obvious. This can be explained by the fact that the
index increased to a level of 1736 points on March 20th of 1717
points on March 30th, so that the option with a strike price of
1700 became in the money. This is also the explanation for the
fact that is the highest for the January-1650 option. There
were only 10 trading days, but the option was out of the money for
several days. For example, the DAX closed on January 8th at 1578
points.
Since in every case TGARCH performs better than GARCH, we conclude that the market follows the asymmetry of the volatility. Therefore, specifying the volatility model correctly plays an important role in determining option prices.