15.3 Application to the Valuation of DAX Calls

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

$\displaystyle Y_t$ $\displaystyle =$ $\displaystyle \lambda \sigma_t + \varepsilon_t$ (15.23)
$\displaystyle {\cal L} ( \varepsilon_t \mid {\cal F}_{t-1})$ $\displaystyle =$ N$\displaystyle (0, \sigma_t^2)$ (15.24)
$\displaystyle \sigma_t^2$ $\displaystyle =$ $\displaystyle \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2$ (15.25)

for the DAX with the Quasi-Maximum-Likelihood-Method - see Section 12.1.6. A possible constant in (14.23) is not significant and is thus ignored from the very beginning. Table 14.3 shows the results of the estimation. All parameters are significantly different from zero. The degree of persistence $ \alpha+\beta = 0.9194$ is significantly smaller than 1 and thus the unconditional variance is finite, see Theorem 12.10. The parameter of the risk premium $ \lambda$ is positive, as is expected from economic theory.


Table 14.3: The GARCH and TGARCH estimation results for DAX returns, January 1, 1988 to December 30, 1991 (QMLE standard error in parentheses)
  GARCH TGARCH
$ \omega$ 1.66E-05 (1.04E-06) 1.91E-05 (1.359E-06)
$ \alpha$ 0.144 (0.006)  
$ \alpha_1$   0.201 (0.008)
$ \alpha_2$   0.045 (0.011)
$ \beta$ 0.776 (0.012) 0.774 (0.016)
$ \lambda$ 0.069 (0.018) 0.039 (0.018)
$ -2 log L$ -7698 -7719


The Quasi-Maximum-Likelihood-Estimator of the TGARCH model

$\displaystyle \sigma_t^2 = \omega + \alpha_1 \varepsilon_{t-1}^2 \boldsymbol{1}...
...repsilon_{t-1}^2 \boldsymbol{1}(\varepsilon_{t-1} \ge 0) + \beta \sigma_{t-1}^2$ (15.26)

is also given in Table 14.3. Taking the value of the log-likelihood into consideration, the ability of the TGARCH model is better than that of GARCH model. A likelihood-quotient test rejects the GARCH model at every rational confidence level. $ \alpha_1$ and $ \alpha_2$ are significantly different; thus the asymmetry of the news impact function is significant. Since $ \alpha_1 > \alpha_2$ we observe the usual leverage effect for financial time series.

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, $ k$ 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 $ t$ the implied volatility at time $ t-1$ is used. To obtain a measure of the quality of the estimate, we define the relative residuals as

$\displaystyle u_{t} \stackrel{\mathrm{def}}{=}\frac{C_{t} - C_{Market,t}}{C_{Market,t}}$

where $ C_t$ is either the Black-Scholes or the GARCH or the TGARCH price and $ C_{Market,t}$ is the price observed on the market. Residuals should be considered as relative values, since a trader would always prefer the cheaper option, which is undervalued by the same amount as a more expensive option, simply because he can multiply his position in the cheaper option. A similar argument holds for the sale of an overvalued option. For reasons of symmetry we use a squared loss criterion, i.e.,

$\displaystyle U = \sum_{t=1}^k u_{t}^2.$

The results for the three models are given in Table 14.4.


Table 14.4: The loss criterium $ U$ for the DAX calls with maturity at $ T$ and a strike price $ K$ using BS, GARCH and TGARCH option prices. The number of observations is given by $ k$.
$ T$ $ K$ $ k$ BS GARCH TGARCH
  1550 10 0.017 0.014 0.014
Jan 1600 10 0.099 0.029 0.028
  1650 10 4.231 1.626 1.314
  1600 47 1.112 0.961 0.954
Mar 1650 53 1.347 1.283 1.173
  1700 56 1.827 1.701 1.649
  1600 53 1.385 1.381 1.373
Jun 1650 56 2.023 1.678 1.562
  1700 51 2.460 2.053 1.913
Sum   346 14.500 10.725 9.980


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 $ U$ 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.