To examine the empirical performance of the GARCH option pricing model, we applied the model to daily closing prices of the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) and its corresponding TAIEX options. We use the index and its corresponding options based on the following consideration. The first reason is that the index and the option data are freely available. The data were obtained from the official website of Taiwan Stock Exchange Corporation (TSEC):
http://www.tse.com.tw
and the official website of Taiwan Futures Exchange (TAIFEX):
http://www.taifex.com.tw .
Second, the TAIEX index option is the most actively traded European-style option in Taiwan. Thus, the TAIEX option market is chosen to test the empirical performance of the Black-Scholes model and the GARCH option pricing model.
We will use the TAIEX index with the sample period from January 4, 2000 to December 10, 2003 to establish the GARCH volatility dynamic. There are 617 observations. Figure 2 describes the evolution of the daily TAIEX index levels, which fluctuated dramatically during this period. One can easily see that the daily TAIEX index levels in Figure 2 demonstrate a decreasing trend. The mean index value is about 5,800, while the maximum index value is more than 10,000, and the lowest value is less than 3,500. Starting year 2000, Taiwanese economy has been affected by many political and non-political issues. The first shift of Taiwanese political party, the suspension and re-establishment of nuclear power station, and the unhealthy financial system had all affected the stock market. Figure 3 describes the daily return for the sample period. It shows that the mean of the return series appears to be constant whereas the variance clearly changes over time. Volatility clustering is also observed in the
plot, a large value tends to follow by another large value. This is known as the conditional heteroscedasticity.
The TAIEX option is a European style option, and it expires at the opening of the Thursday following the third Wednesday of each contract month. The expiration month is the next two calendar months followed by two additional months from the March quarterly cycle (March, June, September, and December). Their strike price intervals are 100 points. On each trading day, we report the first call contract happened in the time interval of 1:00-1:25 which is near the closing time for each strike price and the time-to-maturity. The time-to-maturity is measured as the number of calendar days from the trading date to the Wednesday immediately preceding the Thursday when the option expires date, because TAIEX index options expire at the opening of trading. The reported index level is the closing price of the TAIEX index. After establishing the GARCH dynamic, we will compute the index option prices from September 1, 2003 to December 10, 2003.
The following criteria are employed to filter the option data:
First, general arbitrage violations must be eliminated from the data; otherwise there might be a negative implied volatility. A transaction has to satisfy the following no-arbitrage relationships:
).
, 0
max( r(T t)
t S K e
C ≥ − ⋅ − −
Second, very short-term options and very long-term options are excluded. Options with less than 7 days to expiration are excluded because they are very sensitive to liquidity-related biases and their prices are generally very volatile. Options with time to maturity longer than 40 days are also excluded because they are not actively traded and thus excluded from the sample.
Third, very deep out-of-the-money and very deep in-the-money options are excluded.
This criterion is based on the same considerations as discussed in the above paragraph.
Options with deep out-of-the-money and very deep in-the-money options may contain little information about the volatility process. Moreover, these options are not traded actively. An option is defined as very deep in- or out-of-the-money if its moneyness is greater than 1.2 or lower than 0.8. The option moneyness is defined as the ratio between the TAIEX level and the strike price:
K. S Moneyness=
Our exclusionary criteria yield a final daily sample of 584 observations for 70 days. On average we have about 9 option prices available on each day. Since option prices are not very sensitive to the interest rate and the change of interest rate is small on daily basis, we shall just assume the risk free rate as 2% per year.
It is a common practice in the literature to divide options into different moneyness categories to study their price behavior because option prices are very sensitive to their exercise prices. We divide the option data into 5 categories according to the moneyness.
We define a call option is said to be at-the-money if the moneyness is between (0.98, 1.02), in-the-money if the moneyness is between (1.02, 1.05), out-of-the-money if the moneyness is between (0.95, 0.98) and deep in-the-money if the moneyness is greater than 1.05 and deep out-of-the-money if the moneyness is less than 0.95.
Table 1 provides the average and standard deviation of call option prices reported for each moneyness category, and also shows the numbers of observations in these categories for the period from September 1, 2003 to December 10, 2003. About 25% of the samples are at-the-money, 19% of the samples are out-of-the-money and 19% of the samples are in-the-money. Only 8% of the samples are deep out-of-the-money. The most unusual thing is the high proportion of the deep in-the-money (26%). The demand for in-the-money options is
higher than the demand for out-of-the-money options, which indicates that investors are cynical about future market increases, since in-the-money call options will be valuable only if the market decreases significantly in the future. The overall average call price in our sample period is NT$210.51 with a standard deviation of NT$197.61.