Throughout the thesis, the period of the data is from Jan. 2, 2003 to Aug. 29, 2003. It is the period of initial and terminal stages when epidemic SARS spread globally. During this period, many events related to SARS deserve some discussions.
In chapter 5, the LRT was applied to the TAIEX data and detected several change points. We deeply hope that ignoring the change points will lead to a wrong conclusion. As a matter of fact, it is important to detect change points. Overall, we believe our test constitutes a functional tool for testing for a parameter change in the GARCH(1,1) models. We anticipate that methods in the Western Electric Handbook that combined with the LRT can be extended to other types of time series models.
Finally, empirical results are consistent with our expectation. We should attach importance to structural change in time series models which will lead to better prediction. However, overvalued-volatility is a defect of GARCH models. Perhaps, realized volatility mixed with GARCH model can be a useful model. We leave the task of extension to other types of GARCH models for future research.
References
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Tables
Table 2.1 TAIEX option
Item Description
Underlying Index
Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Ticker Symbol TXO Exercise Style European
Multiplier NT$ 50 (per index point)
Expiration Months
Spot month, the next two calendar months followed by two additional months from the March quarterly Cycle (March, June, September, and December)
Strike Price Interval
100 index points in spot month, the next two calendar months 200 index points in the additional two months from the March quarterly Cycle
Strike (Exercise) Price
When listing series of new expiration months, one series with at-the-money strike price is listed based on the previous day's closing price of the underlying index rounded down to the nearest multiples of 100.
1. For the spot month and the next two calendar months:
Three other series each with in-the-money and out-of-the-money strike prices with price interval of 100 points are listed.
2. For the next two quarter-months: Two other series each with in-the-money and out-of-the-money strike prices with price interval of 200 points are listed.
Up to the 5th business days before expiration,
1. For the spot month, and the next two calendar months:
additional series are added when the underlying trades through the third highest or lowest strike prices available, to maintain at least 3 in- and 3 out-of-the-money strike prices
2. For the next two quarter-months: additional series are
added when the underlying trades through the second highest or lowest strike price available, to maintain at least 2 in-and 2 out-of-the-money strike prices
Premium Quotation
< 10 points: 0.1 point (NT$5)
>=10 points,<50 points: 0.5 point (NT$ 25)
>=50 points, <500 points: 1 point (NT$ 50)
>=500 points, <1,000 points: 5 point (NT$ 250)
>=1,000 points: 10 point (NT$ 500)
Daily Price Limit +/- 7% of previous day's closing price of the underlying index
Position Limit
Individuals: 8,000contracts on either side of the market.
Institutional Investors: 16,000 contracts on either side of the market.
Institutional investors may apply for an exemption from the above limit on trading accounts for hedging purpose.
Exemptions are allowed for Future Proprietary Firms.
Trading Hours
08:45AM - 1:45 PM Taiwan time, Monday through Friday of the regular Taiwan Stock Exchange business days
Last Trading Day The third Wednesday of the delivery month
Expiration Date The first business day following the last trading day
Final Settlement Price
The final settlement price for each contract is computed from the first fifteen-minute volume-weighted average of each component stock's prices in that index on the final settlement day. For those component stocks that are not traded during the beginning fifteen-minute interval on the final settlement day, their last closing prices would be applied instead
Settlement
Cash settlement. An option that is in-the-money and has not been liquidated or exercised on the expiration day shall, in the absence of contrary instructions delivered to the Exchange by the Clearing Member representing the option buyer, be exercised automatically
Data source: http://www.taifex.com.tw/
Table 5.1 Candidates of change points by method 2
(Two out of three consecutive points plot between a distance of 2 sigma and 3 sigma)
Candidates Date LRT value Testing
Change point= 72 Apr. 22, 2003 LRT=4.51952<9.488 Accept H0
Change point= 73 Apr. 23, 2003 LRT=5.19142<9.488 Accept H0
Table 5.2 Candidates of change points by method 3
(Ten out of eleven consecutive points plot on one side of the centerline)
Candidates Date LRT value Testing
Change point= 101 Jun. 3, 2003 LRT=7.5551<9.488 Accept H0
Table 5.3 Candidates of change points by method 4
(A run of five consecutive points of returns plot on one side of the centerline)
Candidates Date LRT value Testing
Change point= 36 Mar. 3, 2003 LRT=3.31969<9.488 Accept H0
Change point= 37 Mar. 4, 2003 LRT=3.21291<9.488 Accept H0
Change point= 87 May 14, 2003 LRT=10.75551>9.488 Reject H0
Change point=106 Jun. 11, 2003 LRT=5.44082<9.488 Accept H0
Change point=107 Jun. 12, 2003 LRT=5.24297<9.488 Accept H0
Change point=119 Jun. 30, 2003 LRT=2.92873<9.488 Accept H0
Change point=120 Jul. 1, 2003 LRT=2.83138<9.488 Accept H0
Table 5.4 Parameter estimates of GARCH (1,1) model
Parameters Alpha0 Alpha1 Beta1
GARCH(1,1) model of period 1 0. 00039566 (0.0000432084)
0. 00446372 (0.002081046)
0. 72666151 (0.0004901274) GARCH(1,1) model of period 2 0. 00015201
(0.0000383383)
1. 01731336 (0.0029162786)
2. 13366277 (0.0688668976) GARCH(1,1) model of period 1+2 0. 00066570
(0.0016216986)
1. 00431641 (0.0208480670)
2. 48087884 (1.1261979466) The numbers in parentheses are the standard errors of the parameters.
Table 5.5 The calculations of accuracy measures in every model Volatility of GARCH(1,1) model
with change point
Volatility of GARCH(1,1) model without change point
RMSE 0.01904166569 0.02349722250
MAE 0.01433519575 0.02296283234
LE 0.81310630801 1.254419791781
Table 5.6 Some statistics of standardized returns GARCH(1,1) with
c. p.*
GARCH(1,1) without c. p.
RV
Mean 0.130611159 0.05201497 0.041427136
Standard deviation 0.745337197 0.450746075 1.180303863
Skewness 0.662977222 0.454234493 -0.318000015
Excess kurtosis 1.399504106 2.020160196 0.542824948
JB test 25.24304206 33.32238285 4.748427444
* c.p. means change point
Table 5.7 The Ljung-Box Q-Statistic
Ljung-Box Q-Statistic Q(10) Q(20) GARCH(1,1) model of period 1 5.5473
(0.85175957)
13.6110 (0.84966122) GARCH(1,1) model of period 2 11.7482
(0.30225856)
19.5059 (0.48919712) GARCH(1,1) model of period 1+2 6.8301
(0.74137724)
11.6695 (0.92698101)
The numbers in parentheses are p value of the test statistic. Thus, these models appear to be adequate.
Figures
Figure 2.1 The SARS daily probable cases happened in Taiwan from Feb. 25, 2003 to Sep. 4, 2003.
Data source: http://sars.health.gov.tw/
Figure 2.2 Stock prices
Data source：Taiwan Stock Exchange Capitalization Weighted Stock Index from 2003.1.2 to 2003.8.29, 162 days in the aggregate. It comes from Taiwan Stock Exchange Corporation. Website: http://www.tse.com.tw/
Figure 3.1 Realized volatility of TAIEX during Jan.2, 2003 to Aug. 29, 2003 with 5-min intraday returns.
Figure 5.1 Log returns under 6 sigma criterion
Figure 5.2 Log returns from Jan. 2, 2003 to Aug. 29, 2003
Figure 5.3.1 QQ-plots of the GARCH(1,1) models versus standard normal distribution
(A)
(B)
Figure 5.3.2 QQ-plot of realized volatility versus standard normal distribution