A special feature of stock volatility is the fact that it is not directly observable.
For example, consider the daily logarithmic rate of returns of Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX). The daily volatility is not directly observable from the returns because there is only one observation in a trading day. If intraday data of the stock, such as 5-minute returns, are available, then one can estimate the daily volatility by realized volatility. The accuracy of such an estimate deserves a careful study. Throughout this thesis, we try to do research in volatility.
We will first start with theoretical illustration of GARCH model that has been popular in financial time series analysis.
3.1 General Autoregressive Conditional Heteroscedastic (GARCH) Model
We assume there is an asset guaranteeing an instantaneously risk-free rate of interest. We have the following characterization of the logarithmic asset price, , where . The continuously compounded return over the time interval
is or is directly written as
, log( t )
The discrete-time models, at a minimum, assume that the correct specification of the h-step ahead conditional mean and variances are known up to a low-dimensional parameter vector.
The general autoregressive conditional heteroscedastic model, in brief, GARCH(m,s) model (Bollerslev, 1986), can be expressed as follows,
2 2
operator. For example, .
0 >0
In most applications, only lower order GARCH models are used frequently, such as GARCH(1,1), GARCH(2,1), and GARCH(1,2) models. We will use the GARCH(1,1) process of Duan(1995) for daily returns in this thesis. The GARCH(1,1) model lists as follows,
2 2 2
where . We use the maximum likelihood (ML) method to estimate the parameters , where is the constant unit risk premium.
(Under conditional lognormality, one plus the conditionally expected rate of return equals
Andersen et al.(2001a, b) propose the sum of squared returns daily realized volatility estimator which sums the squares of intraday returns.
Let , where , represent a set of n+1 intraday returns for day t, and when represents the five minutes commencing at the open, and concluding with
,
rt i
1
0≤ ≤i n i=
the five minutes at the end when i=n.
The realized volatility for trading day t, from the close on day t-1 to the close on day t, is defined by where . The realized volatility is simply the second sample moment of the log return process over a fixed interval.
0≤ ≤i n
We use a sampling frequency of 5-minute returns, which is high enough such that our daily realized volatilities are largely free of measurement error. We show in Figure 3.1.
3.3 Change Point Detection with Methods in Quality Control
In 2003, the most influential event globally is SARS, especially in Asia. The peak period of the high infectious disease is from April to June. Consequently, we must pay attention to the event. However, if we ignore structural change on purpose, in other words, do not readjust the estimation of all parameters in time series models, it is not a reasonable forecasting at all. For this reason, we hope that methods in quality control can help detect some change points. We can then cut a complete period into several subperiods in accordance with these change points. In order to ensure the candidates of change points to be accepted statistically, we will utilize the likelihood ratio test for this purpose in the next section. Once the change points are identified, we estimate parameters and calculate volatility for each subperiod.
Quality control plays an important role in industries of manufacturing. Quality control is a process that measures output relative to a standard, and take action when
output does not meet the standard.
A control chart is essentially a picture of a sampling distribution. That is, it consists of a series of sample values or “statistics” which, if they were gathered together instead of being plotted in sequence, would form a distribution. A natural pattern has three characteristics simultaneously. They can be summarized as follows:
1. Most of the points are near the solid centerline.
2. A few of the points spread out and approach the control limits.
3. None of the points (or at least only a very rare and occasional point) exceeds the control limits.
If any characteristic stated above is missing, the pattern will look unnatural. A control chart may indicate out of statistical control either when one or more points fall beyond the control limits or when the plotted points exhibit some nonrandom patterns of behavior. The Western Electric Handbook (1956) suggests several decision rules for detecting nonrandom points on control charts. It shows that the unnatural consecutive pattern is out of control if there is any one listed below:
1. One point plots outside the 3 sigma control limits;
2. Two out of three consecutive points plot between a distance of 2 sigma and 3 sigma;
3. Ten out of eleven consecutive points plot on one side of the centerline;
4. A run2 of five consecutive points plots on one side of the centerline;
or
5. Five consecutive points are in a rising or falling trend.
__________________________
2A run is a sequence of observations with a certain characteristic.
Under models with Markov properties, each innovation of a financial time series can be treated as independent. Therefore, it is natural to assume that quality control may be a good tool to find nonrandom and unnatural points. We can use it to find candidates of change points, which will be the subject of the next section.
3.4 Likelihood Ratio Test (LRT)
After using several methods of quality control, we can find some candidates of change points. If the Xth day is a candidate, the interval between the first day and the (X-1)st day is period one and interval between the Xth day and the nth day is period two.
Consider the following testing hypothesis problem.
Ｈ０：period one and period two have the same data structure, Ｈ１：period one and period two have different data structures.
For a GARCH(1,1) model, the likelihood function (L) is given by
2
where is the upper 100( point of the chi-square distribution with 4 degrees of freedom
3 The degree of freedom is the amount of difference between parameters under H0 and H1. For instance, there are 4 parameters under H0 and 8 parameters under H1 in the GARCH(1,1) model. Thus, the degree of freedom here equals to 4.
When it comes to the end of hypothesis testing, we can conclude whether the candidates are change points or not from the LRT. Once the change points are established, we estimate the parameters of the GARCH(1,1) model in each subperiod.