Stock Market Volatility – Karachi Stock Exchange
Introduction
The study of volatility is always a serious concern for analysts and researchers because high degree of volatility can affect the smooth functioning of any stock market. It may also affect the economic growth and development of the economy through its effect on investor’s confidence and risk taking ability. The researchers worldwide have attempted to identify the major factors affecting the level of volatility in the stock markets. The available theoretical and empirical literature suggests that the main source of volatility in any stock market is the arrival of new information or news.
There is a general agreement on what constitutes stock market volatility and, to a smaller extent, on how to quantify it, there is far less conventionality on the reasons of changes in stock market volatility .A number of researchers investigated the causes of volatility in the arrival of new, unexpected information that affect expected returns on a stock (Engle and Mcfadden, 1994). Thus, changes in market volatility would just reproduce changes in the domestic or global economic environment. Others maintain that volatility is caused largely by changes in trading volume, practices or tends, which in tum are resolute by factors such as changes in macroeconomic policies, shifts in investor’s risk appetite and growing uncertainty.
Conditional Heteroscedasticity (ARCH) became a very popular method in the modeling of stock market volatility. As comparison to traditional time series models, ARCH models allowed the conditional variances to change during time as functions of precedent errors. First approach was to improve the univariate. ARCH model with a different requirement of the variance function. One development was introduced by Bollerslev (1986) where the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) method was presented. Then after, the Integrated GARCH (IGARCH) Engle and Bollerslev (1994) and the exponential GARCH (EGARCH)Nelson (1991) were significant one wherever respecification of variance equation was considered.
PROBLEM STATEMENT
Objective of study
The purpose of this paper is to understand the daily return data volatility of stocks & to develop an asymmetric GARCH models can explain determination of shock and volatility. This study is an attempt to develop models to elucidate the volatility of the stock in Pakistan. To this end, the study includes main indicex of Pakistan stock market i.e KSE 100. This study uses the Autoregressive Conditional Heteroskedasticity (ARCH) models and its extension, the Generalized ARCH, EGARCH and TARCH models was used to find out the presence of the stock market volatility on Pakistan stock market. The objective is to model the phenomena of volatility clustering and persistence of shock using asymmetric GARCH family of models.
Research Methodology
The study spanned the period from Jan 2003 to Dec 2014. This period of study is selected in order to check volatility in Karachi stock exchange returns. The sample population of the study consists of the daily returns of the most prominent domestic index, KSE. The data was collected from official website of respective stock exchange. Daily closing price of the index was considered for the period of study. This market index was fairly representative of the various industry sectors. The daily stock prices were converted to daily returns. Returns are calculated by difference of old and new.
The econometric software package Eviews 5.0 has been used to do the estimations. Arch and Garch Models Conventional econometric models assume a constant oneperiod forecast variance. To simplify this implausible assumption, Robert Engle presented a set of methods called autoregressive conditional heteroscedasticity (ARCH). These are zero mean, serially uncorrelated methods with non constant variance conditional on the past.
A practical generalization of this model is the GARCH parameterization introduced by Bollerslev (1986). This model is also a weighted average of past squared residuals, but it has waning weights that by no means go entirely to zero.
In the third equation ht= var (€t /CPtI), cptIit is the information prior to time tl. Because GARCH (p,q) is an annex of ARCH model, it has all the properties AsiaPacific Business Review of the original ARCH model. And because in GARCH model the conditional variance is not only the linear function of the square of the lagged residuals, it is also a linear function of the lagged conditional variances, GARCH model is more precise than the original ARCH model and it is easier to compute. The most commonly used GARCH model is GARCH (1,1) model. The (1,1) in parentheses is a standard notation in which the first number refers to how many autoregressive lags, or ARCH terms, come into view in the equation, as the second number refers to how many moving average lags are specified, which here is frequently called the number of GARCH terms. Occasionally models with more than one lag are needed to find better variance forecasts.
GARCH (1,1) is the most extensively used GARCH model because it is correctness and ease. Although GARCH model is very helpful in the predicting of volatility and asset pricing, there are still many problems GARCH model cannot clarify. The main difficulty is that standard GARCH models presume that positive and negative error conditions have a symmetric effect on the volatility.
In other terms, good and bad news have the similar impact on the volatility in this model. In real life this hypothesis is often desecrated, in particular by stock returns, in that the volatility increases more often after a flow of bad news than after good news. According to the challenges in the standard GARCH model, a number of parameterized extensions of the standard GARCH model have been recommended in recent times.
EGarch Model Exponential GARCH (EGARCH) model was first developed by Nelson in 1991. The main purpose of EGARCH model is to explain the asymmetrical response of the market under the positive and negative shocks.. If one compared the above equations with the premises of the conventional GARCH model, one can see that there are no constraints for the parameters. This is one of the biggest benefits of EGARCH model as compared to the standard GARCH model.
TARCH Model Threshold ARCH (TARCH) model was first developed by Zakoian in 1990. It has the conditional variance Where t d is latent variable d, = 1 for fl < 0 and d, = 0 for fl >= 0 because t d is builtin, the rise (0 E) of stock prices will have different impact on conditional variance. When the stock prices increase, cp we say that there is leverage effect.
Analysis and Discussion Arch Test
A descriptive investigation of the plot of daily returns on KSE (Figure 1) reveals that returns incessantly fluctuated about the mean value that was close to zero. The return measures were both in positive and negative area. More fluctuations be tending to cluster together and were alienated by periods of relative calm
This was in agreement with Fama’s (1965) observation of “volatility clustering”. From the time series graph of the returns for KSE market, it is analyzed that high volatilities are followed by high volatilities and low volatilities are followed by low volatilities. That means both time series have important time varying variances. Additionally, it is appropriate to put conditional variance into the function to clarify the impact of risk on the returns. Hence, GARCH class model is the excellent tool for the study
TABLE 1
Return 

Mean  0.003691 
Median  0.004623 
Maximum  0.354357 
Minimum  0.482963 
Std. Dev.  0.055540 
Skewness  0.499174 
Kurtosis  8.308197 
JarqueBera  3601.731 
Probability  0.000000 
Sum  10.93704 
Sum Sq. Dev.  9.136997 
Observations  2963 
Descriptive statistics (Table 1) for KSE returns showed skewness statistic of daily returns different from zero which indicated that the return distribution was asymmetric. In addition, relatively large excess kurtosis recommended that the underlying data was leptokurtic (heavily tailed and sharp peaked).
The Jarque – Bera statistic is calculated to test the null hypothesis of normality rejected the normality assumption. Kse index appeared to have significant strong autocorrelations in oneday lag returns. In addition, the autocorrelation in the squared daily returns suggested incidence of clustering. The results ruled out the independence assumption for the time series of given data set. Stationary of the return series were tested by conducting DickeyFuller. The results of the test confirmed that the series is stationary at level (Table 2).
Table 2
Null Hypothesis: R has a unit root  
Exogenous: Constant  
Lag Length: 0 (Automatic – based on SIC, maxlag=27)  
tStatistic  Prob.*  
Augmented DickeyFuller test statistic  46.96748  0.0001  
Test critical values:  1% level  3.432368  
5% level  2.862317  
10% level  2.567228  
*MacKinnon (1996) onesided pvalues.  
Augmented DickeyFuller Test Equation  
Dependent Variable: D(R)  
Method: Least Squares  
Date: 06/17/15 Time: 11:29  
Sample (adjusted): 1/03/2003 5/12/2014  
Included observations: 2962 after adjustments  
Variable  Coefficient  Std. Error  tStatistic  Prob. 
R(1)  0.854238  0.018188  46.96748  0.0000 
C  0.003153  0.001012  3.115317  0.0019 
Rsquared  0.427017  Mean dependent var  1.78E05  
Adjusted Rsquared  0.426823  S.D. dependent var  0.072602  
S.E. of regression  0.054966  Akaike info criterion  2.963536  
Sum squared resid  8.942867  Schwarz criterion  2.959489  
Log likelihood  4390.997  HannanQuinn criter.  2.962080  
Fstatistic  2205.944  DurbinWatson stat  2.001239  
Prob(Fstatistic)  0.000000  
Before ARCHGARCH is used in the study to approximation the model, the study is required to test whether the data has ARCH effect.
The most commonly used method is Lagrange MUltiplier test (LM).as table below shows that he N*R2 & fstatistic is high with significant pvalue shows that the ARCH effect is present .
The present work used GARCH, GARCHM, TARCH, TARCHM, EGARCH and EGARCHM to estimate the data.
Heteroskedasticity Test: ARCH  
Fstatistic  23.14889  Prob. F(2,2957)  0.0000  
Obs*Rsquared  45.63032  Prob. ChiSquare(2)  0.0000  
ARCH
Dependent Variable: R  
Method: ML – ARCH (Marquardt) – Normal distribution  
Date: 06/17/15 Time: 11:31  
Sample (adjusted): 1/03/2003 5/12/2014  
Included observations: 2962 after adjustments  
Convergence achieved after 14 iterations  
Presample variance: backcast (parameter = 0.7)  
GARCH = C(3) + C(4)*RESID(1)^2  
Variable  Coefficient  Std. Error  zStatistic  Prob. 
C  0.004657  0.000780  5.970079  0.0000 
R(1)  0.189642  0.014004  13.54192  0.0000 
Variance Equation  
C  0.001822  3.57E05  50.98481  0.0000 
RESID(1)^2  0.470400  0.025631  18.35312  0.0000 
Rsquared  0.018415  Mean dependent var  0.003688  
Adjusted Rsquared  0.018083  S.D. dependent var  0.055550  
S.E. of regression  0.055045  Akaike info criterion  3.097226  
Sum squared resid  8.968662  Schwarz criterion  3.089132  
Log likelihood  4590.992  HannanQuinn criter.  3.094313  
DurbinWatson stat  2.086801  

GARCH
Dependent Variable: R  
Method: ML – ARCH (Marquardt) – Normal distribution  
Date: 06/17/15 Time: 11:33  
Sample (adjusted): 1/03/2003 5/12/2014  
Included observations: 2962 after adjustments  
Convergence achieved after 45 iterations  
Presample variance: backcast (parameter = 0.7)  
GARCH = C(3) + C(4)*RESID(1)^2 + C(5)*GARCH(1)  
Variable  Coefficient  Std. Error  zStatistic  Prob. 
C  0.004144  0.000625  6.628444  0.0000 
R(1)  0.108704  0.018898  5.752277  0.0000 
Variance Equation  
C  3.91E05  2.40E06  16.29193  0.0000 
RESID(1)^2  0.156128  0.010756  14.51604  0.0000 
GARCH(1)  0.845044  0.007936  106.4792  0.0000 
Rsquared  0.019628  Mean dependent var  0.003688  
Adjusted Rsquared  0.019297  S.D. dependent var  0.055550  
S.E. of regression  0.055011  Akaike info criterion  3.379572  
Sum squared resid  8.957575  Schwarz criterion  3.369454  
Log likelihood  5010.145  HannanQuinn criter.  3.375930  
DurbinWatson stat  1.925959  
TGARCH
Dependent Variable: R  
Method: ML – ARCH (Marquardt) – Normal distribution  
Date: 06/17/15 Time: 11:35  
Sample (adjusted): 1/03/2003 5/12/2014  
Included observations: 2962 after adjustments  
Convergence achieved after 46 iterations  
Presample variance: backcast (parameter = 0.7)  
GARCH = C(3) + C(4)*RESID(1)^2 + C(5)*RESID(1)^2*(RESID(1)<0) +  
C(6)*GARCH(1)  
Variable  Coefficient  Std. Error  zStatistic  Prob. 
C  0.003833  0.000682  5.623912  0.0000 
R(1)  0.118520  0.019386  6.113554  0.0000 
Variance Equation  
C  4.35E05  2.83E06  15.36350  0.0000 
RESID(1)^2  0.133526  0.015151  8.813000  0.0000 
RESID(1)^2*(RESID(1)<0)  0.059658  0.016272  3.666206  0.0002 
GARCH(1)  0.837064  0.009143  91.54924  0.0000 
Rsquared  0.020387  Mean dependent var  0.003688  
Adjusted Rsquared  0.020056  S.D. dependent var  0.055550  
S.E. of regression  0.054990  Akaike info criterion  3.381526  
Sum squared resid  8.950640  Schwarz criterion  3.369385  
Log likelihood  5014.040  HannanQuinn criter.  3.377156  
DurbinWatson stat  1.946070  
EGARCH
Dependent Variable: R  
Method: ML – ARCH (Marquardt) – Normal distribution  
Date: 06/17/15 Time: 11:37  
Sample (adjusted): 1/03/2003 5/12/2014  
Included observations: 2962 after adjustments  
Convergence achieved after 44 iterations  
Presample variance: backcast (parameter = 0.7)  
LOG(GARCH) = C(3) + C(4)*ABS(RESID(1)/@SQRT(GARCH(1))) + C(5)  
*RESID(1)/@SQRT(GARCH(1)) + C(6)*LOG(GARCH(1))  
Variable  Coefficient  Std. Error  zStatistic  Prob. 
C  0.005133  0.000383  13.41781  0.0000 
R(1)  0.111279  0.018531  6.004927  0.0000 
Variance Equation  
C(3)  0.578039  0.035085  16.47536  0.0000 
C(4)  0.346857  0.020478  16.93799  0.0000 
C(5)  0.075865  0.010874  6.976950  0.0000 
C(6)  0.948154  0.004057  233.6890  0.0000 
Rsquared  0.018936  Mean dependent var  0.003688  
Adjusted Rsquared  0.018605  S.D. dependent var  0.055550  
S.E. of regression  0.055030  Akaike info criterion  3.390574  
Sum squared resid  8.963899  Schwarz criterion  3.378433  
Log likelihood  5027.440  HannanQuinn criter.  3.386204  
DurbinWatson stat  1.929444  
MGARCH
Dependent Variable: R  
Method: ML – ARCH (Marquardt) – Normal distribution  
Date: 06/17/15 Time: 11:41  
Sample (adjusted): 1/03/2003 5/12/2014  
Included observations: 2962 after adjustments  
Convergence achieved after 74 iterations  
Presample variance: backcast (parameter = 0.7)  
GARCH = C(4) + C(5)*RESID(1)^2 + C(6)*GARCH(1)  
Variable  Coefficient  Std. Error  zStatistic  Prob. 
@SQRT(GARCH)  0.027837  0.042554  0.654151  0.5130 
C  0.003251  0.001568  2.073169  0.0382 
R(1)  0.109279  0.019097  5.722392  0.0000 
Variance Equation  
C  3.97E05  2.44E06  16.27203  0.0000 
RESID(1)^2  0.157702  0.010906  14.45994  0.0000 
GARCH(1)  0.843484  0.008057  104.6846  0.0000 
Rsquared  0.018986  Mean dependent var  0.003688  
Adjusted Rsquared  0.018323  S.D. dependent var  0.055550  
S.E. of regression  0.055038  Akaike info criterion  3.379067  
Sum squared resid  8.963444  Schwarz criterion  3.366926  
Log likelihood  5010.398  HannanQuinn criter.  3.374697  
DurbinWatson stat  1.924319  
Table 3
Market  model  rss  AR2  AIC  SC 
KSE  GARCH M  8.963444  0.018323  3.379  3.366 
E GARCH  8.963899  0.0186  3.390  3.378  
T GARCH  8.950  0.02005  3.3815  3.369385  
EGARCHM  8.952  0.019494  3.3911  3.3769 
INTERPRETATION
Following is the table with the results estimated from different models. From this table, one can select the best model for the further forecasting of stock market volatility.
From Table 3, one can see that for kse market TGARCH (l,l) has the lowest RSS and the relative high adjusted 2 R. That means, TGARCH (1,1) is superior to other models in the~timation.
From the standard of AIC and SC, we can see that GARCH (1,1) has the lowest value. That means GARCH (1,1) is also a relative good model for the estimation.
In addition, when the study use GARCH (1,1) to estimate the data, it is found that the kse market is 0.845. They are very close to 1. This demonstrates that there is high durability of the volatilities in market. That means if there is an expected shock in these markets, the sharp movements will not die out in the short run. That is a sign for high risk. At the same time, the study found that the summation of the parameters is less than 1, which indicates that the GARCH process for the stock return is widesense stationary.
When the study used TARCH (1,1) to estimate the model, it is found that the estimate of cps are greater than 0 for stock exchange.
When the study used EGARCH (1,1), it is found the estimates of cps are less than 0 for market. Then one can conclude that there are leverage effects in market. That is to say the volatilities caused by negative shocks are greater than that caused by positive shocks. This is in consistent with most of the existing literature.
The study also used the estimated EGARCH (1,1) to predict the volatilities for KSE. One can see that the model did a great job.
COMPARISION
KSE Return  SENSEX Return  Nifty Return  
Mean  0.003691  0.000613  0.000561 
Median  0.004623  0.001390  0.001540 
Maximum  0.354357  0.079311  0.079691 
Minimum  0.482963  0.118092  0.130539 
Std. Dev.  0.055540  0.015473  0.015388 
Skewness  0.499174  0.537631  0.752137 
Kurtosis  8.308197  7.067238  8.508060 
JarqueBera  3601.731  1463.086  2695.063 
Probability  0.000000  0.000000  0.000000 
Sum  1.93704  1.215901  1.113573 
Sum Sq. Dev.  0.9136997  0.474761  0.469532 
Observations  2963  1984  1984 
By comparing KSE results with base article results (Nifty & SENSEX). At first glance you assume that our market is more volatile then Nifty & SENSEX. While keeping in mind that Indian market is emerging market and Pakistan is also getting a status of Emerging County in 2016 by MSCI. Which show’s that there are a lot of things very common in both markets.
By comparing Mean, Maximum and Minimum values, almost all three markets have same level of fluctuations (Variations). Standard Deviation of KSE 100 Index is higher which shows that market is more volatile as compare to Nifty & SENSEX.
The distribution is negatively skewed for all three markets then the probability density function has a long tail to the left. Kurtosis is positive (It can only be Negative in case of Constant X). Kurtosis > 3 – Leptokurtic distribution, sharper than a normal distribution, with values concentrated around the mean and thicker tails. This means high probability for extreme values.
Sum of Square of Deviations is also high which depict that market is more volatile. The JB statistic is an indication of your distributions deviation of 0 (skewness and Kurtosis if it was truly a normal distribution). Pvalue = 0 indicates that the null hypothesis: “the distribution is normal” is rejected. Probability is less than 5% which shows that our results are significant. We took more number of observations as compare to Nifty & SENSEX.
FINDINGS
The findings of this study suggest that Pakistan stock market has significant ARCH effects and it is appropriate to use ARCH GARCH models to estimate the process. The research found that both TGARCH (1,1) and EGARCH (l,l) did good jobs in fitting the process for exchange. Research also shows that the investors in stock markets are not grown well and they will be heavily influenced by information (good or bad) very easily.
The results of GARCH (1,1) model indicated the presence of time varying volatility. both α and β coefficients were found positive in variance equation. The magnitude of α is lesser than magnitude of β indicating more impact of past volatility on the current volatility in comparison to impact of past shocks/news on the conditional volatility in equity market in Pakistan.
In addition to this, both α and β coefficients have shown high significance at 1 percent and 5 percent levels of significance indicated that past lagged variance has significant impact on the conditional volatility in Pakistan stock market. It indicated that news shocks in past has noteworthy impact on present returns on equity stocks.
CONCLUSION
To conclude it can be stated that the historical performance of any market plays a significant role to determine the investment strategy of that market. Therefore, both the market performance in past as well as market sentiments in the past; which have resulted due to various types of news extended by various market participants; have impact on the forecasting strategies of the investors.
A further study can be conducted to examine whether the impact of flow of capital by foreign institutional investors, introduction of derivatives, and impact of trading volume etc. can redefine the volatility of Pakistan stock market in a better way or not.