MEASURING VOLATILITY USING GARCH MODELS : AN APPLICATION TO SELECTED STOCK OF DHAKA STOCK EXCHANGE

Stock market is an important part of economy of a country. Measuring stock market volatility is an vital issue in finance. There are various models to evaluate volatility. The daily return series shows that there is a variation of closing prices of AB bank. The data of AB bank includes daily closing prices from 01-01-2015 to 05-10-2017 from Dhaka Stock Exchange (DSE) library to forecast phenomena of stock market volatility. We use GARCH models to assess the volatility of stocks from banking sector and find that GARCH (1,1) model is best for measuring volatility of the stocks of AB bank. Once if we measure the volatility then it is possible to make best prediction when to buy and when to sell a stock.

reported that GARCH/ GARCH type models is the best model for forecasting stock market volatility. Liu et al. (2009) investigated the forecast of stock market volatility in China using GARCH model and discovered that volatility forecasts by the GARCH-SGED model are more accurate than those generated using the GARCH-N model. Alberga et al. (2008) searched the predictable stock market volatility in Israel using GARCH model. They found that asymmetric GARCH model with fat-tailed densities improves overall estimation for measuring conditional variance and improve the forecasting performance.

Methodology:-Volatility Model:
The Dhaka stock market is one of the least regulated market. In Asia, the stock exchanges are primarily arms of government, controlled by government appointees. The rapid climb in financial markets and therefore the continual development of latest and more complex financial instruments, there's an ever-growing need for theoretical and empirical knowledge of the volatility in financial time series. It is widely known that the daily returns of monetary assets, especially of stocks, are difficult, if not impossible, to predict, although the volatility of the returns seems to be relatively easier to forecast. Therefore, it is hardly surprising that financial volatility has played such a central role in modern pricing and risk management theories. Several indices are available for Dhaka. The data set we use is the DSE20 capital index, which covers 20 largest and most liquid stocks listed and quoted on the Dhaka Stock Market Exchange (DSE), weighted by the market capitalization without dividends reinvested. Although the method provides a more efficient volatility estimator in terms of approximating the diffusion term in a small sample, it is subject to more biases due to the closure of the stock market overnight [6].

Daily Return Series Measurement:
The daily returns calculated are the logarithmic (continuously compounded) returns, rather than the arithmetic returns, since they are considered symmetric whereas the arithmetic returns are not. Nevertheless, the difference between both types is large only when percent changes are high, as both are approximately equal for small returns.
The logarithmic daily returns are calculated according to the following: . ln where, t P is the daily closing price index, and 1  t P is the yesterday closing price index [7].

Garch :
For the ARCH(q) model, in most empirical studies, q has to be large. This motivates Bollerslev (1986) to use the GARCH(p,q) specification which is defined as Again, the selection of p and q is an important empirical question. As in the ARCHmodel, BIC is used to choose p and q [13].
The conditional variance equation in GARCH(1,1) model process can be modeled as: is a discrete-time stochastic defined to be given ~ iid (0,1) and is the conditional standard deviation of return at time t. All paremeters are non negative. The stationary condition of 805 should hold to ensure weakly stationary of GARCH process. Indicates the short-run persistency of stocks while implies the long-run persistency.

Model Evaluation:
The selection of model is a vital segment of any statistical analysis and is a key to the realization of science. Many authors have examined many tools like R2, Adjusted R2, AIC, SIC, Mallows's Cp criterion and forecast Chi square test for selecting the best model. All these criteria aim at minimizing the residual sum of squares (RSS). Here except R2 all the criteria impose a penalty for including large number of regressor. [5] Akaike's Information Criteria (AIC) and Bayesian Information Criteria (BIC): In Model selection AIC (Akaike's Information Criteria) and BIC (Bayesian Information Criteria) are considered as the widely used criteria. The AIC can be defined as a measure of the goodness of fit of any estimated statistical model. AIC generally tries to find unknown model that has high dimensional reality. This means the models are not true models in AIC. [5][14] AIC= e 2k/n ∑ = e 2k/n where, k is the number of regressors (including the intercept) and n is the number of observations. ln AIC = ( + ln( ) Where, ln AIC = natural log of AIC and 2k/n = penalty factor. In comparing two or more models, the model with the lowest value of AIC is preferred. Akaike's Information Criteria is good for making asymptotically equivalent to cross-validation.
On the other hand, BIC is used for selecting model among a class of parametric models with different numbers of parameters. BIC is consistent and comes across only True models. Bayesian Information Criteria is good for consistent estimation. [14]

Result and Discussion:-
We analyze the data whether it is stationary or not, which is the initial and vital steps to determine the appropriate method by which we can fulfill the study.

Time series plot of closing price of AB Bank:
Year

806
From the above time series plot, we see that over the period of study the time series data seems to be trending, suggesting perhaps that the mean and variance has been changing. So we can say that the daily data of closing price of AB bank of Dhaka Stock Exchange is not stationary. It is visual that mean and variance is not remains constraint from time to time, so we can say that the daily data of closing price is not stationary. The mean of the daily return series appears to be stable with an average return of approximately zero. Also the volatility or variability of the data changes over time. The daily return series of the selected bank shows that there is a small variation of closing price. There is no visual evidence of serial correlation in the return but there is evidence of serial correlation in the amplitude of the return.

ACF and PACF of closing price of AB bank:
Daily volatility of selected Bank: Fig. 4: Daily volatility of AB bank.
From the above graphical plots of daily volatility series shows us several volatility periods for AB bank.

Application of different volatility models:
We consider different volatility models to estimate the parameters. Various deterministic volatility models are fitted to the data. Now the estimated result for the data set is give below: