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International Journal of Research in Economics and Social Sciences (IJRESS) Available online at: http://euroasiapub.org Vol. 8 Issue 12, December - 2018 ISSN(o): 2249-7382 | Impact Factor: 6.939 |

International Journal of Research in Economics & Social Sciences

Email:- [email protected], http://www.euroasiapub.org (An open access scholarly, peer-reviewed, interdisciplinary, monthly, and fully refereed journal.)

13

STOCK PRICE VOLATILITY IN NATIONAL STOCK EXCHANGE OF INDIA

Sumathi D

Research Scholar, Bharathiyar University,

Dr. S.N.S. Rajalakshmi College of Arts and Science (Autonomous), Coimbatore,

Tamil Nadu – 641 049, India

Abstract

Economic status of India is greatly imitated by the introduction of new economic policy in 1991. The

Indian Capital Market has perceived a marvelous progression. There was an outburst of investor

interest during the nineties and an equity cult emerged in the country. Foreign Exchange Regulations

Act is one such legislation in this direction. An important recent development has been the entry of

Foreign Institutional Investors as participants in the primary and secondary markets for industrial

securities. In the past several years, investments in developing countries have increased remarkably.

Among the developing countries, India has received considerable capital inflows in recent years. We

apply the GARCH (1, 1) (General Autoregressive Conditional Heteroscedasticity) framework to on

selected representative stock indices. The findings reveal that the GARCH (1, 1) model successfully

captures nonlinearity and existence of volatility. The analysis suggests indicates a long persistence

of volatility in Indian stock market especially National Stock Exchange (NSE) of India. The

preliminary analysis of data set suggests that volatility in the Indian stock market is time varying in

nature, persist to form clusters and has a long memory process. These findings of the data

characteristics have been consistent with previous studies of Indian markets and justify the

application of GARCH type models. The detailed analysis shows that the TGARCH (1,1) model

outperforms in estimating, predicting and forecasting the stock market volatility.

Keywords: Stock Price volatility, Indian Stock Market, GARCH, EGARCH, TGARCH

Introduction:

A financial system in an institutional arrangement, in which financial surpluses are moved

from the units that are generating surplus income, to the units, that are in need of it. The financial

system plays a vital role in mobilizing the funds from supplying units to the demanding units.

Financial instruments like financial institutions, financial services, financial market, and financial

assets establish the financial system. The activities of the financial system comprised of exchange

and holding of financial assets or monetary resources, in the form of financial institutions, banks

and other intermediaries.

Broadly, the organizational structure of financial system includes the following three

components

Financial Markets,

Financial Institutions and Intermediaries,

Financial Products.

The financial market is the market where the investors buy and sell the financial assets like

stocks, bonds, bills of exchange, commodities and a foreign currency which works as liquid assets.

The financial market plays a pivotal role in the economic development of any country. The large

International Journal of Research in Economics and Social Sciences (IJRESS) Vol. 8 Issue 12, December- 2018 ISSN(o): 2249-7382 | Impact Factor: 6.939



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scale industries in a country mobilize the required resource in the financial market. The financial

markets are broadly classified into following categories.

Money Market: The money market is the market where the short-term needs are achieved

through borrowing and lending of funds, for solving liquidity needs of borrowers and lenders.

Money market instruments are financial claims that have low default risk, maturities under one

year and high marketability.

Capital Market: The capital market is the market where the trade of financial securities like

shares, bonds occurs, where the pricing of financial security is determined by the free market

based on supply and demand. Capital market instruments are financial claims that have a certain

level of risk, long or indefinite maturities and marketability based on supply and demand. The

Capital market comprises of the primary and secondary market. In the primary market,

ownership of the financial securities is issued by companies and government to increase the

capital. And trading of such financial securities occurs in the secondary market.

The primary role of the capital market is to make the investment from the investors who have

surplus funds to the ones who are in need of it. The efficient allocation of the fund by the capital

market depends on the state of the capital market of the country. Therefore, all the countries focus

more on the functioning of the capital market. Indian financial market has faced several challenges

in efficient allocation and mobilization of the capital. Indian financial market has achieved

tremendous growth in the last decade and also continues to achieve the same in the current

decade. The capital market comprises the institutions and mechanisms through which the funds

are brought together and made available to individuals, government and businesses in the public

and private sector. The capital market has two interdependent and inseparable segments they

are, new issues (primary) market and the stock (secondary) market. The sale of new securities

happens in the primary market, whereas, the trade of previously issued securities happens in the

secondary market.

A stock market is a place where buyers and sellers of stocks come together, physically or

virtually. The members range from small individuals investors to large fund managers, who are

present virtually in the market. These members place their requests to the experts of a stock

exchange, who executes this buying and selling of the orders. The stocks are listed and traded on

stock exchanges. Some exchanges are physically located, based on open outcry system where

transactions are carried out on the trading floor. Nowadays it is very common to see a network of

computers to execute the transactions electronically. These are often connected to the internet,

which makes exchanges to be virtual stock exchanges. The whole system is order-driven, the

order placed by an investor is automatically matched with the best limit order. This system

provides more transparency as it shows all buy and sell orders.

Indian Stock Market: The two most prominent stock indices, viz., Bombay Stock Exchange’s

(BSE) Sensitive Index (Sensex) and NSE’s S&P CNX Nifty (Nifty) represents the Indian stock

market. The pioneer of the Indian stock market is the Sensex. It is the older and popular index.

However, of late, with the growing popularity of the NSE, due to its more transparent trading

mechanism and lower trading cost, Nifty has come to be considered as a significant and broader-

based market index. As per SEBI’s Annual Report of 2015-2016 (available at www.sebi.gov.in),

more than 95 per cent of the total business transacted on all the stock exchanges of the India is

from the BSE and NSE. Additionally, according to the data available on the respective exchange

web sites (www.bseindia.com and www.nseindia.com), a major portion (around 75%) of the total

market turnover of the respective stock exchanges is accounted by the index (Sensex and Nifty)

stocks. Regarding market capitalization, BSE and NSE have a place in top five stock exchanges of

developing economies of the world. As of 2015, there is a total of 60 stock exchanges in the world




15

with a total market capitalization of US $65 trillion. Of these, there are 16 exchanges with a market

capitalization of the US $1 trillion or more, and they account for 87% of global market

capitalization. Out of all stock exchanges in the world, BSE stood at 11th position with a market

capitalization of US$1.83 trillion as on December 2016 and NSE at 12th position with a market

capitalization of US$1.81 trillion as on December 2016.

National Stock Exchange (NSE): The National Stock Exchange (NSE) is located in Mumbai. It was

incorporated in 1992 and became a stock exchange in 1993. The fundamental purpose of forming

this exchange was to introduce transparency in the stock markets. It started its operations in the

wholesale debt market in June 1994. The equity market segment of the National Stock Exchange

commenced its operations in November 1994 whereas, in the derivatives segment, it started it

operations in June 2000. It has completely modern and fully automated screen based trading

system. It is playing an important role to reform the Indian equity market to bring more

transparent, integrated and efficient stock market. The National Stock Exchange replaced open

outcry system, i.e. floor trading with the screen based automated system. NSE also created

National Securities Depository Limited (NSDL) which permitted investors to hold and manage

their shares and bonds electronically through the DEMAT account. The electronically security

handling, convenience, transparency, low transaction prices and efficiency in trade which is

affected by NSE, has enhanced the reach of Indian stock market to domestic as well as

international investors.

Stock market volatility: Investment in stock market is always assumed to be risky as the stock

markets are volatile. There is volatility in the stock market because macro-economic variables

influence it and affect stock prices. These factors can have an impact on a single firm’s price and

can be unique to a company. On the other side, some factors like subprime crisis, political

situations and war that commonly affect all the companies. Volatility is the variation in asset

prices change over a particular period. It is challenging to estimate the volatility accurately.

Volatility is responsible for making the stock market risky, but it is this only which provides the

opportunity to earn money to those who can understand it. It gives the investor an opportunity to

take advantage of fluctuation in prices, buy stock when prices fall and sell when prices are

increasing. So, to take advantage of volatility, it is needed to be understood well.

Volatility is measured by variance or the standard deviation of stock returns around their average

value. When measuring the volatility, stock returns are taken rather than stock prices because

mean must be stable at the different period while measuring the dispersion around an average

value. Modeling and forecasting of the volatility of asset returns are important in various

applications related to financial markets such as valuation of derivatives and risk management.

Extensive research has been done the world over in modelling volatility for estimation and

forecasting.

Literature Review:

Uncertainty plays a major role in economic theory. Many economic models assume that the

variance, as a measure of uncertainty, is constant through time. However, empirical evidence

rejects this assumption. Financial time series such as stock returns or exchange rates exhibit so

called volatility clustering. It means that significant changes in these series tend to be followed by

large changes and small changes by minor changes. The technical term given to this behavior is

autoregressive conditional heteroscedasticity (ARCH). As variance (or standard deviation) is

often used as a risk measure in risk management systems, accurate modelling and forecasting of

the variance have received a lot of attention in the investment community for the last two decades.




16

In a seminar paper, (Engle, 1982) for the first time, proposed to model time varying conditional

variance with the ARCH process that uses past disturbances to model the variances of the series

and allows the variance of the error term to change over time. (Bollerslev, 1986) Generalized the

ARCH process by allowing the conditional variance to be a function of prior period’s squared

errors as well as its past conditional variances. Following the introduction of models of ARCH by

(Engle, 1982) and their generalization by (Bollerslev, 1986), there have been numerous

refinements of the approach to modelling conditional volatility to capture the stylized

characteristics of the data. Empirically, the family of GARCH (generalized ARCH) models has been

very successful in describing the financial information. Of these models, the GARCH (1, 1) is often

considered by most investigators to be an excellent model for estimating conditional volatility for

a broad range of financial data.

Though in most of the cases, the ARCH and the GARCH models are apparently successful in

estimating and forecasting the volatility of the financial time series data, they cannot capture some

of the important features of the data. The most interesting feature not addressed by these models

is the ‘leverage effect’ where the conditional variance tends to respond asymmetrically to positive

and negative shocks in errors. To solve this problem, many nonlinear extensions of the GARCH

model have been proposed. (Nelson, 1991) proposed an exponential GARCH (EGARCH) model

based on a logarithmic expression of the conditional variability in the variable under analysis.

Later, some modifications were derived from this method. One of them is the Threshold GARCH

(TGARCH) method which was introduced by (Zakoian, 1994). The model developed by Glosten,

Jagannathan and Runkle (Glosten and Jagannathan, 1993) has been considered the best in

estimating the impact of positive and negative shocks on volatility (Engle and Ng, 1993).

Data and Methodology:

Research methodology is a way of systematically solving the research problem. It deals with the

research design used and methods used to present the study. It refers to the systematic method,

for defining the problem, collecting the facts or data, analyzing the facts and reaching certain

conclusions either in the form of solutions towards the concerned problem. The daily closing price

of the selected sector Indices and its selected companies of the National Stock Exchange of India

(NSE) are collected from 1st January 2006 to 31st December 2016 to analyses the existence of the

price volatility. This research study is based on secondary data, which is mainly collected from

NSE website. CNX Nifty index is used as a proxy for the stock market. The daily closing price of

NSE index Nifty 50 is considered. The NIFTY 50 is a diversified stock index from 50 companies

listed on NSE representing for twelve sectors of the economy. It is utilized for an assortment of

purposes, for example, benchmarking fund portfolios, index based derivatives and index funds.

Methodology

Returns: Daily returns are identified as the difference in the natural logarithm of the closing index

value for the two consecutive trading days. The daily closing price collected during the period of

study for the selected indices are converted into returns. Volatility is defined as:

𝜎 = √1

𝑛−1 ∑ (𝑅𝑖 − �̅�)2 𝑛

𝑖=1 (1)

�̅� – Average log return for the collected samples.

Descriptive Statistics: Descriptive Statistics are used to present quantitative descriptions in a

manageable form. It is used to describe the basic features of the data considered for the study. In




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this study, under descriptive statistics, the Mean, Median, Minimum, Maximum, Standard

Deviation, Skewness and Kurtosis of Daily log returns are calculated.

Econometric Non-Linear Models

GARCH (1, 1): In empirical applications, it is often difficult to estimate models with a large

number of parameters, say ARCH (q). In an ARCH (1) model, next period's variance only depends

on last period's squared residual so a crisis that caused a significant residual would not have the

sort of persistence that we observe after actual crises. To circumvent this problem (Bollerslev,

1986) proposed Generalized ARCH (p, q) or GARCH (p, q) models. The financial modelling

professionals often prefer GARCH (1, 1) because it provides a more real-world context than other

forms when trying to predict the prices and rates of financial instruments. The conditional

variance of the GARCH (1, 1) process is specified as:

𝜎𝑡2 = 𝛼0 + 𝛼1 𝑎𝑡−1

2 + 𝛽1𝜎𝑡−12 (2)

With α0 > 0, α1 ≥ 0, β1 ≥ 0 and (α1 + β1) is less than 1 to ensure that conditional variance is positive.

In GARCH process, unexpected returns of the same magnitude (irrespective of their sign) produce

the same amount of volatility. The large GARCH lag coefficients β1 indicate that shocks to

conditional variance take a long time to die out, so volatility is ‘persistent.’ Large GARCH error

coefficient α1 means that volatility reacts quite intensely to market movements and so if α1 is

relatively high and β1 is relatively low, then volatilities tend to be ‘spiky’. If (α + β) is close to unity,

then a shock at time t will persist for many future periods. A high value of it implies a ‘long

memory.’

The general process for a GARCH model involves three steps. The first is to estimate a best-fitting

autoregressive model; secondly, compute autocorrelations of the error term and lastly, test for

significance. GARCH model is applied for all indices and all the companies considered for the

study.

EGARCH (1, 1): The main drawback of symmetric GARCH is that the conditional variance is

unable to respond asymmetrically to rise and fall in the stock returns. Hence, a number of models

have been introduced to deal with the issue and are called asymmetric models viz., EGARCH,

TGARCH and PGARCH, which are used for capturing the asymmetric phenomena. To study the

relation between asymmetric volatility and return, the EGARCH (1, 1) and TGARCH (1, 1) models

are used in the study. EGARCH model is based on the logarithmic expression of the conditional

variability. The presence of leverage effect can be tested, and this model enables to find out the

best model, which capture the symmetries of the Indian stock market (Nelson, 1991) and hence

the following equation:

ln (𝜎𝑡2) = 𝜔 + β1ln (𝜎𝑡−1

2 ) + 𝛼1 {|𝜀𝑡−1

𝜎𝑡−1− √

𝜋

2|} − γ

𝜀𝑡−1

𝜎𝑡−1 (3)

The left-hand side is the log of the conditional variance. The coefficient γ is known as the

asymmetry or leverage term. The hypothesis can test the presence of leverage effects that γ < 0.

The impact is symmetric if γ ≠ 0.

TARCH (1, 1): In TGARCH model, it has been noticed that positive and negative shocks of equal

magnitude have a different impact on the stock market volatility, which may be attributed to a

“Leverage effect” (Black, 1976). Also, negative shocks show higher volatility than the positive

shock of the same magnitude (Engle and Ng, 1993). The threshold GARCH model was introduced

by (Zakoian, 1994) and (Glosten and Jagannathan, 1993). The main target of this model is to

capture the asymmetry effect regarding positive and negative shocks. The generalized

specification of the threshold GARCH for the conditional variance is given by:

𝜎𝑡2 = 𝜔 + 𝛼1 𝜀𝑡−1

2 + γ 𝜀𝑡−1 2 𝐼𝑡−1 + β1 𝜎𝑡−1

2 (4)




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The γ is known as the asymmetry or leverage parameter. In this model, good news (𝜀𝑡−1 > 0) and

the bad news (𝜀𝑡−1< 0) have differential effect on the conditional variance. Good news has an

impact of𝛼𝑖, while bad news has impact on 𝛼𝑖 + 𝛾𝑖. Hence, if γ is significant and positive, negative

shocks have a larger effect on 𝜎𝑡2 than the positive shocks.

Empirical Analysis:

The descriptive statistics of the Nifty 50 is presented in Table 1. The mean values of the selected

indices are positive during the period of the study from January 2006 to December 2016,

indicating the fact that price has increased over the period. The Nifty 50 has negative (Left skewed

distribution) skewness, which means that the selected indices are having the higher possibility of

getting positive returns, which may be higher than the average returns. Kurtosis of the returns for

the selected indices is higher than 3. This Higher kurtosis value indicates that the unexpected

return distributions are not normal, which implies that the return series is fat tailed and does not

follow a normal distribution. Also, the return series is sharply peaked about the mean when

compared with the normal distribution.

Table 1: Descriptive statistics of Nifty 50

Index /

Statistics

Mean Median Standard

Deviation

Skewness Kurtosis

Nifty 50 0.00042 0.00082 0.01574 -0.02249 9.03610

Figure 1 shows the daily closing index value of Nifty index Nifty 50 from 01-Jan-2006 to 31-Dec-

2016. The index shows substantial growth in 2006 and 2007. In 2008 and 2009 the index was

returned to the original value in 2006. This is due to subprime financial crisis in US Stock markets,

which influenced the other stock markets in the world. After 2009 the index sees ups and downs,

but tendency shows that there is a continuous growth. This is one of the layman’s evidence of

volatility in stock price for the companies listed in the index Nifty 50. Figure 2 shows the log return

of daily closing index value of Nifty index Nifty 50 from 01-Jan-2006 to 31-Dec-2016. It is

undeniable that the magnitude of the everyday stock returns is changing in Nifty 50. The extent

of this shift is not consistent over the time, but it is sometimes large and sometimes small. This

phenomenon depicts the existence of volatility clustering in the selected return series of Nifty 50.

Figure 1: Daily closing index value of Nifty 50 Figure 2: Log return of Nifty 50

GARCH (1, 1)

The assumption of GARCH (1, 1) model is that the future volatility declines geometrically

over time and has less influence of a return shock. This GARCH (1, 1) model is further predictable

2000

3000

4000

5000

6000

7000

8000

9000

2006 2008 2010 2012 2014 2016

Clo

se

Nifty 50 - Daily Close

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

2006 2008 2010 2012 2014 2016

LogRetu

rnClo

se

Nify 50 - Log return




19

concerning volatility clustering, where the substantial changes in stock returns are probably

followed by further huge variations in the stock returns. The three coefficients in the variance

equation are listed as α0, the intercept; ARCH (1) - α1, the first lag of the squared return; and β1 -

GARCH (1), the first lag of the conditional variance. The statistical parameters like Standard

errors, Z-statistics and p-values computed in Gretl and populated in the table.

Table 2: GARCH (1, 1) estimates for Nifty 50

Source: Computed based on secondary data using Gretl; α0: Long run average of the variance; α1:

The ARCH term which is first lag of the squared return, represents news about volatility from the

previous period; β1: The GARCH term which is the first lag of the conditional variance.

From Table 2, it is clear that around 89% (β1) of the information comes from the past information.

The effect of new information has minimal impact of about 9.9% (α1). The long run average

variance (α0) is close to zero, and its implications are substantially negligible. There is an

interesting feature in the above graphs that the volatility is higher when prices are falling and

lower when prices are rising. This means that negative returns result in higher volatility whereas

positive returns lead to lower volatility. This is called asymmetric volatility effect. And, this is not

captured by GARCH (1, 1) model. Hence, EGARCH (1, 1) model is used for stock return volatility

estimation.

EGARCH (1, 1)

Table 3 shows the estimates of coefficients using EGARCH (1, 1) model for the index Nifty 50 with

the given data. In this model, α is the GARCH term that measures the impact of last period’s

variance of the forecast. A positive α indicates volatility clustering implying that positive stock

price changes are associated with further positive changes and the other way around. The ARCH

term β is the measure of the effect of news about volatility from the previous period on current

period volatility. The term gamma (γ) is the measure of leverage effect. The null hypothesis may

test the presence of leverage effect that the coefficient of the last term in the regression is negative

(γ < 0). Thus, for a leverage effect, we would see γ > 0. The impact is asymmetric if this coefficient

is different from zero (γ ≠ 0). Ideally, γ is expected to be negative implying that bad news has a

bigger impact on volatility than the good news of the same magnitude. The sum of the ARCH and

GARCH coefficients, (α + β) specifies that the volatility shock is persistent over time.

Statistics coefficient Std. error z-Statistic p-value

α0 0.0000029 0.0000074 3.938 ~0.000

α1 0.09939 0.01190 8.341 ~0.000

β1 0.89088 0.01220 72.850 0.000




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Table 3: EGARCH (1, 1) estimates for Nifty 50

Source: Computed based on secondary data using Gretl ω: Constant in the model which

represents the long run average; α: The ARCH term which is first lag of the squared return,

represents news about volatility from the previous period; β: The GARCH term which is the first

lag of the conditional variance; γ: is the measure of leverage effect, which represents the

correlation between realized volatility and the historical return.

The condition for stationary is that the sum of ARCH and GARCH coefficient is less than 1 (α +β

<1). Hence the stationary condition is not met for the selected return series using EGARCH model.

Therefore this model is not suitable for modelling and forecasting of the selected return series. As

the value of Gamma (γ≠ 0) is non-zero, the results of the EGRACH model identifies the presence

of asymmetry in volatility in the selected return series. But this model cannot provide information

whether good news or bad news that increases or decreases the volatility. However, this feature

of volatility modelling is captured by TGARCH model.

TGARCH (1, 1)

Table 4 shows the estimates of coefficients using TGARCH (1, 1) model for the index Nifty 50 with

the given data. In this model, good news, 𝜀𝑡−1 > 0 and bad news,𝜀𝑡−1 < 0 have differential effects

on the conditional variance.

Table 4: TGARCH (1, 1) estimates for Nifty 50

Source: Computed based on secondary data using Gretl ω: Constant in the model which

represents the long run average; α: The ARCH term which is first lag of the squared return,

represents news about volatility from the previous period; β: The GARCH term which is the first

lag of the conditional variance; γ: identifying “good news” and “bad news” have a different impact.

The good news has an impact of the factor α, while bad news has an impact α + γ. The value γ > 0,

indicates that the bad news increases volatility. This shows the presence of leverage effect in the

selected return series. Also the, value of Gamma (γ ≠ 0) not equal to zero, conveys that the news

impact is asymmetric. The estimated form of TGARCH model for Nifty 50 is:

𝜎𝑡2 = 0.0000037 + 0.0926 𝜀𝑡−1

2 + 0.4743𝜀𝑡−1 2 𝐼𝑡−1 + 0.8815 𝜎𝑡−1

2 (5)

It shows that the good news has an impact of 0.0926 magnitudes and the bad news has an impact

of 0.0926 +0.3817 = 0.4743 magnitudes in the Nifty 50. Thus, it is inferred that in this index, the

bad news increases the volatility substantially. Also, the time varying stock return volatility is

asymmetric. From the econometric analysis of GARCH family of models, it is evident that TGARCH

model outperforms the other GARCH models in estimation and prediction of the market volatility

for the return series considered for this study.


ω -0.36028 0.03440 -10.470 ~0.000

α 0.20009 0.01492 13.430 ~0.000

γ -0.11431 0.01101 -10.390 ~0.000

β 0.97602 0.00320 300.200 0.000


ω 0.0000037 0.0000014 2.671 0.007

α 0.09268 0.02130 4.337 ~0.000

γ 0.38173 0.14391 2.652 0.008

β 0.88153 0.02010 44.030 0.000




21

Conclusion:

In today’s scenario, volatility is calculated for various types of financial variables, such as stock

return, interest rate and exchange rate. Stock return volatility measures the variability of the stock

return around the average value of the return. More specifically it is the standard deviation of the

stock return. It is found that the volatility prevails in all the stock market around the world. Due

to the arrival of the new information which are available publicly or privately the expected value

of the stock change. This will result into either increase or decrease in the prices and thereby

volatility enters in the stock return. Volatility is natural phenomenon in the stock market but

excessive volatility is a matter of concern which arises due to the irrational behavior of the trader,

investor and lack of transparency in the operations of the stock market. The excessive volatility

may lead to loss of the investor’s life time savings and market traders insolvent. The preliminary

analysis of data set suggests that volatility in the Indian stock market is time varying in nature,

persist to form clusters and has a long memory process. These findings of the data characteristics

have been consistent with previous studies of Indian capital markets and justify the application

of GARCH type models for the case. The detailed analysis shows that the TGARCH (1,1) model

outperforms in estimating, predicting and forecasting the stock market volatility. Considering the

factors like ability to capture the Leverage effect, ability to distinguish between good news and

bad news (asymmetric effect) and forecast accuracy, TGARCH (1,1) model is found to be best

model in the study.

Reference:

1. Black, F., 1976. Studies of stock price volatility changes. In: Proceedings of the 1976

Meeting of the Business and Economics Statistics Section in American Statistical

Association. pp.177–181.

2. Bollerslev, T., 1986. Generalized autoregressive conditional heteroscedasticity. Journal of

Econometrics, 31, pp.307–327.

3. Engle, R. and Ng, V., 1993. Measuring and testing the impact of news on volatility. The

journal of finance, 48(5), pp.1749–1778.

4. Engle, R.F., 1982. Autoregressive Conditional Heteroscedasticity with Estimates of the

Variance of United Kingdom Inflation. Econometrica, 50(4), p.987.

5. Glosten, L. and Jagannathan, R., 1993. On the relation between the expected value and the

volatility of the nominal excess return on stocks. The journal of finance, 48(5), pp.1779–

1801.

6. Nelson, D.B., 1991. Conditional Heteroskedasticity in Asset Returns: A New Approach.

Econometrica, 59(2), p.347.

7. Zakoian, J.M., 1994. Threshold heteroskedastic models. Journal of Economic Dynamics and

Control, 18(5), pp.931–955.

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