iic paper on modelling the intra day activity in stock market

8/14/2019 IIC Paper on Modelling the Intra Day Activity in Stock Market

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INDIAN INSTITUTE OF MANAGEMENT CALCUTTA

WORKING PAPER SERIES

WPS No. 588/ March 2006

Modeling daily volatility of the Indian stock market using intra-day data

by

Ashok Banerjee & Sahadeb Sarkar

[email protected] [email protected]

Professors, IIM Calcutta, Diamond Harbour Road, Joka P.O., Kolkata 700 104 India


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Full Title: Modeling daily volatility of the Indian stock market using intra-day data

Authors:

Ashok Banerjee

Professor (Finance & Control)

Indian Institute of Management Calcutta

Diamond Harbour Road

Joka

Kolkata-700 104

INDIA

Email: [email protected]

Phone Number: 91-33-2467-8300

Fax Number: 91-33-2467-8307

Sahadeb SarkarProfessor (Operations Management)

Indian Institute of Management Calcutta

Diamond Harbour Road

Joka

Kolkata-700 104

INDIA

Email: [email protected]


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ABSTRACT

The prediction of volatility in financial markets has been of immense interest among

financial econometricians. Using data collected at five-minute intervals, the present

paper attempts to model the volatility in the daily return of a very popular stock market

in India, called the National Stock Exchange. This paper shows that the Indian stock

market experiences volatility clustering and hence GARCH-type models predict the

market volatility better than simple volatility models, like historical average, moving

average etc. It is also observed that the asymmetric GARCH models provide better fit

than the symmetric GARCH model, confirming the presence of leverage effect. Finally,

our results show that the change in volume of trade in the market directly affects the

volatility of asset returns. Further, the presence of FII in the Indian stock market does

not appear to increase the overall market volatility. These findings have profound

implications for the market regulator.

KEY WORDS: Volatility; GARCH models; MDH; half-life of volatility shock;

leverage.


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1. INTRODUCTION

The magnitude of fluctuations in the return of an asset is called its volatility. The

prediction of volatility in financial markets has been of immense interest among

financial econometricians. This interest is further rekindled by Bollerslev et al. (1994)

when they established that financial asset return volatilities are highly predictable. It is

true that unlike prices, volatilities are not directly observable in the market, and it can

only be estimated in the context of a model. However, Andersen et al. (2001) concluded

that by sampling intra-day returns sufficiently frequently, the realized volatility

(measured by simply summing intra-day squared returns) can be treated as the observed

volatility. This observation has profound implication for financial markets (Brooks

1998) in that (a) the realized volatility provides a better measure of total risk (value at

risk) of financial assets, and (b) it can lead to better pricing of various traded options.

It has been observed in early sixties of the last century (Mandelbrot, 1963) that stock

market volatility exhibits clustering, where periods of large returns are followed by

periods of small returns. Later popular models of volatility clustering were developed by

Engle (1982) and Bollerslev (1986). The autoregressive conditional heteroskedastic

(ARCH) models (Engle, 1982) and generalized ARCH (GARCH) models (Bollerslev,

1986) have been extensively used in capturing volatility clusters in financial time series

(Bollerslev et al., 1992). Using data on developed markets, several empirical studies

(Akgiray, 1989; West et al., 1993) have confirmed the superiority of GARCH-type

models in volatility predictions over models such as the nave historical average,

moving average and exponentially weighted moving average (EWMA). GARCH

models can replicate the fat tails observed in many high frequency financial asset return

series, where large changes occur more often than a normal distribution would imply.


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Financial markets also demonstrate that volatility is higher in a falling market than it is

in a rising market. This asymmetry or leverage effect was first documented by Black

(1976) and Christie (1982). Three popular GARCH formulations for describing this

asymmetry are Power GARCH model (Ding et al., 1993), Threshold GARCH model

(Glosten et al., 1993) and Exponential GARCH model (Nelson, 1991). We have used all

these three models in the present study.

Empirical results also show that augmenting GARCH models with information like

market volume or number of trades may lead to modest improvement in forecasting

volatility (Brooks, 1998; Jones et al, 1994). The association between stock return

volatility and trading volume was analyzed by many researchers (Karpoff, 1987). The

initial research on price-volume relation can be attributed to Osborne (1959) who

attempted to model stock price change as a diffusion process with the variance

dependent on the number of transactions. Later research on the empirical relationship

between daily price volatility and daily trading volume was based on Clarks (1973)

mixture of distribution hypothesis (MDH). The essence of MDH is that if the stock

return follow a random walk and if the number of steps depends positively on the

number of information events, then stock return volatility over a given period should

increase with the number of information events (e.g., trading volume) in that period. In

a recent study on individual stocks in the Chinese stock market, Wang et al. (2005)

showed that inclusion of trading volume in the GARCH specification reduces the

persistence of the conditional variance dramatically, and the volume effect is positive

and statistically significant in all the cases for individual stocks. However, another study

on the Austrian stock market (Mestel et al., 2003) found that the knowledge of trading

volume did not improve short-run return forecasts. Most of the studies on the


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relationship between return volatility and trading volume have used volume levels.

However, few studies (Ying, 1966; Mestel et al., 2003) have used a relative measure

(turnover ratio) for volume. We have used a relative measure of volume in the present

paper. The present paper also uses trading volume of foreign institutional investors as

another exogenous variable.

There have been a few attempts to model and forecast stock return volatilities in

emerging markets. For example, Gokcan (2000) finds that for emerging stock markets

the GARCH(1,1) model performs better in predicting volatility of time series data. In

another market specific study, Yu (2002) observes that the stochastic volatility model

provides better volatility measure than ARCH-type models. A few studies were

conducted (e.g., Varma, 1999 and 2002; Kiran Kumar and Mukhopadhyay, 2002; Raju

and Ghosh, 2004; Pandey, 2005; Karmakar, 2005) on modeling stock return volatility in

the worlds largest democracy, India. Varma (1999) showed, using daily data from

1990-1998 of an Indian stock index (Nifty), that GARCH (1, 1) with generalized error

distribution performs better than the EWMA model of volatility. In a later study,

Pandey (2005) showed that extreme value estimators perform better than the conditional

volatility models. In another recent study, Karmakar (2005) used conditional volatility

models to estimate volatility of fifty individual stocks and observed that the GARCH

(1,1) model provides reasonably good forecast. However, none of the studies, based on

Indian stock markets, attempted to fit a mean equation for the stock return series before

modeling volatility of stock returns. The present paper determines the best-fit mean

model for the index return, which is then used in GARCH model specifications.

The present paper attempts to model the daily volatility, using high frequency intra-

day data, in the stock index return of a very popular stock market in India, called the


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National Stock exchange (NSE). The financial markets in India have gone through

various stages of liberalization that has increased its degree of integration with the world

markets. Some instances of new policy reforms introduced in the Indian stock markets

include introduction of trading in index futures in June 2000, trading in index options in

June 2001, trading in options on individual securities in July 2001, introduction of VaR

(value at risk)-based margin, and introduction of the T+2 settlement system from April,

2003. After implementation of such reforms, the Indian securities market has now

become comparable with securities markets of developed and other emerging

economies. In fact, India has a turnover ratio, which is comparable with that of other

developed markets and also one of the highest in the emerging markets (NSE, 2005).

These developments in the Indian securities market have drawn attention of researchers

from across the globe to look at the price behaviour of the Indian securities market. The

daily gross activity (purchase and sales) of the Foreign Institutional Investors (FII) in

the Indian stock market has increased almost three-fold in three and half years, namely,

from Indian Rupees (Rs.) 6 billion in October 2000 to Rs. 17 billion by the end of

January 2004. The increasing interests of foreign investors in the Indian market call for

greater research on various properties of this market. The present paper examines the

evidence of stylized properties in the Indian stock market. The findings of this study

would greatly help fund managers have a better understanding of the Indian stock

market volatility.

This paper also attempts to forecast volatility using various competing models and

measure their predictive power using standard evaluation measures. Results show that

the Indian stock market experiences volatility clustering and GARCH-type models

predict the market volatility far better than the simple volatility models, like historical


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average, moving average, EWMA etc. It is also observed that the asymmetric variants

of GARCH models give better fit than the symmetric GARCH model, confirming the

presence of the leverage effect. Finally, our results show that the change in volume of

trade in the market directly affects the volatility of asset returns. However, we did not

find similar convincing evidence in support of the FII investment affecting market

volatility.

The remainder of the paper is organized as follows. The next section discusses the

data structure and methodology used in the study. The third section briefly describes

various competing volatility models. The fourth section presents the results and

interpretations and the final section summarizes the findings and indicates scope for

further research.

2. DATA AND METHODOLOGY

The Indian capital market has witnessed significant regulatory changes since 1992 with

the creation of an independent capital market regulator, the Securities and Exchange

Board of India (SEBI). Subsequent changes (e.g., screen based trading, derivatives

trading, trading cycles etc.) have further developed the market and brought it in line

with international capital markets. Presently only two exchanges in India, the NSE and

the BSE (Stock Exchange, Mumbai) provide trading in the security derivatives. We

have used the most popular index of the National Stock Exchange in India, called S&P

CNX NIFTY (Nifty) to model the volatility in the Indian capital market. The Nifty, a

market capitalization weighted index, is an index of 50 scrips accounting for 23 sectors

of the Indian economy. Although the BSE, the oldest stock exchange in India (and also

in Asia), has been in existence for more than 100 years, the reason for choosing the


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Nifty is its increasing popularity. The BSE was established in 1875 and the NSE started

its capital market (equities) segment in late 1994. However, the market capitalization of

this segment in March 2000, before the launch of stock index futures contracts, was

Rs.10204 billion as against the BSEs Rs. 9128 billion. The average daily turnover

during May 2005 was around Rs. 40 billion in the NSE as against Rs. 20 billion in the

BSE.

Our sample contains a total of 60,631 data points consisting of the Nifty values at

five-minute intervals from 01 June 2000 through 30 January 2004. The choice of the

period is guided by the fact that a lot of policy reform initiatives in the Indian securities

market have started during this period. For example, trading on index futures was

allowed in India since June 2000. High-frequency data have now become a popular

experimental bench for analyzing financial markets (Dacorogna et al., 2001). High

frequency data are direct information from the market. Hence, instead of using the daily

closing value of the index, this paper uses the directly observable data. It cannot be

denied that very high frequency data have microstructure effect (e.g., how the data are

transmitted and recorded in the data base). In order to avoid serious microstructure

biases and at the same time reduce the measurement error due to data generation at low

frequency; we have used data at regularly spaced five-minute intervals (Andersen et al.,

2001). The present paper uses index values rather than stock prices and thus there are no

bid and ask prices. We have used the last quoted value of the index at five-minute

intervals. The daily index return is estimated using these five-minute interval values.

A large part of the data set, from June 01, 2000 through December 16, 2003, is used

to model volatility using various established volatility models. The remaining data set,


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from December 17, 2003 through 30 January, 2004, is used to test the efficacy of

various models using one-day-ahead volatility forecasts.

Let ,,,1, tti mir L= denote log of price relatives at an intra-day time-point i on day t,

where tm is the number of return observations obtained by using prices m times per

day. Then daily return on day tis calculated as ==tm

i titrr

1. Following Andersen et al.

(2001), we define the daily realized volatility (2

tV ) as the sum of squares of returns

collected at 5-minute intervals:

==tm

itit rV

1

22

(1)

For example, on June 01, 2000, the first 5-minute price observation was at 9.55AM and

the trading on that day ended at 3.25PM, giving us m = 67 prices and hence tm = 66

return observations. So, the realized volatility for day 1 is computed as ==66

1

21

21 i i

rV .

Let tV denote the positive square root of2

tV . The realized volatility represents a natural

approach to measure actual (ex post) realized return variation over a given period of

time (Andersen et al., 2005).


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Fig.1:Daily Realized Volatility

00.0020.0040.006

0.008

Date

20001122

20010518

20011109

20020509

20021031

20030505

20031025

Real-vol

We have computed 913 daily return ( tr) and realized volatility (2

tV ) values using

the 5-minute interval intra-day data. The first 883 daily return points are used to model

Nifty volatilities and the remaining 30 return points are used for volatility prediction

using various models. The realized volatility series (Fig.1) shows that the volatility in

the Nifty has significantly reduced over a period of time. One of the reasons for the

decline in market volatility could be the launch of stock index futures in June 2000.

However, some sharp spikes can be seen during 2001. The maximum daily volatility

was around 0.00576, which happened in March 2001. India witnessed multi-billion

rupees stock market scam in March 2001 which led to a freeze on the flagship scheme

of Indias largest mutual fund (Unit Trust of India) in June 2001. The market witnessed

a 15% decline in value in just one month! This aberration could prompt any researcher

(Alexander, 2001) to remove outliers and then model volatility. However, it cannot be

denied that stock market scams are results of systemic failure and hence these may

recur. Therefore, any robust volatility model should be able to capture this phenomenon

and hence we have not removed these extreme observations from our sample.


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In order to test whether the volume of trade influences the volatility of the Indian

stock market, we have included traded volume in the variance equation of competing

GARCH models. The volume tX1 , at time t, is defined as the change in the daily traded

volume:

111 /)( = tttt vvvX ,

where v = traded volume measured in Rs. billion. Similarly, we have used FIIs

activities in the Indian stock market to capture their influence, if any, on the market

volatility. The FIIs have, in the recent past, shown tremendous interest in the Indian

stock markets. These investors regularly participate in the Indian capital market by

buying and selling large volume of stocks. The FIIs are currently net buyers in the

Indian stock market on most of the days. We have collected the daily purchase and sale

volume of FIIs in the stock market from the official website of the capital market

regulator in India (SEBI). In order to capture the activity of the FIIs in India, we have

considered the daily gross trade (sum of purchase and sale volume). We did not have

data on the FII trade for all 883 days that we have considered for GARCH models

without exogenous variable(s). We have used a total of 798 data points on FII trade data

from October, 2000 through mid-December, 2003. In other words, while the other

GARCH models (Table 4) use 883 data points, the models with FII trade (as an

exogenous variable in the variance equation) use 798 data points. In the conditional

variance equation of competing GARCH models, the FIIs daily gross trade

volume tX2 , at time t, is defined as follows:

)log(2 ttX =

where t = sum of purchase and sales volume. To compare predictive power of the

competing volatility models, we have predicted daily conditional volatility for thirty


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days forward. Under each model, all the associated parameters are estimated each time

the day-ahead forecast is made. The first day-ahead forecast is made using parameter

estimates from Table 4. The sample is then rolled forward by removing the first

observation of the sample and adding one to the end to get the next day-ahead forecast.

This process is repeated to calculate the subsequent day-ahead forecasts. Such daily

volatility forecasts are compared with the realized volatility for the designated thirty

days.

Let 2 t denote volatility forecast. One can assess the accuracy of the daily volatility

forecasts under a model by considering the simple linear regressions (Andersen et al.,

2005) ofyt on2 t :

ttt bay ++=2

where2

ttVy = or 2tr , and then computing the coefficient of determination

2R . The

model with the highest2

R value may be treated as the best model for predicting yt. On

the other hand, if the2

R value turns out to be generally higher for one choice ofyt, then

that choice (2

tV or

2

tr ) may be considered to be a better measure of observed volatility.

In addition to the regression-based framework, we have used the standard measures

of predictive power of a model the root mean square error (RMSE), the mean absolute

error (MAE), and the Theil-U statistic. They are defined as follows:

=

=

==

=== Kk K

kkkK

K

k

kkK

kkKKk kkK

VV

V

UTheilVMAEVRMSE 1

1

2221

1

1

2221

2211

2221

)(

)(

|,|,)(

3. VOLATILITY MODELS


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We have fitted various competing volatility models to the daily Nifty series, and

predicted the one-day-ahead volatility of the series for the last thirty days and compared

relative performance of the models in estimating the daily realized volatility.

Let

)|( 1= ttt FrE , )|( 12

= ttt FrV = )|)(( 12

ttt FrE (2)

where 1tF is the information made available up to time (t1). The return series { tr }

may be either serially uncorrelated or may have minor lower order serial correlations,

but it may yet be dependent. Volatility models are used to capture such dependence in a

return series. Generally, the conditional mean t of such return series { tr } can be

modelled using a simple time series model such as a stationary ARMA(p,q) model, i.e.,

ttt ar += , = = +=q

j jtj

p

i ititar

110 , (3)

where the shock (or mean-corrected return) ta represents a white noise series with

mean zero and variance 2a , and p, q are non-negative integers. From equations (2)

and (3) we have

)|( 12

= ttt FrV = )|( 1tt FaV (4)

The equation for t is called the mean equation for the tr , and that for2t its

volatility (or conditional variance)equation.A GARCH model expresses the volatility

evolution through a simple parametric function. To compute the predicted value 2 1 +T of

21+T at t= (T+1), parameters are first estimated by fitting an appropriate mean and

volatility equations jointly to the data upto time point t=T and then an appropriate

prediction formula used. The most commonly employed method of estimation is the

maximum likelihood estimation under certain regularity conditions and a detailed


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discussion on this is given in Andersen et al. (2005). Exogenous explanatory variables

such as the change in daily traded volume (X1) and the FII investment (X2) may be

included in the volatility equationto improve prediction accuracy. We now discuss the

models briefly.

Random walk: The random walk model is the simplest of the models considered

and it is given by: ttt += 2

12 , where t iis a white noise series.

Historical Average: The historical average model (Yu, 2002) is given by:

==1

1i

22

1

1 tit

t .

EWMA: The exponentially weighted moving average (EWMA) model is given by:

))(1()( 212

12

+= ttt V , where 10 is estimated by maximizing an appropriate

likelihood function (Hull, 2003).

Next we briefly discuss the popular GARCH-type models. The GARCH model

focuses on the time-varying variance of the conditional distributions of returns. The

GARCH and related models capture the stylized fact that financial market volatility

appears in clusters, where tranquil periods of small returns are interspersed with volatile

periods of large returns. The feature of volatility clustering or volatility persistence went

unrecognized in traditional volatility models such as the historical average, which

assumed market volatility to be constant.

GARCH: The volatility model for the tr or ta is said to follow a GARCH(m, s)

model (Bollerslev, 1986; Bollerslev et al., 1994) if

ttta = ,2

11

20

2jt

s

j j

m

i itita == ++= , (5)

where 0 > 0, i 0, j 0,


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= +),max(

)(sm

ii ii < 1, with i = 0 for i>m and j = 0 forj >s,

and }{ t is a sequence of iid random variables with mean 0 and variance 1, which is

often assumed to have a standard normal or standardized Student-tdistribution.

An exogenous explanatory variableXk may be included in the GARCH model. For

example, the GARCH(1,1) model can be augmented as

ttta = , ktttt Xa 12

112

1102

+++= (6)

Moving average based on half-life of the volatility shock: The high or low

persistence in volatility is generally captured in the GARCH coefficient(s) of a

stationary GARCH model. For a stationary GARCH model the volatility mean-reverts

to its long-run level, at the rate given by the sum of ARCH and GARCH coefficients,

which is generally close to one for a financial time series. The average number of time

periods for the volatility to revert to its long run mean level is measured by the half life

of the volatility shock and it is used to forecast the Nifty series volatility on a moving

average basis.

A covariance stationary time series { ty } has an infinite order moving average

representation of the form (Fuller, 1996)

=

=


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GARCH(1,1) process 2 112

1102

++= ttt a is done as follows. For this model the

mean-reverting form is given by

,))(()(11

22

111

22

++=

ttttuuaa

where )1/( 1102 = is the unconditional long-run level of volatility, and

)( 22 ttt au = is the volatility shock. The half-life of the volatility is given by the

formula (Zivot and Wang, 2002): halfL = )ln(/)ln( 1121 + .

EGARCH : A stylized fact of financial volatility is that bad news (negative shocks)

tend to have a larger impact on volatility than good news (positive shocks). However, a

GARCH model fails to respond differently to positive and negative shocks. Nelson

(1991) developed the exponential GARCH (E-GARCH) model, which corrects this

weakness by appropriately weighting innovation t .

|)](||[|)( tttt Eg += , (7)

where and are parameters. An EGARCH(m, s) model is of the form

ttta = , )g()1

1()ln( 1-t

1

10

2

ms

sst

BB

BB

++++=

L

L , (8)

where 0 is a constant,B is the back-shift or lag operator such that )())(( 1= tt ggB ,

and )1( 1s

sBB +++ L , )1( 1m

sBB L are polynomials in B. The EGARCH

model uses logged conditional variance to relax the constraint of model coefficients

being nonnegative, and through )( tg can respond asymmetrically to positive and

negative lagged values ofat. An alternative representation for EGARCH(m, s) model is

given by

2

1102 ln)|(|lnand, jt

s

j jitim

i ititttta == +++== ,


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where is are the leverage parameters.

TGARCH: Another GARCH variant which is capable of responding

asymmetrically to positive and negative innovations is the threshold GARCH

(TGARCH) model (Zakoian, 1991; Glosten et al., 1993). The TGARCH(m,s) model has

the form

ttta = ,2

11

2

1

20

2 )( jts

j jm

i ititim

i ititaSa == = +++= , (9)

where

iic paper on modelling the intra day activity in stock market

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