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5Model for Forecasting Volatility in Indian Stock Market: The Volatility Index 2008 The Icfai University Press. All Rights Reserved.

Model for Forecasting Volatility

in Indian Stock Market: The Volatility Index

At present, Indias economy looks upbeat with an increase in inve stment in the stock marke t. At the same

time, it is also seen that the market fluctuations are very high. Such high levels of fluctuations or volatility

make it diff icul t to predict the futur e expected values of the marke t and make investment deci sions.

Similarly, volatility is an important component for estimating the option price using the Black Scholes

model. In such a situa tion, having an estimate of futur e volatility is very useful. Simila r to the pric e indices

that are used as indicators of the overall market value and returns, volatility indices are used as indicators

of expected market volatility over a future period. Such indices exist in the US and the Europeancountries. The CBOE Volatility Index or VIX is the first volatility index of its kind providing volatility

forecas ts on a cont inuous basis for the Chicago Board o f Exchange. The present study looks into the aspec t

of construction of such an index in the Indian context. The study aims to construct an Indian Market

Volatility Index (IMVI) for the Indian stock market using the Nifty option series which are based on the

S&P CNX Nifty Index. The study also aims to validate the predictive properties and effectiveness of

IMVI. The results of this study are expected to offer guidance for the future researchers in the area of

marke t microstructure. Though this study focuses on cons titut ing the index on a daily bas is, the index

can be further developed for continuous prediction.

Jadhav Aditya Mohan* and Chakrapani Venkata Chaturvedula**

Introduction

Volatility is the tendency for the prices to change with respect to new information. Price

changes occur due to access to new information regarding values of the underlying asset

or due to the demand for liquidity by impatient traders. In the areas of investment, security

valuation, risk management, and financial intermediation, forecasting volatility is the

central idea of research, because to assess investment risk, a good forecast of the volatility

of asset price serves as a key input. Volatility affects the investment returns as well as the

risk from the investment. Excessive volatility in the market indicates that there is an

erratic demand and supply in the market and hence the markets are not functioning well.

(Harris Larry, 2002)

Black Fischer and Myron Scholes (1973) valued the European option contracts on

non-dividend paying stocks on the basis of five variablesspot price, strike price, interest

rate, time to maturity, and volatility of underlying asset prices. The only input in the

framework of an option pricing, which cannot be directly observed by a trader, is volatility.

Over the years, historical volatility was used in place of expected volatility in valuing

option contracts (Hull, 2003). Historical volatility is induced by new information or

* Research Scholar, The Icfai Institute for Management Teachers, Hyderabad, India. E-mail: [email protected]

** Faculty Member, The Icfai Business School (IBS), Hyderabad, India. He is the corresponding author.

E-mail: [email protected]

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The Icfai Journal of Applied Finance, Vol. 14, No. 7, 20086

liquidity demand in the market at that specific point of time and is expected to die down

with time as it is transitory in nature. Hence, using historical volatility value in valuingoption contracts may not give proper value of the options.

In practice, the volatility implied in the option can be calculated with the help of

backward induction and iteration technique using the Black Scholes model. This value of

volatility is known as implied volatility. The value or price of the option, spot price,

strike price, interest rate, and time to maturity are known from the market. In such a case,

by inserting some expected value of implied volatility, the theoretical option value can be

calculated using the Black Scholes model. The value of implied volatility at which the

theoretical call option value converges with the market value of the call option is the

implied volatility in the call option. (Hull, 2003)

The implied volatility is often interpreted as market expectation of volatility over theoptions maturity for the underlying asset (Poon and Granger, 2003). The first volatility

index using implied volatility was developed by the Chicago Board of Exchange (CBOE)

based on the Dow Jones Index. This index is known as CBOE DJIA Volatility Index

(VXD). The index is the weighted average of eight different OEX option series. It is

calculated in such a way that at any point of time it forecasts the stock market volatility

over the next 30 calendar days. (Whaley, 1993)

The S&P CNX Nifty index is a good indicator of the Indian stock market. Similarly,

the Nifty option series which uses the S&P CNX Nifty index as the underlying asset is

one of the most traded and liquid options and represents the market consensus regarding

the future values of the stock market. The main objective of this study is to compute theimplied volatility of the Nifty option series and use it to predict the market volatility for

a specific period in the future.

Literature Review

Studies looking into the predictive ability of implied volatility include Chiras and

Manaster (1978); Lamoureux and Lastrapes (1993); Xu and Taylor (1995); Vasilellis and

Meade (1996); Blair et al. (2001); Edrington and Guan (2002); and Poon and Granger

(2003) among many others.

Whaleys (1993) study was the first one which discussed the construction of the

volatility index. It explained the construction of VIX in detail. VIX is a weighted averageof the implied volatilities of eight near-the-money nearby and second nearby OEX option

series. The implied volatilities are weighted in such a way that VIX represents the implied

volatility of a hypothetical 30-calendar-day at-the-money OEX option. The model

proposed by Whaley is:

12

12

12

21

2222

,t,t

,t

,t,t

,t

NN

N

NN

NVIX

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7Model for Forecasting Volatility in Indian Stock Market: The Volatility Index

where 1

is the weighted implied volatility of nearby option series, 2

is the weighted

implied volatility of second nearby option series,Nt,1 is the time to expiry of nearby optionseries, and N

t,2is the time to expiry of the second nearby option series.

Fleming et al. (1995) explain the construction of VIX in detail as well as investigate

the statistical properties of VIX and its predictive powers. They have analyzed VIX for its

univariate properties, its relationship with the stock index returns, and its ability to

forecast the realized volatility for the period. They analyzed VIX on a daily basis for a

period of 7 years from 1986 to 1992. VIX shows a negative autocorrelation at lag-1 and

negative correlation with index returns indicating its competency as a volatility index.

They also conclude that implied volatility index (VIX) performs well as a forecast for

future realized stock market volatility. Whaley (2000) has also stated that VIX reflects the

consensus view of investors regarding the expected future stock market volatility.

Christensen and Prabhala (1998) analyzed the relationship between implied andrealized volatility. They analyzed the implied volatility in OEX option series over a period

of 12 years for its ability to predict the realized volatility over the life of the option.

The study revealed that implied volatility has higher means and lower variance than

realized volatility indicating that implied volatility is a smoothed expectation of realized

volatility. They used the two-stage least squares regression model to analyze the predictive

properties of implied volatility to overcome the errors-in-variable problem that occurs in

a single equation model. The study concluded that implied volatility is able to predict the

future realized volatility in isolation as well as in conjunction with the history of past

realized volatility.

However, the study by Maheshwaran and Ranjan (2006) has contradictory observations

to those mentioned above. The study was aimed to understand the ability of the impliedvolatility in Nifty options to predict the realized volatility. Their study estimated the

implied volatility for the Nifty call option series and Nifty put option series and

analyzed their capability to predict the realized volatility separately. The study also

compared the information content of the implied volatility of Nifty options with the

three other Asian indices, viz., Hang Seng (Hong Kong), KOSPI (South Korea), and

TWSEW (Taiwan). While the R-square for the regression equations for India is 3%, it is

7% for Korea, 38% for Hong Kong, and 39% for Taiwan. Though the coefficient for implied

volatility is not significant for India, it is significant for others. The study concluded that

the implied volatility is a poor and biased estimator of realized volatility in the Indian and

South Korean markets but useful in the other two marketsHong Kong and Taiwan.

Though Maheshwaran and Ranjans (2006) study puts forth the inability of the Niftyoption series to predict the realized volatility, this exercise has not been conducted for an

index as such. Similarly, other studies support the construction of an implied volatility

index to forecast the realized volatility in the market. Therefore, this study aims to

construct such an index for the Indian market.

Data

The study is conducted in two parts. The first part discusses the modeling of the volatility

index for the Indian stock market using the implied volatilities of Nifty option contracts.

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The second part deals with the analyses regarding the statistical and predictive properties

of volatility index.

The analysis is conducted on daily frequency for a period of four and half years from

June 6, 2001 to November 18, 2005. The daily near-the-money call and put option series

data regarding the strike price, premium, and period to maturity is collected from

bhav-copies provided by the NSE. The interest rate on the 91-day Treasury bill is

considered as the risk-free interest rate and the daily closing prices of the Nifty index are

used to compute the realized volatility. The daily closing prices of Nifty were collected for

the period from the CMIE PROWESS database. Days when the market crashed were

identified and the data for those periods has not been considered while conducting the

analysis. Due to the illiquid derivatives market, out of the 1,089 trading days, data for only

521 days was used for analysis.

Methodology

The Indian Market Volatility Index (IMVI) was calculated using the implied volatilities

of four nearby at-the-money or near-the-money Nifty option series which are derived

using the S&P CNX Nifty index as the underlying asset. The four option contract series

consist of two nearby call option contracts and two nearby put option contracts. The two

call option contracts are selected such that one has the exercise price (XL) just below the

index value (S) and the other having exercise price (XU) just above the (S). Similarly, the

put option contracts are selected such that one has the exercise price (XL) just below the

index value (S) and the other having exercise price (XU) just above the (S).

Implied volatilities for these four option series were calculated using the Black Scholes

option pricing model. Table 1 presents the details of these four option contracts.

Table 1: Four Nearby Near-the-Money Nifty Option Series Considered for IMVI

Option CategorizationNearby Series

Call Put

SXL Strike PriceC

,LX 1P

,LX 1

Implied Volatility LX

,C 1LX,P 1

SXU Strike Price C,UX 1 P ,UX 1

Implied Volatility UX

,C 1UX,P 1

The implied volatilities for the near-the-money call option series and put option series

both having the strike price (XL) just below the spot price (S) are averaged as:

2

111

LL

L

X,P

X,CX

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Similarly, the implied volatilities for the near-the-money call option series and put

option series both having the strike price (XU) just above the spot price (S) are averagedas:

2

111

UU

U

X,P

X,CX

Finally, the IMVI is computed as the weighted average of these implied volatilities.

The at-the-money implied volatility for nearby contract is calculated by interpolating the

near-the-money implied volatilities of low strike price nearby option series and high strike

price nearby option series. The model for IMVI is:

LU

LX

LU

UX

XX

XS

XX

SX

IMVIUL

11

The implied volatility was computed for options having at least 8 calendar days or

6 trading days to expiry. In case any option series has less than 8 days to expiry, the second

nearby contract was used for computing the IMVI. This enables the index to predict the

volatility for at least the next 8 days at any point of time.

Results and Analysis

IMVI was analyzed for its statistical and predictive properties in three phases.

The analysis has been conducted for the complete period (June 2001 to November 2005)

first and then the period under analysis has been divided into two parts and IMVI was

analyzed separately for both the periods. The first period is from June 2001 to December2002 and the second period is from January 2003 to November 2005.

The realized volatility for the period (T) from the current day (t) to the day of expiry

of option over which the implied volatility is estimated is calculated as the standard

deviation of the index returns for that period.

21

1

2

1

1

T

i

Tit RRT

V

The series of IMVI, index returns, and realized volatility were tested for their

stationarity property using the Augmented Dickey-Fuller test. The Augmented Dickey

Fuller statistic for all the three series for the three periods under analysis was lower than

the tabulated statistic suggesting that all the series were stationary.

IMVI and S&P CNX Nifty index returns were analyzed for univariate measures such

as mean, variance and standard deviation, and autocorrelation at the first, second, and

third lag were estimated. Table 5 provides these results. The mean and standard deviation

of IMVI are slightly lower than that of realized volatility for all the three periods.

This indicates that IMVI is biased on the lower side of the realized volatility and as it is

a smoothened estimate of the realized volatility.

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Cross correlations between IMVI and S&P

CNX Nifty returns at various lags (from +2to 2) were estimated. The volatility index is

expected to have a negative correlation with the

contemporary stock market returns when the

stock prices are expected to fall as the volatility in

the market increase. This is the result of the

higher returns demanded by the investors in the face of high volatility and uncertainty.

Table 2 presents the cross-correlation analysis between IMVI and Index Returns.

As IMVI shows a continuous negative correlation with the index returns up to two lags

on both sides, we can say that IMVI shows the requisite characteristics necessary for a

volatility index.

Christensen and Prabhala (1998) analyzed the predictive capability of the implied

volatility from the OEX option series using the two-stage least square regression method.

The performance of IMVI as a forecast of realized volatility was analyzed using a similar

two-stage least square regression method.

The first stage of the two-stage least square regression method involves conducting a

regression with implied volatility (I) as a function of the lag values of implied volatility

and lag (1) value of realized volatility (V).

'tt

't

''t eVII 1211

The second stage of the two-stage least square regression method involves conductinga regression with realized volatility (V) as a function of the implied volatility which is

estimated from the first stage of the two-stage least square regression method and the lag

values of the realized volatility.

tttt eVIV 1431

The lag values of IMVI in the first stage and lag values of realized volatility in the

second stage were included to remove the autocorrelation existing in these equations.

In case IMVI has information content than the coefficient should be significant and good

forecast will have coefficient value almost 1 for 3

and the values for 1

and 4

should

be insignificant or almost equal to zero.

Table 3 provides the results of the two-stage least squares regression model for the

complete period from June 2001 to November 2005.

It is noted that though the coefficient of IMVI in the second stage of the regression

process is significant, the value of the coefficient is very low (0.1542) indicating that IMVI

has very low information content regarding the realized volatility and is not an efficient

estimator of the realized volatility. These results are in line with those of Maheshwaran

and Ranjan (2006).

Table 2: Cross-Correlation Analysis

between IMVI and Index ReturnsLag Negative Positive

0 0.1842 0.1842

1 0.0892 0.1885

2 0.0085 0.0857

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Table 3: Two-Stage Least Squares Regression Model (Full Period)

Stage I: Dependent Variable: IMVI

Variable Coefficient Std. Error t-Statistic Prob.

C 0.009482 0.000754 12.576590 0.0000

RELVOL(1) 0.142966 0.052703 2.712682 0.0069

AR(1) 0.585587 0.036421 16.078180 0.0000

R-squared 0.406588 F-statistic 176.7740S.E. of Regression 0.004041 Prob (F-statistic) 0.0000

Stage II: Dependent Variable: RELVOL


C 0.010035 0.000900 11.15152 0.0000

IMVIEST 0.154169 0.031587 4.880712 0.0000

AR(1) 0.840788 0.024010 35.01812 0.0000

R-squared 0.731939 F-statistic 705.8320

S.E. of Regression 0.003010 Prob (F-statistic) 0.0000

While Table 4 provides the results of the two-stage least squares regression model for

the first period from June 2001 to December 2002, Table 5 furnishes the results for the

second period from January 2003 to November 2005.

Table 4: Two-Stage Least Squares Regression Model (First Period)



C 0.008425 0.000979 8.602778 0.00000

RELVOL(1) 0.102908 0.087697 1.173445 0.24230

AR(1) 0.329908 0.072435 4.554528 0.00000



Stage II: Dependent Variable: RELVOL


C 0.009084 0.00139 6.521266 0.00000

IMVIEST 0.058047 0.03996 1.452431 0.14820

AR(1) 0.858782 0.03903 22.002140 0.00000



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Table 5: Two-Stage Least Squares Regression Model (First Period)



C 0.011186 0.001024 10.91878 0.00000

RELVOL(1) 0.067008 0.061965 1.081398 0.28030

AR(1) 0.681006 0.040494 16.817320 0.00000



Stage II: Dependent Variable: RELVOL2


C 0.010089 0.001091 9.245417 0.00000

IMVIIMP3 0.223652 0.044625 5.011845 0.00000

AR(1) 0.818799 0.031280 26.176280 0.00000



Table 6 provides the univariate analysis of IMVI, corresponding market returns, and the

corresponding realized volatility values.

Property IMVI Realized Volatility Index Returns

Full Period (June 2001 to November 2005)Mean 0.01116 0.01178 0.00041

SD 0.00524 0.00580 0.01374

Autocorrelations

Lag 1 0.626 0.848 0.10800

Lag 2 0.518 0.757 0.00600

Lag 3 0.465 0.687 0.00500

Partial Autocorrelations

Lag 1 0.626 0.848 0.10800

Lag 2 0.206 0.136 0.01800

Lag 3 0.134 0.063 0.00800

I Period (June 2001 to December 2002)

Mean 0.0096 0.0098 0.00134

SD 0.0045 0.0049 0.00988

Autocorrelations

Lag 1 0.322 0.856 0.04700

Lag 2 0.186 0.734 0.17400

Lag 3 0.103 0.586 0.00100

(Contd...)

Table 6: Univariate Analysis of IMVI, Realized Volatility and Nifty Index Returns

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Table 6: Univariate Analysis of IMVI, Realized Volatility and Nifty Index Returns

(...contd)Partial Autocorrelations

Lag 1 0.322 0.856 0.04700

Lag 2 0.092 0.004 0.17600

Lag 3 0.021 0.163 0.02000

II Period (January 2003 to November 2005)

Mean 0.01195 0.01273 0.00005

SD 0.00541 0.00595 0.01527

Autocorrelations

Lag 1 0.700 0.832 0.11800

Lag 2 0.592 0.743 0.02600

Lag 3 0.548 0.692 0.00400

Partial Autocorrelations

Lag 1 0.700 0.832 0.11800

Lag 2 0.200 0.165 0.01200

Lag 3 0.155 0.130 0.00100

Note: Values in bold are not significant at 95% confidence level.

The lagged realized volatility is a poor estimate of the implied volatility during the firstperiod (R-square of 12.52% in Stage I). On the other hand, lagged realized volatility is agood estimate of implied volatility for the second period which is in line with the analysisof the full period. Similarly, the coefficient of IMVI in Stage II of the regression processis not significant for the first period but is significant for the second period. It is also notedthat though the coefficient of IMVI in Stage II of the regression process is significant inthe second period, the value of the coefficient is very low (0.2236) indicating that IMVIhas low information content regarding the realized volatility and is not an efficientestimator of the realized volatility. These results are in line with those of the completeperiod and also with Maheshwaran and Ranjan (2006).

Conclusion

The study shows that IMVI and hence the implied volatility has low information contentregarding the realized volatility in the Indian market. This can be attributed to the lack ofliquidity in the options market which has started only in 2001. The option trading is notas frequent as the spot market trading in India which is reverse in the case of the Westerncountries. The option is not traded at high frequency which reflects in the available datapoints for analysis. This indicates that the market efficiency of the derivatives market is verylow or the market is not efficient (Christensen and Prabhala, 1998). While the impliedvolatility index is effective in the Western bourses like CBOE, it is ineffective in India. Thiscan be attributed to the idea that option pricing contains a market risk premium not onlyon the asset itself but also on the volatility of the asset prices. Hence, implied volatilitymight be a poor estimate of realized volatility in India (Maheswaran and Ranjan, 2006). But,at the same time, it can be noted that the coefficient has increased from the first period tothe second period and the effectiveness of implied volatility to forecast realized volatility in

the future when the markets become more efficient cannot be ruled out.

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Acknowledgment: The authors would like to thank A V Vedpurishwar and seminar participants at

International Conference on Business and Finance 2008, conducted by The Icfai Business School,Hyderabad and Bentley College, Boston, for their valuable suggestions and comments.

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