Testing Volatility for Selected Indian Indices
By
A.MUTHUSAMY
Associate Professor
Department of International Business and Commerce
Alagappa University
Karaikudi603003
EMil:
muthuroja67@rediffmail.com
Phone: + (91) 9042736251
and
S.VEVEK
M.Phil Research scholar
Department of International Business and Commerce
Alagappa University
Karaikudi603003
Email:
vevekpdy@gmail.com
Phone: + (91) 9843030935
Testing Volatility for Selected Indian Indices
MUTHUSAMY. A
[1]
VEVEK. S
[2]
Abstract
Volatility can be estimated in two ways viz., historical and implied, whereas historical volatility shows the past and the implied volatility shows the
future movements of the market. Estimation of implied volatility (IV) will be helpful for the market participants including hedgers, arbitrageurs and
speculators. The study attempts to approximate IV of Index options of National stock exchange of India using three different models viz. the
CorradoMiller’s Model (1996), BrennerSubrahmanyam’s Model (1988) and BharadiaChristopherSalkin’s Model (1996) and to estimate the market behaviour
by comparing calculated IV with the India VIX on four option index namely S&P CNX Nifty options, Nifty Midcap 50 options, Bank Nifty options and
CNX IT options. Near month Atthemoney contract were chosen for the period of five years and ten months from 2^{nd} March 2009 to 31 ^{st} December 2014 for all the four indices. The findings of the study reveal that in most of the contracts the calculated IV is different
from the India VIX. The market participants could make their investment strategies based on the calculated volatility appropriately (whether
underpriced or overpriced). The speculators especially are interested in the volatility trading this could help them in a big way to position themself
in the market movement.
Keywords:
Implied volatility (IV), India Volatility Index (VIX), CorradoMiller’s Model, BrennerSubrahmanyam’s Model, BharadiaChristopherSalkin’s Model
Introduction
The capital market is an incorrigibly uncertain. This uncertainty makes the investors in blues. There are few adept investors who win even in the
sudden mishap of capital market. The adept investors do hedge, arbitrage and speculates by measuring volatility. The investors’ guard their securities
from the volatility through the help of derivative instruments such as Futures, Options, Forward and SWAPS. In the derivatives market future price can
be revealed due to the estimation of expected increase in the price level of stock or index or commodity. The derivatives provide the investor to
anticipate the shortterm risk. Among the derivative instruments, Options provide leverage and right to execute according to the will of writer and
holder, apart from these Options market allows estimating implied volatility. The volatility is of two types historical and implied. Whereas,
historical volatility deals with standard deviation of historical return of index return or stock return. The implied volatility is the market’s
assessment of underlying asset’s volatility, as reflected in the Options pricing. Through the implied volatility probability of gauge in near future
movement is possible. We can see the difference between the historical and implied volatility, historical volatility provides the historical volatile
and the implied volatility provides exact pulse of the capital market but measuring the implied volatility is little complicated, actually by reversing
the famous benchmark model so called BlackScholesMerton (1973) Options pricing model (BSM) is used to estimate the implied volatility because of
its complicated calculations the academicians tried to provide the simple calculation to estimate the implied volatility. There are several other
models exists which approximates the implied volatility among them the CorradoMiller’s Model (1996), BrennerSubrahmanyam’s Model (1988) and
BharadiaChristopherSalkin’s Model (1996) were used to approximate implied volatility for atthemoney contract. Similar to this study the BSu Model
(1988) used atthe money contract to estimate the implied volatility, to an extension of the BSu approach the BCS model deals with inor outof the
money and CM Model is also an extension of BSu approach. All these three models provide simplified volatility approximation approach for the
investors and the academicians.
Generally Volatility Index (VIX) is known as fear index. In India it is called as India VIX, it is based on Nifty Index Options Prices, calculated for
30 days of best bidask prices (Put) of Nifty Options. If more Put options purchased on Nifty the VIX will increase. Basically the Put Options were
purchased by the investor when the market is in turmoil. VIX measures the behavioural pattern of the investors. When the VIX is skyscraping buy the
Nifty, when VIX is near to the ground sell the Nifty or components of Nifty. The VIX act inverse to the market, there is always lead lag or lag lead
relationship between the VIX and the S&P CNX Nifty. The VIX is calculated using outofthe money contract.
For further understanding, IV means Implied Volatility, VIX is Volatility Index, CM Model is CorradoMiller Model, BSu Model is BrennerSubrahmanyam
Model and BCS Model is Bharadia and Christopher and Salkin Model.
In this paper an attempt was made to approximate implied volatility using three different models they are CM Model (1996), BSu Model (1988), BCS
Model (1996) and compared the each estimated implied volatility separately with the volatility index (Indian VIX).
Literature Review
The financial engineers and practitioners have been continuously trying to estimate the volatility and it became at most impossible to measure because
the variables of volatiles are unobservable. But through the help of variables available in BSM Options pricing the implied volatility can be
approximated. The relevance of calculating the IV using various models are tested and relevant modes has been suggested. Some of the important studies
in this area are discussed below:
A formula was built to find the European option price, the formula has a lognormal process and applied stochastic calculus to calculate Options price.
This study gave a significant impulse to the Options trade all over the world, because it developed a generally applicable method to calculate European
Options prices. Still now the BSM model is popular model to estimate the European Options price (Black, Scholes, Merton, 1973). An attempt is made to
develop more realistic Options pricing models. BSM formula and also checked with the BSM model and founded that the model is better in practical
implementation on European Options price (Gurdip Bakshi, Charles Cao, zhiwu Chen, 1997). Strike price biases in BS Options pricing model with small
errors in the riskfree rate and standard deviation proxies found that the small errors in the riskfree rate and standard deviation proxies can
produce the same systematic biases that empirical studies of the BlackScholes Options pricing model report. Using of implied volatility reduces the
standard deviation error (Jerry A. Hammer, 1989). BlackScholes pricing formula can be approximated in closedform for the strike price equals to the
future price of underlying asset. An interesting result is that the derived equation is not only very simple in structure but also that it can be
immediately inverted to obtain an explicit formula for implied volatility. The comparison was made on the accuracy of three approximation formulas,
through the analysis said that the first order approximations are close only for small maturities, P´olya approximations are remarkably accurate for a
very large range of parameters (Paolo Pianca, 2005). The solution for investors’ problem of Options pricing with classic BSM model was found by
finding the Gamma, strike price with timetomaturity, spot prices of Options and using BSM model an exact expression of implied volatility can be
calculated (Philippe Jacquinot and Nikolay Sukhomlin, 2010). Testing the accuracy of these approximation methods (BS Model) using call only and
putcall average elicitation of an implied volatility estimate and the results of the analysis conducted for approximations using averages from implied
volatilities derived from calls and puts were remarkably different (Olga IsengildinaMassa et al., 2007). However having analysed the three different
models (Chance’s (1993, 1996) model, Corrado and Miller’s (1996) model and Bharadia, Christofides and Salkin’s (1996) model for approximating implied
volatility, Corrado and Miller’s (1996) model is the best model even without an additional information. But, Chance's model, especially as extended in
study, has relatively simple and accurate for most cases (Donald R. Chambers and Sanjay K. Nawalkha, 2001). Identifying the implied volatility errors
in BS formula, when the Options price is away from the money, to measure the implied volatility errors the GLS (Generalized Least Squares) estimator
used to reduce the noise and bias in implied volatility calculation (Ludger Hentschel, 2003). The study on empirical performance of GARCH model for
Options pricing and comparing with volatility index (VIX)found that nonaffine models clearly outperform affine models (Juho Kanniainen et al., 2014).
There are studies justified that the Corrado & Miller (1996) model is the best alternative to estimating the implied volatility using BSM
framework, may be in the future improvements can be done (Winfried G. Hallerbach, 2004). The efficiency of S&P CNX Nifty index Options in Indian
securities market, reveal that implied volatilities do not hold all the information available in the past returns so these are indicative of the
violation of efficient market hypothesis in the case of S&P CNX Nifty index Options market in India (Alok Dixit et al., 2010). Understanding the
implied volatility with respect to macroeconomic announcements were examined and found that inthemoney and outofthe money Options have different
characteristics in their responses, leading to the conclusion that heterogeneity in investor beliefs and preferences affect Options implied volatility
through the state price density (SPD) function (Hassan Tanha et al., 2014). Testing the volatility smile with the core assumption of the Black–Scholes
Options pricing model with the Options data gives a classical Ushaped volatility. Indeed, there is some evidence that the “volatility smirk” which
pertains to 30day Options and also implied volatility remain higher for the shorter maturity Options and decrease as the timetoexpiration increases.
The results lead us to believe that inthemoney calls and outofthemoney puts are of higher volatility than atthemoney Options (Imlak Shaikh and
Puja Padhi, 2014). Therefore the study intends to approximate the IV with three different models viz. CM Model (1996), BSu Model (1988) and BCS
Model (1996) for the every near month atthemoney contract of Options index. The calculated IV is individually compared with the India VIX to identify
the best possible implied volatility measure.
Methodology
In order to study the best measure for Implied Volatility with the three classic models viz. CM Model (1996), BSu Model (1988) and BCS Model (1996)
secondary data comprises of four Options indices such as CNX Nifty, Midcap50, Bank Nifty and IT index Options for both call and put Options for the
period of 6 years from January 2009 to December 2014 daily options price data of nearmonth atthemoney contract of European Options includes S&P
CNX Nifty, CNX Midcap 50, Bank Nifty and IT index for the study. The implied volatility (IV) depends on several inputs from BSM Options pricing
model. Mumbai InterBank Offer Rate (MIBOR) is used as proxy for riskfree rate. The IV is calculated using three different models on daily Options
price, they are CorradoMiller’s Model (1996), BrennerSubrahmanyam’s Model (1988) and BharadiaChristopherSalkin’s Model (1996). The data source for
The Volatility Index (VIX) and other data required for the IV are downloaded from the NSE website (www.nseindia.com). MIBOR is downloaded from debt
segment of NSE. The calculated IV prices of three models are individually compared with the VIX using Independent sample ttest to identify which model
is best model to estimate VIX.
Models and formula used to calculate the implied volatility (IV)

The CorradoMiller’s (CM) Model (1996) is
Where, ; ;
C = [S * N (d_{1})] – [Ke^{rt} * N (d_{2})]; d1 = , d2 = d1 –
· The BrennerSubrahmanyam’s (BSu) Model (1988) is
· The BharadiaChristopherSalkin’s (BCS) Model (1996) is
The implied volatility (IV) depends on several inputs from BSM Options pricing model like Ln = Natural Logarithm, S = Spot price of the underlying
asset, K = Exercise or Strike price of the Options, r = Annual Risk free rate of return, t = Time to expiry of the Options, N = Cumulative standard
normal Distribution, e = Exponential term (2.7183), σ= Standard deviation of the continuously compounded annual rate of return of the underlying asset,
P = Theoretical price of Put option, C = Theoretical price of Option Price/ Call Price/ Premium Price. In the above formulas 2 refers to α. The CM
Model (1996) had originally suggested the value of α = 1.88 would be optimal, for simplicity settled on α = 2 in their equation.
Hypothesis of the Study
H_{0}^{1}: The means of the NIFTY’s IV and VIX are not significantly different.
H_{0}^{2}: The means of the Midcap50’s IV and VIX are not significantly different.
H_{0}^{3}: The means of the BANK’s IV and VIX are not significantly different.
H_{0}^{4}: The means of the IT’s IV and VIX are not significantly different.
ANALYSIS AND INTERPRETATION
In the study, IV is estimated through CM Model (1996), BSu Model (1988) and BCS Model (1996) for the every near month contract over a period of 6
years for 4 Options index and atthemoney contracts is chosen. The calculated IV is compared individually and with the Indian VIX using independent
sample ttest.
INDEPENDENT SAMPLES TTEST OF CNX NIFTY, MIDCAP50, BANK, IT FOR THE YEAR 2009 TO 2014
Table – 1A Group Statistics of CNX Nifty

Mode

N

Mean

Std. Deviation

Std. Error Mean

IV & CM Model

VIX

1442

22.5319

7.83084

.20622

IV

1442

20.4541

13.55058

.35684

IV & BSu Model

VIX

1442

22.5319

7.83084

.20622

IV

1442

24.4929

17.89648

.47129

IV & BCS Model

VIX

1442

22.5319

7.83084

.20622

IV

1442

25.0637

30.88056

.81321

Source: computed as per data taken from NSE
Table – 1B Independent Samples tTest of CNX Nifty

Levene's Test for Equality of Variances

ttest for Equality of Means

F

Sig.

t

df

Sig. (2tailed)

Mean Difference

Std. Error Difference

95% Confidence Interval of the Difference

Lower

Upper

IV & CM Model

Equal variances assumed

50.422

.000

5.041

2882

.000

2.07778

.41214

1.26965

2.88590

Equal variances not assumed



5.041

2306.911

.000

2.07778

.41214

1.26957

2.88598

IV & BSu Model

Equal variances assumed

149.487

.000

3.812

2882

.000

1.96108

.51443

2.96977

.95240

Equal variances not assumed



3.812

1973.280

.000

1.96108

.51443

2.96996

.95220

IV & BCS Model

Equal variances assumed

280.154

.000

3.018

2882

.003

2.53187

.83895

4.17687

.88687

Equal variances not assumed



3.018

1625.565

.003

2.53187

.83895

4.17741

.88634

Source: computed as per data taken from NSE
Table – 1A shows the CM Model mean of VIX is 22.5319 and the mean of IV is 20.4541, with the mean difference of 2.07778 which is insignificant. As per
table – 1B the significant value is 0.000 which is lesser than 0.05. Hence, the null hypothesis cannot be accepted and the mean of VIX and IV is
statistically not equal.
In BSu Model the mean of VIX is 22.5319 and the mean of IV is 24.4929 (as per table – 1A) with the mean difference of 1.96108 which is significant.
As per table – 1B the significance value is 0.000 which is lesser than 0.05. Hence, the Null hypothesis cannot be accepted and the mean of VIX and IV
is statistically not equal.
In BCS Model the mean of VIX is 22.5319 and the mean of IV is 25.0637 (as per table –1A) with the mean difference of 2.53187 which is insignificant.
As per table – 1B the significant value is 0.003 which is lesser than 0.05. Hence, the Null hypothesis cannot be accepted and the mean of VIX and IV is
statistically not equal.
Table – 1B shows the Levene’s Test for Equality of Variances. The result shows that the significant value for all the three models is 0.00 which means
both group (VIX and IV) are homogenous. Thus the ttest for equal variance not assumed is considered.
The following Table – 2A pre–highlights the CM Model mean of VIX is 22.4521and the mean of IV is 26.4508, with the mean difference of 3.99862 which
is insignificant. As per table – 2B the significant value is 0.000 which is lesser than 0.05. Hence, the null hypothesis cannot be accepted and the
mean of VIX and IV is statistically not equal.
In BSu Model the mean of VIX is 22.4521 and the mean of IV is 32.1177 (as per table – 2A) with the mean difference of 9.66552 which is significant.
As per table – 2B the significant value is 0.000 which is lesser than 0.05. Hence, the Null hypothesis cannot be accepted and the mean of VIX and IV is
statistically not equal.
In BCS Model the mean of VIX is 22.4521 and the mean of IV is 33.2083 (as per table –2A) with the mean difference of 10.75621 which is
insignificant. As per table – 2B the significance value is 0.000 which is lesser than 0.05. Hence, the Null hypothesis cannot be accepted and the mean
of VIX and IV is statistically not equal.
Table – 2B shows the Levene’s Test for Equality of Variances. The result shows that the significant value for all the three models is 0.00 which means
both group (VIX and IV) are homogenous. Thus the ttest for equal variance not assumed is considered.
Table – 2A Group Statistics of NIFTY MIDCAP 50

Mode

N

Mean

Std. Deviation

Std. Error Mean

IV & CM Model

VIX

1438

22.4521

7.69409

.20290

IV

1438

26.4508

17.62729

.46484

IV & BSu Model

VIX

1438

22.4521

7.69409

.20290

IV

1438

32.1177

23.72926

.62576

IV & BCS Model

VIX

1438

22.4521

7.69409

.20290

IV

1438

33.2083

41.70749

1.09985

Source: computed as per data taken from NSE
Table – 2B Independent Samples tTest of NIFTY MIDCAP 50

Levene's Test for Equality of Variances

ttest for Equality of Means

F

Sig.

t

df

Sig. (2tailed)

Mean Difference

Std. Error Difference

95% Confidence Interval of the Difference

Lower

Upper

IV & CM Model

Equal variances assumed

115.817

.000

7.884

2874

.000

3.99862

.50719

4.99312

3.00412

Equal variances not assumed



7.884

1965.379

.000

3.99862

.50719

4.99332

3.00393

IV & BSu Model

Equal variances assumed

216.748

.000

14.693

2874

.000

9.66552

.65783

10.95538

8.37565

Equal variances not assumed



14.693

1735.854

.000

9.66552

.65783

10.95573

8.37530

IV & BCS Model

Equal variances assumed

373.750

.000

9.617

2874

.000

10.75621

1.11841

12.94918

8.56324

Equal variances not assumed



9.617

1534.695

.000

10.75621

1.11841

12.94998

8.56243

Source: computed as per data taken from NSE
Table – 3A Group Statistics of CNX BANK

Mode

N

Mean

Std. Deviation

Std. Error Mean

IV & CM Model

VIX

1443

22.5289

7.82891

.20610

IV

1443

30.3405

24.78701

.65252

IV & BSu Model

VIX

1443

22.5289

7.82891

.20610

IV

1443

36.3443

32.82095

.86401

IV & BCS Model

VIX

1443

22.5289

7.82891

.20610

IV

1443

38.9515

59.28762

1.56074

Source: computed as per data taken from NSE
Table – 3B Independent Samples tTest of CNX IT

Levene's Test for Equality of Variances

ttest for Equality of Means

F

Sig.

t

df

Sig. (2tailed)

Mean Difference

Std. Error Difference

95% Confidence Interval of the Difference

Lower

Upper

IV & CM Model

Equal variances assumed

121.361

.000

11.416

2884

.000

7.81158

.68429

9.15332

6.46983

Equal variances not assumed



11.416

1726.871

.000

7.81158

.68429

9.15370

6.46945

IV & BSu Model

Equal variances assumed

182.623

.000

15.554

2884

.000

13.81540

.88825

15.55706

12.07373

Equal variances not assumed



15.554

1605.565

.000

13.81540

.88825

15.55765

12.07315

IV & BCS Model

Equal variances assumed

257.953

.000

10.432

2884

.000

16.42255

1.57429

19.50939

13.33570

Equal variances not assumed



10.432

1492.273

.000

16.42255

1.57429

19.51060

13.33449

Source: computed as per data taken from NSE
Table – 3A shows the CM Model mean of VIX is 22.5289 and the mean of IV is 30.3405, with the mean difference of 7.81158 which is insignificant. As
per table – 3B the significance value is 0.000 which is lesser than 0.05. Hence, the null hypothesis cannot be accepted and the mean of VIX and IV is
statistically not equal.
In BSu Model the mean of VIX is 22.5289 and the mean of IV is 36.3443 (as per table – 3A) with the mean difference of 13.81540 which is significant.
As per table – 3B the significance value is 0.000 which is lesser than 0.05. Hence, the Null hypothesis cannot be accepted and the mean of VIX and IV
is statistically not equal.
In BCS Model the mean of VIX is 22.5289 and the mean of IV is 38.9515 (as per table – 3A) with the mean difference of 16.42255 which is
insignificant. As per table – 3B the significance value is 0.000 which is lesser than 0.05. Hence, the Null hypothesis cannot be accepted and the mean
of VIX and IV is statistically not equal.
Table – 3B shows the Levene’s Test for Equality of Variances. The result shows that the significant value for all the three models is 0.00 which means
both group (VIX and IV) are homogenous. Thus the ttest for equal variance not assumed is considered.
The following Table – 4A prehighlights the CM Model mean of VIX is 22.5289 and the mean of IV is 25.4070, with the mean difference of 2.87810 which
is insignificant. As per table – 4B the significance value is 0.000 which is lesser than 0.05. Hence, the null hypothesis cannot be accepted and the
mean of VIX and IV is statistically not equal.
In BSu Model the mean of VIX is 22.5289 and the mean of IV is 30.4572 (as per table – 4A) with the mean difference of 7.92822 which is significant.
As per table – 4B the significance value is 0.000 which is lesser than 0.05. Hence, the Null hypothesis cannot be accepted and the mean of VIX and IV
is statistically not equal.
In BCS Model the mean of VIX is 22.5289 and the mean of IV is 32.2913 (as per table – 4A) with the mean difference of 9.76233 which is
insignificant. As per table – 4B the significance value is 0.000 which is lesser than 0.05. Hence, the Null hypothesis cannot be accepted and the mean
of VIX and IV is statistically not equal.
Table – 4B shows the Levene’s Test for Equality of Variances. The result shows that the significant value for all the three models is 0.00 which means
both group (VIX and IV) are homogenous. Thus the ttest for equal variance not assumed is considered.
Table – 4A Group Statistics of CNX IT

Mode

N

Mean

Std. Deviation

Std. Error Mean

IV & CM Model

VIX

1443

22.5289

7.82891

.20610

IV

1443

25.4070

15.80167

.41598

IV & BSu Model

VIX

1443

22.5289

7.82891

.20610

IV

1443

30.4572

20.69614

.54482

IV & BCS Model

VIX

1443

22.5289

7.82891

.20610

IV

1443

32.2913

37.89065

.99747

Source: computed as per data taken from NSE
Table – 4B Independent Samples tTest of CNX IT

Levene's Test for Equality of Variances

ttest for Equality of Means

F

Sig.

t

df

Sig. (2tailed)

Mean Difference

Std. Error Difference

95% Confidence Interval of the Difference

Lower

Upper

IV & CM Model

Equal variances assumed

114.560

.000

6.200

2884

.000

2.87810

.46423

3.78836

1.96784

Equal variances not assumed



6.200

2109.699

.000

2.87810

.46423

3.78850

1.96770

IV & BSu Model

Equal variances assumed

215.835

.000

13.611

2884

.000

7.92822

.58250

9.07038

6.78606

Equal variances not assumed



13.611

1846.405

.000

7.92822

.58250

9.07065

6.78579

IV & BCS Model

Equal variances assumed

413.313

.000

9.585

2884

.000

9.76233

1.01854

11.75946

7.76519

Equal variances not assumed



9.585

1564.897

.000

9.76233

1.01854

11.76017

7.76448

Source: computed as per data taken from NSE
Conclusion
This paper attempts to find out the best model for estimating IV, among selected three models viz. CM Model (1996), BSu Model (1988) and BCS Model
(1996) is calculated for total of 5 year 10 month from 2^{nd} March 2009 to 31^{st} December 2014. For estimating IV, four Options index
were chosen they are NIFTY, MIDCAP 50, BANK and IT Options index, the calculated IV are separately compared with the Indian VIX. The previous studies have
analysed various implied volatility models under different market conditions. Most studies have accepted the above three selected models under different
market conditions because of its implementation and calculation thus, it was used to approximate the IV of NIFTY, MIDCAP50, BANK and IT Index Options.
Whereas calculated IV Index Options and India VIX are significantly different in all the Options Indices in the study period because the IV of Index
Options are calculated using atthemoney contract but the India VIX is calculated using outofthe money contract. The difference in IV can differ with
other contracts viz. inor atthemoney, which are not taken for the study and the results may change accordingly. Hence, the study concludes that the
market participants could make their investment strategies based on the calculated volatility appropriately (whether underpriced or overpriced). The
speculators especially are interested in the volatility trading this could help them in a big way to position themself in the market movement
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[1]
Associate Professor, Department of International Business and Commerce, Alagappa University, Karaikudi603003, EMil: muthuroja67@rediffmail.com , M:9042736251
[2]
M.Phil Research scholar, Department of International Business and Commerce, Alagappa University, Karaikudi603003, Email: vevekpdy@gmail.com , M: 9843030935
