## 3

### The Information Content

### of Indian Implied

### Volatility Index

**3.1 Introduction**

Most of the investors believe that stock prices, even when rising, climb a wall of uncertainty and worry. When volatility and investors sentiment about the future go hand in hand, the forward looking measure of volatility implied by the option prices is often quoted as bona fide investors fear gauge (Whaley, 2000). Implied volatility has become so important that it is being reported on routine basis by financial news agencies and is closely followed by investors for making investment decisions. As a result, the study of information content and forecast quality of implied volatility has become an important topic in financial research.

The researchers study the information content of implied volatility in two ways. The first class deals with the contemporaneous association between implied volatility index and market returns, which is rather limited in the finance literature. Second, category is focused on analyzing its ability to forecast future volatility and to empirically examine the relationship between realized and implied volatility. The present chapter aims to study the first dimension for the financial market of an emerging economy like India.

These volatility indices can be used (i) to make inferences on current stock market returns, (ii) to learn about the current volatility expectations quoted in the market, and/or (iii) to anticipate future uncertainty in market returns. A negative relationship

between changes in implied volatility index and market returns has been documented in the literature for the various financial markets. These results give foundation to the interpretation of implied volatility index as a measure of capturing market sentiments and risks. The arrival of news to the market, a sudden increase in the trading volume and the number of orders crossed may produce a negative relationship between volatility changes and index returns. In particular, an increased level of uncertainty in markets, due to the release of some economic data figures or some political announcements or policy changes that increase risk, may cause an upward surge in financial market volatility. At the same time, these changes induce pressure on the selling decisions of the investors; that may lead to generate negative returns. Whaley (2000), Simon (2003) and Giot (2005) found a negative contemporaneous relationship between volatility changes and index returns in American markets. They found that arrival of bad news may induce a larger volatility increase than the arrival of good news of same relevance. Therefore, if this asymmetric negative relationship between volatility changes and index returns is confirmed, the information in volatility index can become an important element in portfolio management.

In the light of the above uses of volatility index as a measure of expected volatility, the empirical analysis of the present study is initiated by studying the information content of implied volatility index of India (henceforth IVIX). The present chapter is devoted to investigate the following issues:

(i) The major macroeconomic events and volatility index; (ii) Statistical properties and seasonal patterns in IVIX;

(iii) Leverage effect and asymmetry in risk-return relationship and (iv) Short-term asymmetric return-volatility relation.

The above issues are analyzed by developing the following hypothesis:

**Hypothesis I: A negative contemporaneous correlation exists between innovations in **
IVIX and S&P CNX Nifty returns.

**Hypothesis II: Contemporaneous negative and positive index returns are the most **
important determinants of changes in implied volatility.

**Hypothesis III: **The lagged returns and/or changes in the past implied volatilities
determine current changes in the implied volatility.

**Hypothesis IV: Asymmetry exists for the return-volatility relation, that is, negative **
returns have much higher impact on volatility than the positive returns.

**Hypothesis V: In short-term return-volatility relation, contemporaneous returns are **
the most important determinants of the largest changes in the implied volatility index.
The asymmetry is implied to vary across the quantiles of IV change distribution,
particularly from the median quantile to the uppermost quantile of IV change
distributions.

The given hypotheses are tested empirically using the analysis structured on bi-partite
levels. Firstly, the univariate properties and seasonality of time-series history of IVIX
are examined. Secondly, the temporal relationship between volatility and stock market
returns is examined using three different methodologies. The changes in implied
volatility index are regressed on: lead-lag, contemporaneous and contemporaneous
absolute stock market returns; on positive and negative returns; and finally, the
short-term return-volatility relationship is deshort-termined using the quantile regression method.
**3.2 Empirical analysis **

**3.2.1 Daily and weekly movements of IVIX: The current section examines daily and **
weekly movements of implied volatility Index of India and its underlying index S&P
CNX Nifty during sample period since March, 2009 till June, 2012 (Figure 3.1 and
Figure 3.2). The preliminary analysis intends to capture the movements of IVIX and
S&P CNX Nifty during the sample period and their behavior surrounding some
important economic events. The univariate properties of the changes in IVIX and
S&P CNX Nifty are computed, along with IVIX cross-correlations with its underlying
index returns of S&P CNX Nifty. The approach followed here is based on the
methodology adopted by Fleming *et al. (1995) for assessing the behavior of CBOE *
Market volatility index. It is the conventional technique of examining the risk-return
relation and is widely used in many studies to analyze the intertemporal relationship
between the implied volatility indices and market returns of different countries
(Dowling & Muthuswamy, 2005; Frijns et al. 2010)

**Figure 3.1: Daily movements of S&P CNX Nifty and IVIX **

*Source: Authors*

**Figure 3.2: Weekly movements of S&P CNX Nifty and IVIX **

*Source: Authors*

**3.2.2 Worldwide macroeconomic shocks and IVIX: **Figure 3.3 plots the daily
closing IVIX at levels against absolute magnitude of S&P CNX Nifty index returns.
The absolute returns can be interpreted as a measure of risk i.e. volatility of an asset
(Granger & Sin, 2000; Taylor, 1986). Thus, Figure 3.3 shows the movements in the
Indian volatility index in accordance with the key events that occurred in the financial
markets at global and domestic levels. Figure 3.3 shows that several periods of

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**S&P CNX Nifty** **IVIX**

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relatively high volatility and large stock market moves stand out. IVIX has reached the highest value of 56.07 on May 19, 2009 and the lowest value of 15.22 on September 6, 2010, thus, dropping almost 3 times in this period.

At the start of data period (April, 2009), there was an atmosphere of political uncertainty due to which IVIX rose to 56. This peak value of IVIX was indicating a high level of nervousness in the markets. In November 2009, debt crisis enveloped Dubai markets; which spread a panic alarm across the world stock markets. Indian markets also took a sharp plunge, with banking and realty sectors being the worst sufferers.

The Greek debt crisis in May, 2010 resulted in rise of more than 23% in IVIX, as it stayed in a narrow range for the preceding couple of months. This reflected a spurt in uncertainty and panic in the Indian stock markets and IVIX reached the level of 34.37 in May, 2010. After this, the index declined to the lowest level of 15.22 in Sept, 2010 and attained a plateau level 22 in the end of year 2010. The Ireland crisis coupled with hike in the interest rates by the Chinese government also affected volatility of the Indian markets.

For the first six months of 2011, India registered an impressive growth in FDI inflows. Stock market volatility represented by VIX declined from 26 levels at the beginning of 2011 to around 20 levels, indicating that investor were gaining confidence in the Indian markets. But, in Aug, 2011 S&P lowered its long-term sovereign on the US from AAA to AA+. This move triggered a fresh round of shockwaves in global markets and IVIX witnessed the sharpest rise of over 30% in the preceding two years. The index shot up 23% to close at 24.90, compared to previous day close of 20.22. In October 2011, there was a return of optimism in the equity markets on account of positive news flow from the Eurozone and IVIX dropped to the levels that one saw only prior to the US downgrade. At this point the IVIX dropped by 7.93% to close at 20.89.

Apart from the global factors, India’s domestic problems were also accountable for the under performance of the markets in 2011. The main issues were the supply constraints in important sectors such as energy, food and mineral resources and policy paralysis which brought a halt to the fresh investments in the country.

**Figure 3.3: Daily closing IVIX and S&P CNX Nifty index return absolute values during the period from March, 2009 through June, **
**2012* **

*Source: Authors *

**Note: The S&P CNX Nifty returns are reported on the basis of the log index relative multiplied 100 *

0 10 20 30 40 50 60 0 2 4 6 8 10 12 14 16 18 In d ian Im p li ed V o latil it y In d ex S & P CNX Nif ty In d ex Re tu rn s (A b so lu te V A lu e, % ) Date

S&P CNX Nifty Returns IVIX
**UPA Govt **
**election **
**victory in **
**India **
**Dubai debt **
**worries/ **
**restructuring **
**Greek **
**government **
**Ireland's **
**crisis & **
**Chinese **
**stock fell **
**rate hike **
**Highest **
**inflow of **
**FDI in 39 **
**months in **
**Indian **
**stock **
**markets **
**S& P **
**Downgrade **
**US credit **
**rating ** **GDP **
**slumped **
**to a 9yr **
**-low of **
**5.3% & ₹ **
**depreciat**
**es against **
**$ **
**Satyam **
**Accounting **
**IIP for **
**Apr-Oct, **
**2011-12 **
**stood at **
**3.5% **
**against **
**8.7%last **
**yr **
**CAG unearths 2G scam **

**which leads to arrest of **
**ex-telecom minister A **
**Raja and cancellation of **

In 2012, India’s economic expansion plunged to a near-decade low of 5.3% in March quarter as the debt crisis in Europe, its top trading partner, curbed exports and policy gridlocks deterred investments.

Until mid-2012, the IVIX behavior pattern suggested a bullish trend in the markets. IVIX in April-May had retested an intermediate high of 29.12. From the first week of June, it started declining, and had broken below the crucial support level of 20. After this level, bulls started gaining strength.

During the period covered, the spikes in IVIX were usually accompanied by large moves, up and down in the underlying stock index levels. However, there were few exceptions for instance there was no change in IVIX value at the time of Satyam accounting fraud. Again in October 2012, the Index of Industrial Production (IIP) witnessed a negative growth of (-5.6%) and IVIX declined by 0.88% to 29.27. Thus, barring few exceptions, the IVIX remained fairly stable when higher changes were observed in S&P CNX Nifty.

**3.2.3 IVIX normal range: In this section, the normal and abnormal behavior of IVIX **
shall be studied in a probabilistic sense (Table 3.1). The low, high and normal
readings of IVIX vary in different kinds of market conditions. Over the entire sample
period, the median daily closing level of IVIX is 24.15. In the sample set, 50% of the
times IVIX closed between 20.7225 and 28.60750 (a range of 7.885 points), 75% of
the times IVIX closed between 18.919 and 37.659 (a range of 18.740 points) and 90%
times IVIX closed between 17.9845 and 42.4975 (a range of 24.513 points). Table 3.1
also reports a great variation in what is considered normal from year to year. For
example in 2010, the median closing level of IVIX is 20.980 and 50% of the time
IVIX remained in the range of 19.360 and 23.028 and 90% of the time between
29.933 and 16.996. The widest range was experienced in 2009 when the IVIX closed
between 25.314 and 51.8795 (range of 26.57 index points). The second widest range
was experienced in 2011 – the year in which stock markets followed the bearish trend.
The 5% and 95% percentile values indicate that the range of daily IVIX levels was
from 19.131 to 27.277. These results are similar to the findings of Dhanaiah *et al. *
(2012)for the Indian stock market.

**Table 3.1:** **Normal range for daily levels of IVIX over sample period, March, **
**2009 through June, 2012 and year-wise **

**Period ** **Obs ** **5% ** **10% ** **25% ** **50% ** **75% ** **90% ** **95% **
**Mar’09-Jun’12 ** 830 17.985 18.919 20.723 24.15 28.608 37.66 42.498
**Sub-periods**
**Mar’09-Dec’09 ** 204 25.314 26.242 28.305 36.575 40.908 48.395 51.880
**2010 ** 252 16.996 17.632 19.360 20.980 23.028 28.087 29.933
**2011 ** 247 18.203 18.650 20.410 23.160 26.790 29.374 32.007
**Jan’12- Jun’12 ** 127 19.131 20.066 21.740 23.550 25.095 26.362 27.277

An important way of judging market anxiety is to examine the persistence with which IVIX remains above the extraordinary levels during the entire sample period. From Table 3.1, it is observed that the chance of IVIX remaining above 42.498 is 5%. The history of IVIX is re-examined to count the number of consecutive days IVIX remained above 42.498 index points. Three periods: March 2, 2009 to March 4, 2009 (3 days); April 8, 2009 to May 26, 2009 (29 days) and June 17, 2009 to July 3, 2009 (10 days) can be demarcated for demonstrating abnormal behavior in the history of IVIX.

**3.2.4 Statistical properties of Indian volatility index: **In this section the statistical
properties of implied volatility index changes and its underlying index returns are
described. The empirical analysis of volatility index is based on the changes or
innovations in the implied volatility denoted by:

_{ }………...………(1)
where, *Vt* denotes implied volatility index at level at time *t. There are three *
considerations which support the use of “change” in the implied volatility index.
Firstly, the academicians and practitioners are interested in changes or innovations in
the expected volatility. They want to know how security valuation is influenced with
changes in expected volatility. Secondly, if stock prices follow a random walk (Figure
3.1 & 3.2), estimation of relationship between volatility and stock indexes at level
may produce spurious results. Thirdly, Figure 3.1 shows that IVIX at levels appear to
follow a near-random walk pattern. Further, Table 3.2 & 3.3 states the descriptive
properties and unit root statistics of daily IVIX and S&P CNX Nifty at levels. The
value of autocorrelation function are positive and significant 1% levels, which means

that the series is first order serially correlation i.e. autoregression of order one (AR (1)) or volatility persistent behavior. The unit root statistics show that daily IVIX levels series is non-stationary and first difference series are stationary (Table 3.3).Thus, the presence of high degree of autocorrelation may affect the inference in finite samples.

**Table 3.2: Descriptive statistics for daily S&P CNX Nifty and IVIX at levels **

**Series ** **S&P CNX Nifty ** **IVIX **

** Mean ** 5104.957 26.158
** Median ** 5186.100 24.150
** Maximum ** 6312.450 56.070
** Minimum ** 2573.150 15.220
** Std. Dev. ** 646.393 7.730
** Skewness ** -1.401 1.447
** Kurtosis ** 6.174 4.896
** Jarque-Bera ** 619.944 414.200
** Probability ** 0.000 0.000
** Observations ** 830 830
**ρ(1) ** 0.986* 0.978*
**ρ(2) ** 0.971* 0.958*
**ρ(3) ** 0.956* 0.939*

*Note: *, **, *** represents 1%, 5% and 10% level of significance *

**Table 3.3: Unit root test statistics **

**Test/ Series ** **ADF ** **PP ** **KPSS **

**Levels ** **1 ** **2 ** **3 ** **1 ** **2 ** **3 ** **1 ** **2 **
**S&P CNX Nifty ** -2.878
(1)
-3.449
(1)
0.800
(1)
-2.675
(12)
-3.308
(13)
0.806
(9)
1.296
(23)*
0.673
(23)*
**IVIX ** -3.064
(1)
-3.353
(1)
-1.243
(4)
-2.873
(12)
-3.183
(10)
-1.432
(17)
1.430
(23)*
0.611
(23)*
**First Difference**
**ΔS&P CNX Nifty ** -7.960
(9)*
-11.035
(6)*
-7.852
(9)*
-27.746
(11)*
-27.922
(14)*
-27.665
(9)*
0.645
(14)
0.131
(14)
**ΔIVIX ** -21.239
(1)*
-21.232
(1)*
-21.239
(1)*
-31.384
(16)*
-31.386
(16)*
-31.366
(16)*
0.066
(18)
0.030
(18)
**Asymptotic critical values**

**1% level ** -3.463 -4.004 -2.576 -3.462 -4.003 -2.576 0.739 0.216
**5% level ** -2.876 -3.432 -1.942 -2.875 -3.432 -1.942 0.463 0.146
**10% level ** -2.574 -3.140 -1.616 -2.574 -3.139 -1.616 0.347 0.119

*Note: ADF is the Augmented Dickey-Fuller, PP is the Phillips-Perron, and KPSS is Kwiatkowski, Phillips, *
*Schmidt, and Shin test. Model Specification: 1.Intercept, 2. Intercept+trend, 3.None. The Null hypothesis for ADF *
*and PP test: H0 = Variable is non-stationary and for KPSS: H0 = Variable is stationary. *, ** and *** indicate the *

*rejection of the null hypothesis at the 1%, 5% and 10% significance levels, respectively. The proper lag order for *
*ADF test is chosen by considering Akaike Information Criteria, representing in parenthesis. For KPSS and PP *
*tests, the bandwidth is chosen using Newey–West method and spectral estimation uses Bartlett kernel represented *
*in parenthesis.*

The univariate properties of changes in implied volatility index are considered first. Summary statistics for daily and weekly IVIX changes over the entire sample period and for calendar-year sub-periods are estimated.

Table 3.4 reports the properties of change in daily closing implied volatility index. The mean volatility change over the entire sample period is 0.0291 and ranges from -0.098 (2009) to 0.043 (2011) for the calendar sub-periods. The standard deviation of the daily volatility changes (volatility of volatility) for the entire sample period is 1.4929. The standard deviation in the calendar sub-periods ranges from 1.168 in 2012 to 1.855 in 2009. Not surprisingly, higher value of standard deviation reported in the year 2009 is due to financial crisis of a premier bank of America, Lehman Brothers, which enveloped the whole world in 2008.

The autocorrelation structure of the daily IVIX for lag one, two and three are provided in the Table 3.4. The autocorrelation structure of changes in daily IVIX varies substantially from year to year, unlike mean and standard deviation which are found to be relatively stable during the calendar-year sub-periods. The first-order autocorrelations for the sub-periods, range from -0.153 in 2012 to -0.029 in 2011 and is significant at 10% level in the year 2012. The autocorrelation functions (ACF) for the entire sample period at one through three lags are -0.063, -0.011 and -0.071 respectively and are significant at 10% level for lag one and three. The values for the ACF are near to zero for all lags which indicates randomness in the dataset. The autocorrelation at lag 1 is negative and significant in the year 2012 and at lag 2 it is positive in the year 2010, 2011 and 2012. Similarly, ACFs at lag 3 is negative and significant in 2011, hence indicating that changes in volatility follow a mean reverting process. Similar results have been documented by Kumar (2012).

Table 3.4 also summarizes the properties of daily S&P CNX Nifty 50 index returns. The mean value of index returns during the entire sample period is 0.0820 and ranges from -0.114 in 2011 to 0.328 in 2009. The volatility of market returns represented by standard deviation during the entire sample is 2.099.The volatility of returns is changing over the sample period, providing some preliminary evidence of time-varying volatility. The stock return distribution is found to be non-normal and is positively skewed. The ACF for one through two lags is near zero and positive.

**Table 3.4: Statistical properties of daily closing Indian volatility index level changes and S&P CNX Nifty returnsa **

**Volatility Index Changes ** **S&P CNX Nifty ** **Cross Correlationsb**

**Period ** **Obs. ** **Mean ** **SD ** **ρ(1) ** **ρ(2) ** **ρ(3) ** **Mean ** **SD ** **ρ(1) ** **ρ(2) ** **ρ(3) ** **-2 ** **-1 ** **0 ** **1 ** **2 **

**Entire Sample **
**Mar,09-Jun, 12 ** 829 -0.0291 1.4929 -0.063*** -0.011 -0.071*** 0.0820 1.459 0.035 0.001 -0.03 0.0041 0.165 -0.482 -0.017 -0.046
**Year wise **
**2009 ** 203 -0.098 1.855 -0.076 -0.120 0.041 0.328 2.099 0.018 -0.070 -0.022 0.015 -0.011 -0.238 0.293 -0.023
**2010 ** 252 -0.027 1.233 -0.058 0.085 -0.055 0.066 1.024 -0.036 0.041 -0.007 0.026 0.060 -0.677 0.021 -0.108
**2011 ** 247 0.043 1.554 -0.029 0.037 -0.245* -0.114 1.321 0.102*** 0.045 -0.114*** 0.056 0.025 -0.711 -0.081 -0.076
**2012 ** 127 -0.063 1.168 -0.153*** 0.030 0.031 0.104 1.118 -0.073 0.081 -0.004 -0.022 0.2646 -0.614 0.1154 -0.098

*Note: a The mean, standard deviation (SD), and autocorrelation (ρ ) for the first three lags are provided for the changes in the volatility index and S&P CNX Nifty returns *
*along with the cross correlation between the volatility index innovations and S&P CNX Nifty index returns. bCorrelations for negative (positive) lags/lead denotes *
*correlation between the volatility index changes and past (future) S&P CNX Nifty index returns.*, **, *** denotes 1%, 5% and 10% level of significance *

**Table 3.5: Statistical properties of weekly Indian volatility index level changes and S&P CNX Nifty returnsa **

**Volatility Index Changes ** **S&P CNX Nifty ** **Cross Correlationsb**

**Period ** **Obs. ** **Mean ** **SD ** **ρ(1) ** **ρ(2) ** **ρ(3) ** **Mean ** **SD ** **ρ(1) ** **ρ(2) ** **ρ(3) ** **-2 ** **-1 ** **0 ** **1 ** **2 **

**Entire Sample **
**Mar,09-Jun, 12 ** 173 -0.124 2.386 0.043 -0.021 -0.069 0.39 2.799 0.172** 0.138** 0.077** 0.0896 -0.0361 -0.3824 -0.144 -0.039
**Year wise **
**2009 ** 43 -0.413 2.767 0.208 -0.153 -0.145 1.581 3.930 0.005 0.159 0.033 0.1863 -0.1178 -0.069 -0.109 0.015
**2010 ** 52 -0.127 2.220 0.038 0.119 -0.042 0.299 1.833 0.281** 0.111*** -0.003 -0.0792 0.0089 -0.612 -0.081 -0.0956
**2011 ** 52 0.178 2.469 -0.127 -0.033 -0.029 -0.496 2.535 0.137 -0.137 -0.007 0.0993 0.146 -0.66 -0.212 -0.004
**2012 ** 26 -0.243 1.865 -0.021 0.089 -0.046 0.375 1.970 0.326*** 0.263*** 0.200*** 0.1866 -0.1118 -0.449 -0.013 -0.115
*Note: a The mean, standard deviation (SD), and autocorrelation (ρ ) for the first three lags are provided for the changes in the volatility index and S&P CNX Nifty returns *
*along with the cross correlation between the volatility index innovations and S&P CNX Nifty index returns .bCorrelations for negative (positive) lags/lead denotes *
*correlation between the volatility index changes and past (future) S&P CNX Nifty index returns.*, **, *** denotes 1%, 5% and 10% level of significance *

Consistent with Hypothesis I, there exists negative contemporaneous correlation of
-0.482 between changes in IVIX and S&P CNX Nifty returns. The highest negative
correlation of -0.711 is found in the year 2011 and lowest of -0.238 in year 2009.
Positive correlations are reported between IVIX and lead and lagged index returns.
However, the magnitude of correlation in the intertemporal periods is small in
significant number of cases. As the theory predicts, there is fall (rise) in stock prices
when expected volatility rises (falls). The correlation dampens out quickly to small
levels at non-contemporaneous lags. As per Fleming *et al. (1995) and Dowling & *
Muthuswany (2005) the magnitude of negative correlation reported for America and
Australia is higher, being –0.615 and –0.453 respectively. According to Fleming et al.
(1995) the intertemporal correlations are positive and small across all the sub-periods.
A similar phenomenon is observed for the Indian markets - indicating that they are
heading to become efficient as IVIX tries to adjust rapidly with the stock index
returns.

The statistical properties of weekly volatility changes and market returns are summarized in Table 3.5. The mean and standard deviation of weekly data for changes in volatility index and stock market returns exhibit patterns similar to those apparent in daily data. The mean of weekly volatility index changes is -0.124 for the entire sample and ranges from -0.413 in 2009 to 0.178 in 2011, which is approximately four times their corresponding level in Table 3.4. The volatility of the weekly IVIX changes ranges from a low of 1.865 in 2012 to 2.767 in 2009. The mean of market returns ranges from -0.496 in 2011 to 1.581 in 2009.

The autocorrelation structure of the weekly changes in IVIX and index returns are also provided in Table 3.5. The weekly autocorrelation function of changes in IVIX at lag one is positive and insignificant. The autocorrelation function for market returns for lags one through three is positive and significant at 5% level for the entire sample. Unlike daily autocorrelations, the weekly correlation for market returns is positive and significant in most of the cases.

The temporal association between volatility changes and stock market returns, reported in Table 3.4 for the daily observations, is also present in Table 3.5 for weekly observations. The contemporaneous correlations is -0.3824 for the entire sample period, and a moderate range of -0.4482 in 2012 to -0.66 in 2011 is observed in the calendar-years except for 2009 in which very small value of correlation is reported. Positive relationship between implied volatility and past-week and future-week stock

market returns is also reported in Table 3.5. The cross-correlations for the non-contemporaneous lags show greater variability from year to year.

**3.2.5 Distributional properties - Non-parametric test: **Figure 3.4, 3.5 and 3.6
draws the histograms, corresponding kernel density function and Q-Q plot
respectively to present the distributional properties of IVIX, changes in IVIX, S&P
CNX Nifty and S&P CNX Nifty returns. Histograms provide three very important
pieces of information about distribution of data values: shape, central location (the
middle), and spread. The histograms are constructed in order to make them
comparable with kernel density functions, for the relative frequency scaled by each
bin width. The kernel densities is non-parametric test in which window width, or the
smoothing parameter or bandwidth consists of Silverman bandwidth (Silverman,
1986), and the linear binning method is chosen. The Epanecknikov kernel function is
chosen for optimal bandwidth as it tends to minimize the Approximate Mean
Integrated Squared Error (AMISE criterion) (Wand & Jones, 1995).

In addition, the Q-Q Plots are drawn, which depicts of the percentiles (or quintiles) of a standard normal distribution against the corresponding percentiles of the observed data. If the observations follow approximately a normal distribution, the resulting plot should be roughly a straight line with a positive slope.

**Figure 3.4: The histograms of IVIX, changes in IVIX, S&P CNX Nifty and S&P **
**CNX Nifty returns. **
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**Figure 3.5: The kernel density distributions of IVIX, changes in IVIX, S&P CNX **
**Nifty and S&P CNX Nifty returns. **

**Figure 3.6: The Q-Q Plot of IVIX, changes in IVIX, S&P CNX Nifty and S&P **
**CNX Nifty returns. **
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.0000
.0002
.0004
.0006
.0008
.0010
2,000 3,000 4,000 5,000 6,000 7,000
D
e
n
s
it
y
S&P CNX Nifty
.0
.1
.2
.3
.4
-10 -5 0 5 10 15 20
D
e
n
s
it
y

S&P CNX Nifty returns

0 10 20 30 40 50 60 10 20 30 40 50 60 Quantiles of IVIX Q ua nt ile s of N or m al IVIX -6 -4 -2 0 2 4 6 -8 -4 0 4 8

Quantiles of Changes in IVIX

Q ua nt ile s of N or m al Changes in IVIX 2,000 3,000 4,000 5,000 6,000 7,000 8,000 2,000 3,000 4,000 5,000 6,000 7,000 Quantiles of S&P CNX Nifty

Q ua nt ile s of N or m al S&P CNX Nifty -6 -4 -2 0 2 4 6 -10 -5 0 5 10 15 20 Quantiles of S&P CNX Nifty returns

Q ua nt ile s of N or m al

The combination of histograms, kernel density and Q-Q plots (Figure 3.4, 3.5 and 3.6) are the graphical methods of studying the probability distributions of a data series. The histograms (Figure 3.4) show that the IVIX series at levels has a large positive skewness as its tails extends toward the right side whereas the S&P CNX Nifty at levels is skewed towards left side. The histograms for the changes in IVIX and index returns show that the series follow a normal distribution. The bump of kernel density functions (Figure 3.5) of IVIX and S&P CNX Nifty at levels is more widened to that of first difference series of IVIX and S&P CNX Nifty.

The Q-Q plots (Figure 3.6) of the data at level show deviation from the straight line, which depicts that the data is non-normal. These Q-Q plots confirm the results of histograms of IVIX and S&P CNX Nifty at levels. It shows that the data series of IVIX is skewed toward right and S&P CNX Nifty is skewed towards left and both have heavier tails as compared to normal distributions. The Q-Q plots for first difference series have approximately a normal distribution, as both series follow a straight line with positive slope pattern.

**3.2.6 Seasonal patterns in IVIX: ** Having considered some of the statistical
properties of IVIX, the issue of whether the IVIX contains any seasonal or predictable
patterns is addressed here. Figure 3.7 shows the intraweek volatility index behavior; it
shows the average IVIX on different days of a week. The average level of IVIX starts
high on Monday and falls systematically during the week. The total drop in average
level of IVIX from Monday to Friday is 0.65 basis points. The average volatility
index level increases from Tuesday to Wednesday and decreases from Thursday to
Friday. However this increase and decrease is around 0.20 and 0.07 basis points
respectively.

A regression equation in which daily changes in IVIX on dummy variables for the different days of the week is performed to determine the seasonality in IVIX changes is:

∑_{ } _{ } ………...……….(2)
where is close-to-close changes in the volatility index, i indicates the day of the
week (i = 1 is Monday, etc.) and is the dummy variable for each day of the week,
Monday through Friday, which is equal to 1 on day *i and 0 otherwise. Therefore, the *
regression indicates whether the average change from one day to the next is
significant for different days of the week.

**Figure 3.7: Intraweek behaviour of IVIX **

Regression results reported in Table 3.6 show that two days in a week namely
Monday and Thursday, are associated with significant changes in IVIX. The
coefficient of Monday is significant in the entire sample which indicates the presence
of weekend effect in IVIX returns, a finding similar to that of Kumar (2012) and
Fleming *et al. (1995). It is found that IVIX decreases significantly from Monday to *
Friday, similar to the findings of Frijns *et al. (2010) for Australian markets. *
However, presence of seasonality is weak considering the adjusted R2 of the
regression (0.020). The regression results for various sub-periods show that in 2009
and 2010 a significant Monday and Thursday effect is found, Friday effect is found
significant in 2009 and Wednesday effect is not found significant in the entire sample
periods. The range of adjusted R2 is found to be 0.001 to 0.062 for various calendar
years. The poor values of R2 which is representative of goodness of fit of a model
given in Table 3.6, reflect that the results of this model should be read with caution.
**Table 3.6: Day-of-the-week effect in Indian Volatility Index Level changesa**
**Period ** **Obs. ** **Monday ** **Tuesday Wednesday Thursday ** **Friday ** **R2** **Adj. R2**

**All ** 829 0.430*
(3.217)
-0.167
(-1.472)
-0.093
(-0.880)
-0.220**
(-2.095)
-0.093
(-0.800)
0.024 0.020
**Mar. – 2009 ** 203 0.930**
(2.562)
-0.083
(-0.313)
-0.267
(-0.999)
-0.513**
(-2.074)
-0.506**
(-1.955)
0.081 0.062
**2010 ** 252 0.248***
(1.637)
-0.040
(-0.243)
0.113
(0.643)
-0.398*
(-2.488)
-0.046
(-0.220)
0.031 0.016
**2011 ** 247 0.252
(0.947)
-0.082
(-0.347)
-0.296
(-1.583)
0.187
(1.029)
0.159
0.723
0.017 0.001
**Jun-12 ** 127 0.372
(1.366)
-0.737*
(-3.289)
0.188
(1.098)
-0.100
(-0.384)
-0.070
(-0.342)
0.104 0.074

*Note: a In this table the parameter estimated are presented for the regression of daily volatility index changes on *
*day-of-the-week dummy variables (equation 2). t-statistics are provided in the parentheses and are based on *
*Newey-West(1987) method for correction of heteroscedasticity and autocorrelation. The levels of significance are *
*represented with asterisks (* = 1%, ** = 5% and *** = 10%). *

25.4 25.6 25.8 26 26.2 26.4 26.6

Monday Tuesday Wednesday Thursday Friday

**Av**
**er**
**a**
**g**
**e **
**IVIX**
**Day **

**Average IVIX **

Average IVIX

**3.2.7 Inter-temporal return-volatility relation: Fleming ***et al. (1995) were the first *
to measure the kinked yet linear inter-temporal risk-return relation for changes in
CBOE implied volatility index and stock market returns. This methodology has been
adopted by various academicians (Frijns *et al., 2010; Sarwar, 2010) to examine the *
volatility-return relationship in various countries. A similar methodology has been
adopted here to analyze the relationship between the innovations in implied volatility
index of India and S&P CNX Nifty returns. The following conventional multivariate
regression model proposed by Fleming *et al. (1995) in which changes in volatility *
index is regressed on two lag, two lead, and the contemporaneous stock market
returns, as well as the absolute value (magnitude) of the contemporaneous return, is
used for the Indian markets:

∑_{ } _{ } _{ } _{| |}| _{ }| …..…………..(3)
This model focuses on inter-temporal relationship between stock market returns and
changes in implied volatility. The regression model is estimated across different
sub-periods for daily and weekly data through various calendar years, from March, 2009
to June, 2012. The weekly observations are calculated from the daily data on the basis
of average of closing values of trading days of each week. For estimated model, a
significant negative (positive) _{ }, i= -2… 2, coefficient indicates that increases
(decreases) in expected volatility at time t are accompanied by stock market declines
(advances) at time *t+i. If coefficient * _{| |} is positive (negative) and significant, it
indicates that stock markets are independent of their direction and are associated with
increase (decreases) in expected volatility. Further, summation of _{ } and _{| |}
measures the asymmetry in the relationship of changes in expected volatility and
stock market return.

As CBOE VIX is regarded as an investor’s fear gauge, it is expected that India VIX
can also show similar characteristics. The theories of CAPM models contributed by
Merton (1973), Sharpe (1964) and Linter (1965) suggest that with rises in expected
volatility, stock prices fall - a prediction consistent with the negative values of _{ }.
Black (1976) and Christie (1982) suggest that drop in stock prices increase the
leverage which is associated with increase in the implied volatility. Banerjee *et al. *
(2007) reported an inverse relationship between return and implied volatility index.

The estimated results for equation (3) using daily data are reported in Table 3.7. The
t-statistic provided for regression is based on the Newey & West (1987) method for
correction of heteroscedasticity and autocorrelation in the residuals. The results show
that for the entire sample period there is a significant negative relationship between
market returns and changes in IVIX (the coefficient of _{ } is significant at 1%). The
coefficient is consistent with the cross-correlation results reported in Table 3.4, as a
negative correlation is found between and ΔIVIX. The coefficient of
contemporaneous returns is negative and significant at 1% level for the sub-periods
2010, 2011 and 2012 and range between 0.622 to 0.857 except for the year 2009, in
which the coefficient is negative and significant at 10% level ( ̂_{ } = 0.272).

The coefficient at lag one for the entire sample is positive and significant but is
smaller in magnitude as compared to that of contemporaneous coefficient. Thus, the
contemporaneous negative relationship between changes in the IVIX and market
returns dominates the relationship of expected volatility with past and future stock
market returns. This pattern is consistent for all the sub-periods also, as the lag one
coefficient is significant in the year 2011 and 2012, and lag two is significant only for
the year 2011. Finally, the value of ̂_{| |} coefficient is 0.277 and is significant at 5%
level. Apparently, the changes in the expected stock market volatility are positively
affected by the size of a stock market.

The estimates of ̂_{| |} in Table 3.7 indicate a significant asymmetric contemporaneous
volatility-return relation. If the stock market generates positive returns, the coefficient
impacting the change in volatility is _{ } _{| |}, or -0.273. An increase in stock
market returns is associated with a decrease in expected volatility. On the other hand,
if market generates negative returns, the coefficient is | |, or -0.827.
This implies that a decrease in stock market returns is expected to bring an increase in
future volatility. The difference in the magnitude of coefficients ( is more than
three times the size of ), indicates an asymmetry in volatility-return relation. The
asymmetry in risk-return relation implies that negative stock market moves generate
changes in volatility index that are much larger in magnitude than those generated by
the positive stock market moves. This asymmetric relationship also has been observed
by Whaley (2000), Giot (2005) and Carr & Wu (2006) for American markets.

**Table 3.7: Intertemporal relationship between daily Indian volatility index level changes and S&P CNX Nifty index returnsa**

**Parameters **

**Period ** **Obs. ** ̂ ̂_{ } ̂_{ } ̂_{ } ̂_{ } ̂_{ } ̂_{| |} ̅ **Adjusted R2** **DW **

**Mar,2009-Jun,2012 ** 829 -0.283**
(-2.484)
0.016
(0.208)
0.193*
(5.409)
-0.550*
(-7.271)
-0.017
(-0.513)
-0.035
(-1.021)
0.277**
(2.213)
0.298 0.293 1.967
**2009 ** 199 -0.330***
(-1.638)
-0.030
(-0.239)
0.264*
(5.359)
-0.272***
(-1.854)
-0.007
(-0.132)
0.013
(0.217)
0.171)
(0.900)
0.160 0.134 1.995
**2010 ** -0.233*
(-2.833)
0.094
(1.419)
0.068
(1.085)
-0.805*
(-12.000)
-0.034
(-0.586)
-0.091
(-1.489)
0.321*
(3.282)
0.498 0.486 2.015
**2011 ** 242 -0.203***
(-1.829)
0.105***
(1.876)
0.108**
(2.060)
-0.857*
(-12.235)
0.005
(0.077)
-0.041
(-0.831)
0.156
(1.394)
0.531 0.519 2.156
**2012 ** 123 -0.179
(-1.571)
0.043
(0.594)
0.249**
(2.712)
-0.622*
(-7.938)
0.043
(0.575)
-0.041
(-0.544)
0.173
(1.557)
0.431 0.402 1.868

*Note: aThis table shows the parameter estimates for the regression of daily volatility index changes on two lags, two lead, and the contemporaneous S&P CNX Nifty index returns, as well as *
*magnitude (absolute value) of contemporaneous market returns (equation 3). t-statistics are provided in the parentheses and are based on Newey-West(1987) method for correction of *
*heteroscedasticity and autocorrelation . *, **, *** presents 1%, 5% and 10% levels of significance respectively. *

**Table 3.8: Intertemporal relationship between weekly Indian volatility index level changes and S&P CNX Nifty index returnsa **

**Parameters **

**Period ** **Obs. ** ̂ ̂_{ } ̂_{ } ̂_{ } ̂_{ } ̂_{ } ̂_{| |} ̅ **Adjusted R2** **DW **

**Mar,2009-Jun,2012 ** 169 -0.399***
(-1.790)
0.115***
(1.868)
0.046
(0.378)
-0.401*
(-3.696)
-0.070
(-1.120)
0.011
(0.134)
0.211
(1.534)
0.205 0.176 2.023
**2009 ** 39 0.309
(0.461)
0.172***
(1.826)
-0.134
(-0.710)
0.034
(0.152)
-0.052
(-0.462)
0.153
(1.560)
-0.270
(-0.911)
0.116 -0.049 1.520
**2010 ** 48 0.080
(0.198)
-0.105
(-0.733)
0.280
(1.614)
-0.818*
(-4.264)
0.133
(1.061)
-0.151
(-1.216)
0.076
(0.247)
0.436 0.354 1.572
**2011 ** 47 -0.616
(-1.209)
-0.038
(-0.419)
0.267**
(2.053)
-0.664*
(-4.236)
-0.064
(-0.618)
-0.110
(-0.996 )
0.245
(0.938)
0.4995 0.426 2.466
**2012 ** 21 0.575
(0.616)
0.319
(1.523)
0.061
(0.287)
-0.549*
(-2.708)
0.151
(0.811)
0.077
(0.407)
-0.288
(-0.631)
0.409 0.173 2.085

*Note : a _{This table shows the parameter estimates for the regression of weekly volatility index changes on two lags, two lead, and the contemporaneous S&P CNX Nifty index returns, as well as }*

*magnitude (absolute value) of contemporaneous market returns (eq 3).The weekly observations are average of closing values of trading days of each week. t-statistics are provided in the *
*parentheses and are based on Newey-West (1987) method for correction of heteroscedasticity and autocorrelation . *, **, *** presents 1%, 5% and 10% levels of significance respectively. *

The Table 3.8 reports the results of regression for the weekly IVIX changes and S&P
CNX Nifty returns. The results for the weekly data show that there is negative relation
between changes in implied volatility index and market returns. These results are in
line with the findings of analysis done using daily data. The value of
contemporaneous coefficient is found to be ̂_{ } = 0.401, which reveals a strong
volatility-return relationship. The coefficient of contemporaneous index return is
found to be negative and significant in all sub-periods expect for 2009 and the value
of R2 ranges from 40% to 49% approximately for the years 2010, 2011 and 2012.
The value of ̂_{| |} coefficient is 0.211 (t-statistic = 1.534) indicating a positive
relationship between the size of weekly stock market moves and changes in expected
market volatility. The and , show an asymmetric relation for the weekly data
also, where = -0.190 and = -0.612. Overall, the lagged coefficients in eq (3) are
insignificant or marginally significant, the leverage effect explanation does not hold
true for the daily and weekly data in the Indian markets.

**3.2.8 Multiple regression model for return-volatility relationship: The asymmetry **
in return-volatility relationship for financial market can be studied using the two
partition asymmetric equation with returns being separated into upside and downside
partitions (Low, 2004). This regression equation is adopted for Indian stock market to
investigate the relationship between changes in risk perception and stock index
returns. The relationship between risk and return is estimated using the various forms
embodied in the functional equation 4 to equation 9. The following linear OLS
regression is specified:

………...(4) where, represents percentage change in IVIX and represents S&P CNX Nifty returns. For accommodating the asymmetry in Indian markets, the two partition asymmetric equation with positive and negative returns is used. The multiple regression equation is estimated:

………(5) The above regression equation is equivalent to two linear regression equations (upside and downside partition) of the following form:

……….(6)

_{ } ……….(7)
where, and are %ΔIVIX and reduced by removing the days when < 0
and where, and are %ΔIVIX and reduced by removing the days when
≥ 0.

**Table 3.9: Results of regressions of percentage changes in daily Indian volatility index on contemporaneous S&P CNX Nifty returns **

**Regression ** **Obs. ** **Constant ** **Rt** **(** **R2** **Adjusted R2** **DW **

**A ** 829 0.199
(1.328)
-1.879*
(-5.176)
0.259 0.258 1.988
**B ** 432 -1.819*
(-3.726)
-0.575
(-1.595)
0.024 0.022 1.878
**C ** 397 0.148
(0.268)
-2.589*
(-4.034)
0.182 0.180 1.743
**D ** 432 -1.086*
(-3.239)
-1.489*
-3.702
0.102*
(4.325)
0.060 0.056 1.844
**E ** 397 -1.218**
(-2.502)
-5.418*
(-4.712)
-0.853**
(-2.036)
0.232 0.228 1.946

*Note: This table shows the parameter estimates for regressing of daily percentage changes in the implied volatility index on contemporaneous returns. the regression equations are A: *

*; B:*

*;*

**C:***;*

**D**:*;and*

**E**:*.*

*and*

*are*

*%ΔIVIX and * * reduced by removing the days when * * < 0 and * * and * * are %ΔIVIX and * * reduced by removing the days when * * ≥ 0. t-statistics are provided in the parentheses and are *
*based on Newey-West (1987) method for correction of heteroscedasticity and autocorrelation . The levels of significance are represented with asterisks (* = 1%, ** = 5% and *** = 10%). *

**Table 3.10: Results of regression of percentage changes in weekly Indian volatility index on contemporaneous S&P CNX Nifty returns **

**Regression ** **Obs. ** **Constant ** **Rt** **(** **R2** **Adjusted R2** **DW **

A 173 0.491 (0.767) -1.311* (-3.861) 0.165 0.161 2.038 B 99 -3.281* (-3.754) -0.078 (-0.250) 0.001 -0.010 1.879 C 74 0.319 (0.196) -2.236** (-2.084) 0.158 0.146 2.451 D 99 -3.192* (-2.719) -0.149 (-0.186) 0.006 (0.123) 0.001 -0.020 1.873 E 74 -0.620 (-0.300) -3.439 (-1.169) -0.230 (-0.322) 0.162 0.139 2.602

*Note: This table shows the parameter estimates for regressing of weekly percentage changes in the implied volatility index on contemporaneous returns. The regression equations are A: *

*; B:*

*;*

**C:***;*

**D**:*;and*

**E**:*.*

*and*

*are %ΔIVIX and * * reduced by removing the days when * * < 0 and * * and * * are %ΔIVIX and * * reduced by removing the days when * * ≥ 0. t-statistics are provided in the parentheses and *
*are based on Newey-West (1987) method for correction of heteroscedasticity and autocorrelation . The levels of significance are represented with asterisks (* =1%,**=5%, and *** = 10%)*

A quadratic term is added to the upside and downside return partition to estimate non-linear return-volatility relation. The non-non-linear regression equation estimated is:

………(8)

……….(9) The results obtained for daily and weekly data by estimating equations 4, 6, 7, 8 and 9 are reported in Table 3.9 and Table 3.10 respectively. Table 3.9 reports the results obtained by regressing percentage changes in daily IVIX on contemporaneous S&P CNX Nifty 50 returns. The slope in the univariate linear regression for daily and weekly data is negative and significant at 1% level (Row 4 in table 3.9 and 3.10). This implies that when index returns are negative, IVIX goes up and when it is positive, IVIX goes down.

The results from the two partition asymmetric regression are reported in Tables 3.9 and 3.10. The R2 in the downside return partition (Row 7 in Tables 3.9 and 3.10) is higher than the R2 reported for the upside returns partition (Row 6 in Tables 3.9 and 3.10). The difference in the slopes (β+ - β- = 2.014 and β+ - β- = 2.158) is statistically significant at 1% level and 5% level for the daily and weekly data respectively. The same has been examined using the incremental regression equation (Eq. 10 for daily data and 11 for weekly data):

where, is a dummy variable which is equal to 1 when > 0 otherwise 0. The t-statistic reported in the parenthesis are obtained using the Newey and West (1987) method.

This asymmetric relationship for Indian markets is consistent with the notion of loss aversion phenomenon proposed in Kahneman & Tversky’s (1979) Prospect Theory. Here, value function mentioned in the Prospect theory, is replaced by a metric called implied volatility index, that is often considered as the investor’s fear gauge or a sentiment index. This concept had been also validated for American markets volatility index (Low, 2004).

**3.2.8.1 Reclined S-curve: Tables 3.9 and 3.10 also report the results of the non-linear **
return-volatility relation estimate of daily and weekly data. A quadratic term is added
to each of downside and upside return partition equations. The results of quadratic
equations for daily data are reported in Table 3.9. The results show that the downside
partition has convex profile ( > 0, <0 and significant) and the upside partition
has a concave profile ( < 0, <0 and significant). The results shown in Table 3.10
presents a convex profile for the downside partition and concave profile for the upside
partition but the coefficients are found to be insignificant and the value of R2 is very
less. This non-linear relation is best described by Low as a downward-sloping
reclined S-curve. Similar results are found in the context of the Indian markets but
the degree of association is less when compared to the American markets. In case of
Indian markets, the quadratic fit for the downside return partition equation has a lower
value (R2 = 0.22) in comparison with American markets (R2 = 0.77). The convexity in
the downside return partition indicates accelerating increases in the IVIX; and
concavity in the upside returns partition indicates accelerating decreases in the IVIX.
**3.2.9 QRM for return-volatility relation: To assess the short-term asymmetric **
return-volatility relationship at different levels of changes in implied volatility
distribution, a quantile-regression model is applied. This model is a generalization of
the multiple regression model (MRM) given by Low (2004), Goit (2005), Hibbert *et *
*al. (2008) and Badshah (2010). In this study, the standard MRM is extended by *
modelling the asymmetric return-volatility relation using the QRM to analyse the
impact of negative and positive returns across the various quantiles of implied
volatility changes. The major drawback of MRM specification is that the effect of
negative and positive returns is static across various levels of IVIX changes. So the
MRM model would miss important information across quantiles of IV changes that
could otherwise be captured using the QRM. Further, the descriptive statistics show
that stock returns and changes in IVIX have fat tails and the series are not symmetric.
Thus, the QRM can be used for modelling conditional quantiles in IVIX as a function
of independent variables. QRM method effectively estimates the rate of change in all
parts of distribution of a response variable. This framework allows for the
heteroscedasticity in the error terms in order to obtain different coefficients at

different quantiles. The QRM equation is estimated using the quantile-regression method proposed by Kronear and Bassett (1978).

For analysis, the following regression framework is formulated in which is daily percentage change in volatility index, IVIX and is the daily percentage continuously compounded returns of S&P CNX Nifty index:

{ _{ } _{ } _{ } { _{ } _{ } _{ }

Thus, the standard MRM for asymmetric relation has the following form:

∑
∑ _{ }
∑ _{ }

where, is the intercept term, is the coefficient for lagged implied volatility
index *IVIX, * L=1-3, and are the coefficients for positive and negative stock
market returns, and L=0-3 for both type of returns. is the error term which is
assumed to be independent and identically distributed (iid) with mean zero. The
following *qth QRM specification which is generalization of MRM specification (12) *
for measuring the asymmetric relation is estimated:

_{ ∑ } _{ }
∑ _{ }
∑ _{ }

where, is the intercept term, is the coefficient for lagged implied volatility index IVIX, L=1-3. and are the coefficients for positive and negative stock market returns and L=0-3 for both type of returns. is the error term which is assumed to be independent and derived from the error distribution Φq ( with qth quantile equal to zero. The prime feature of QRM is that the coefficients

_{ } _{ }

capture the effect of independent variable on dependent variable across each of the qth quantile within the range q ϵ (0, 1).

The results of MRM and QRM for asymmetric return-volatility relation between IVIX changes and S&P CNX Nifty returns are presented in Table 3.11. The estimated model includes 11 co-variates and an intercept term. The following inferences can be drawn from the estimated values:

The results show that the sign of slope coefficients are significantly negative in
either direction around the centre of the distribution (q= 0.50). This implies that
there is an inverse relation between returns and changes in implied volatility index.
The results are in concurrence with other studies investigating return-volatility
relation, using traditional multiple regression models (Fleming *et al., 1995 and *
Whaley, 2008) or quantile regression models (Kumar, 2012).

The estimated coefficients of and in Table 3.11 present the contemporaneous return-volatility relation. Table 3.11 also presents their comparison with the coefficients of lagged covariates i.e. lagged positive returns, negative returns and VIX changes. The results show that the contemporaneous returns coefficients are significant at 1% level across all quantiles except q=0.05 for market declines and q= 0.70,…, 0.95 in upward direction. It is also observed that the coefficients of lagged covariates are mostly insignificant or contribute marginally to changes in implied volatility index.

The above results supports that the behavioural explanations, such as
representativeness and affect heuristic dominates the return-volatility relation. This
confirms the hypothesis II that the contemporaneous returns are important
determinants of changes in IVIX. However, the fundamental theories of leverage
hypothesis and volatility feedback hypothesis fail to explain the short-term
return-volatility relationship. These findings are consistent with the results of Hibbert *et *
*al. (2008) and Badshah (2012). Further, this relationship is not statistically *
significant at the tails and the independence is more pronounced at right tail rather
than left tail.

The absolute difference in the estimated coefficient of covariates and in Table 3.11 depicts that there is evidence of asymmetry in all quantile estimates including the OLS estimate. The absolute value of is higher than the absolute values of . The Wald test (Table 3.12) is applied to determine the statistical difference between the coefficients and in equation (13). The null hypothesis (contemporaneous positive and negative returns coefficients are equal) is rejected at each quantile except at extreme quantiles (q= 0.05,…0.25). This shows that asymmetric relationship is observed at q= 0.30,…0.95. In these

**Table 3.11: Quantile regression results of changes in Indian volatility index and S&P CNX Nifty returns **

**Q ** **Intercept ** **Pseudo R2** **Adjusted R2**

**0.05 ** -4.616*
(-5.459)
-2.156
(-1.442)
-2.210*
(-2.872)
0.551
(-0.773)
1.092
(-1.431)
-0.244
(-0.462)
-0.128
(-0.274)
-0.416
(-0.663)
-0.73
(-1.228)
-0.150**
(-1.775)
-0.001
(-0.013)
-0.147**
(-1.940)
0.166 0.155
**0.1 ** -3.584*
(-5.092)
-2.370*
(-3.320)
-2.476*
(-4.998)
0.719**
(-2.111)
0.109
(0.260)
-0.028
(-0.079)
-0.127
(-0.333)
0.064
-0.127
-0.727
(-1.410)
-0.035
(-0.633)
0.002
-0.037
-0.102
(-1.581)
0.192 0.181
**0.15 ** -3.114*
(-5.843)
-2.362*
(-6.716)
-2.128*
(-5.263)
0.625
(-2.153)
0.214
(-0.585)
-0.096
(-0.344)
0.214
(-0.843)
-0.062
(-0.137)
-0.184
(-0.356)
-0.028
(-0.590)
-0.034
(-0.599)
-0.056
(-1.089)
0.211 0.2
**0.2 ** -2.634*
(-5.850)
-2.431*
(-6.886)
-1.970*
(-6.892)
0.748*
(-2.822)
0.324
(0.988)
-0.123
(-0.553)
0.244
(-1.072)
-0.101
(-0.240)
-0.108
(-0.240)
-0.032
(-0.835)
-0.017
(-0.386)
-0.05
(-1.216)
0.225 0.215
**0.25 ** -2.358*
(-6.487)
-2.694*
(-8.000)
-1.930*
-9.178
0.695**
(-2.068)
0.476
(1.414)
-0.053
(-0.237)
0.264
(-1.207)
-0.002
(-0.004)
-0.017
(-0.043)
-0.02
(-0.518)
-0.003
(-0.069)
-0.049
(-1.358)
0.231 0.221
**0.3 ** -2.203*
(-5.756)
-2.901*
(-8.751
-1.877*
-7.361
0.431
(-1.1)
0.413
(1.117)
0.085
(0.374)
0.353
(1.486)
0.157
(-0.389)
0.017
-0.049)
-0.021
(-0.457)
-0.005
(-0.140)
-0.031
(-0.852)
0.228 0.218
**0.35 ** -1.783*
(-4.183)
-3.088*
(-8.238)
-1.819*
(-6.19)
0.266
(0.687)
0.683**
(2.086)
0.097
(0.388)
0.552*
(2.65)
-0.009
(-0.024)
0.032
(-0.095)
-0.026
(-0.465)
0.014
(-0.379)
-0.025
(-0.642)
0.227 0.216
**0.4 ** -1.24*
(-2.908)
-3.193*
(-8.046)
-1.773*
(-5.538)
0.282
(0.783)
0.535***
(1.669)
0.238
(1.069)
0.531**
(2.575)
0.132
(-0.413)
-0.125
(-0.382)
-0.028
(-0.490)
0.008
(-0.229)
-0.022
(-0.556)
0.228 0.218
**0.45 ** -1.236*
(-3.157)
-3.584*
(-9.499)
-1.484*
(-4.361)
0.409
(1.422)
0.411
(1.361)
0.245
(1.255)
0.509**
(2.473)
0.274
(-0.881)
-0.023
(-0.077)
-0.012
(-0.222)
0.016
(-0.42)
-0.029
(-0.807)
0.232 0.222
**0.5 ** -0.882**
(-2.218)
-3.733*
(-10.397)
-1.395*
(-4.202)
0.255
(0.949)
0.521***
(1.818)
0.331***
(1.863)
0.491**
(2.352)
0.287
(-0.981)
-0.078
(-0.263)
-0.003
(-0.064)
0.014
(-0.352)
-0.026
(-0.781)
0.236 0.226
**0.55 ** -0.757*
(-1.922)
-3.961*
(-11.549)
-1.365*
(-4.312)
0.264
(1.003)
0.406
(1.236)
0.387**
(2.131)
0.466*
(2.005)
0.448
(-1.568)
-0.02
(-0.075)
0.003
(-0.056)
0.032
(-0.716)
-0.021
(-0.586)
0.239 0.229
**0.6 ** -0.355
(-0.910)
-4.061*
(-11.920
-1.210*
(-3.432)
0.36
(1.217)
0.54***
(1.692)
0.315
(1.386)
0.626**
(-2.531)
0.371
(-1.285)
0.03
-0.112
-0.022
(-0.418)
0.063
(-1.352)
-0.011
(-0.284)
0.24 0.229
**0.65 ** -0.307
(-0.835)
-4.201*
(-12.192
-0.995*
(-2.591)
0.286
(0.908)
0.462
(1.476)
0.348
(1.4)
0.781*
(-2.619)
0.529**
(-2.031)
0.029
(-0.11)
-0.01
(-0.201)
0.069
(-1.371)
0.003
(0.070)
0.241 0.231
**0.7 ** -0.034
(-0.100)
-4.455*
(-13.175)
-0.648
(-1.544)
0.207
(0.676)
0.696**
(2.203)
0.382
(1.464)
0.933*
(-3.018)
0.601**
(-2.384)
-0.05
(-0.165)
-0.023
(-0.495)
0.109**
(-2.162)
0.002
(0.051)
0.245 0.235
**0.75 ** 0.398
(-0.993)
-4.603*
(-12.984)
-0.38
(-0.893)
0.369
(1.341)
0.827**
(2.29)
0.417
(1.452)
1.03*
(-3.25)
0.65**
(-2.372)
-0.132
(-0.401)
-0.007
(-0.141)
0.139**
(-2.27)
-0.026
(-0.586)
0.252 0.242
**0.8 ** 0.84***
(-1.664)
-4.598*
(-11.625)
-0.099
(-0.231)
0.456***
(-1.714)
0.359
(0.842)
0.499
(1.359)
0.947*
(-2.819)
0.688**
(-2.05)
-0.322
(-0.863)
0.011
-0.232
0.132**
(-1.887)
-0.037
(-0.704)
0.261 0.251
**0.85 ** 1.19**
(-2.004)
-5.001*
(-10.348)
-0.161
(-0.486)
0.665***
(-1.807)
0.329
(-0.604)
-0.018
(-0.046)
0.893**
(-2.22)
0.64
(-1.584)
0.309
(-0.834)
-0.032
(-0.525)
0.104
(-1.38)
-0.016
(-0.256)
0.278 0.268
**0.9 ** 1.975* -5.312* -0.225 0.669*** 0.252 0.117 1.004** 0.722*** 0.502 -0.018 0.116 0.068 0.298 0.289

(-3.226) (-9.203) (-0.621) (-1.637) (-0.381) (-0.279) (-2.283) (-1.727) (-1.175) (-0.251) (-1.398) (-0.998)
**0.95 ** 3.315*
(-4.531)
-6.088*
(-9.663)
0.351
(-0.724)
1.459*
(-2.851)
-0.023
(-0.032)
-0.159
(-0.298)
1.297*
(-3.225)
0.657
(-1.456)
0.23
(-0.438)
0.127
(-1.42)
0.214**
(-2.194)
0.09
(-1.132)
0.333 0.324
**OLS **
-0.763***
(-1.826)
-3.431*
(-5.995)
-0.844**
(-2.126)
0.573*
(2.828)
0.425
1.475
0.006
0.026
0.521*
(3.112)
0.021
(0.060)
-0.299
(-0.981)
-0.012
(-0.304)
0.032
0.832
-0.06***
(-1.604) 0.339 0.330

**Panel B: Quantile slope equality test results: Only significant results of asymmetry are reported **

0.2-0.3** 0.2-0.4** 0.2-0.4** 0.25-0.5* 0.25-0.5* 0.25-0.5* 0.3-0.5* 0.3-0.5** 0.4-0.5** 0.4-0.5** 0.4-0.6* 0.4-0.6** 0.5-0.75* 0.5-0.75* 0.5-0.75* 0.5-0.75* 0.6-0.7** 0.6-0.8** 0.75-0.95** 0.75-0.95* 0.75-0.95** 0.75-0.95** 0.75-0.95*** 0.8-0.95** 0.8-0.95** 0.8-0.95** 0.8-0.95** 0.8-0.95** 0.9-0.95*** 0.9-0.95*** 0.9-0.95**

*Note: the table shows the parameters estimated for the MRM and QRM specification 12 and 13 respectively, for the asymmetric return-volatility relation between changes in the IVIX and S&P *
*CNX Nifty returns. In QRM specification 13, the standard errors are obtained using the bootstrap method; therefore robust t-statistics (in parenthesis) are computed for each of the quantile *
*estimates. The MRM specification 12 is computed with Newey and West, 1987 correction for heteroscedasticity and autocorrelation. In last panel the results of quantile slope equality test are *
*reported which rejects the null hypothesis of the equality of coefficients across quantiles. *,** and *** denotes the rejection of null hypothesis at 1%, 5% and 10% level of significance. *

**Table 3.12: Wald tests for the equality of coefficients **

**Quantiles ** **0.05 ** **0.10 ** **0.15 ** **0.20 ** **0.25 ** **0.30 ** **0.35 ** **0.40 ** **0.45 ** **0.50 ** **0.55 ** **0.60 ** **0.65 ** **0.70 ** **0.75 ** **0.80 ** **0.85 ** **0.90 ** **0.95 **

**F-statistic ** 0.00 0.01 0.14 0.91 1.67 3.33 5.39 5.96 9.85 15.49 23.48 26.85 28.75 38.14 37.38 190.25 139.55 75.55 45.26

**p-value ** 0.96 0.92 0.71 0.34 0.19 0.07 0.02 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

*Note: this table shows the parameters estimated for the Wald test. The null hypothesis, Ho = The coefficients for the contemporaneous positive and negative returns are equal i.e. *

quantiles the representative heuristic which is reinforced by affect heuristic, gets dominates because the investor associates higher levels of volatility with higher risk and negative returns with least benefits. This confirms the Hypothesis IV that the asymmetry exists for return-volatility relation.

More specifically, looking at each row of Table 3.11 (i.e. each quantile of estimates), the impacts of positive and negative returns on IVIX distribution is changing and highly asymmetric. The dynamic nature quantile estimates provide an interesting picture of how the changes in the IVIX distribution depends on the contemporaneous and lagged covariates. The absolute value of is increasing monotonically when moving from a median quantile towards an upper quantile; that is the marginal effect of negative returns is much larger in upper quantiles (i. e. q= 0.95) and vice-versa for positive returns. The equality of slopes across quantiles is also tested using Wald Test. To examine this equality, quantile slope equality test is used in which coefficients of each variable across quantiles are compared; the null hypothesis that the coefficients of a particular covariate across quantile are same is tested. The test results are reported in Panel B of Table 3.11.

Thus, these asymmetric responses across the quantiles of changes in IVIX confirm the Hypothesis V: that the contemporaneous negative returns are the major determinants of IV changes in the context of short-term return-volatility relationship. From this it can inferred that in short-term and during the period of extremely volatile market conditions, the affect heuristic and time pressure dominates the investor’s judgement, and this is found to be consistent with the behavioural explanations of Finucane et al. (2000).

The coefficients of contemporaneous positive and negative returns, lagged returns and lagged implied volatility index covariates, with their 19 quantile-regression estimates, are plotted in Figure 3.8. Each plot on the x-axis represents the quantile (or q) scale, and y-axis indicates the percentage-point change of the covariate on volatility changes, holding other covariates constant.

**Figure 3.8: Graphical representation of the coefficients estimated for contemporaneous and lagged covariates across various quantiles **
**for the response variable IVIX **

*Note: In graph the coefficients are represented as: c= intercept term; PIVIX(-1) = * _{ }*; PIVIX(-2) =* _{ }*; PIVIX(-3) =* _{ }*; RNSEN = * *; RNSEN(-1) = * _{ }*; *
*RNSEN(-2) = * _{ }*; RNSEN(-3) =* _{ }*; RNSEP =* *; RNSEP(-1) =* _{ }*; RNSEP(-2) = * _{ }*; and RNSEP(-3) = * _{ }

-8 -6 -4 -2 0 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0 Quantile C -.3 -.2 -.1 .0 .1 .2 .3 0.0 0.2 0.4 0.6 0.8 1.0 Quantile PIVIX(-1) -.2 -.1 .0 .1 .2 .3 .4 0.0 0.2 0.4 0.6 0.8 1.0 Quantile PIVIX(-2) -.3 -.2 -.1 .0 .1 .2 .3 0.0 0.2 0.4 0.6 0.8 1.0 Quantile PIVIX(-3) -8 -6 -4 -2 0 0.0 0.2 0.4 0.6 0.8 1.0 Quantile RNSEN -0.5 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 Quantile RNSEN(-1) -2 -1 0 1 2 3 0.0 0.2 0.4 0.6 0.8 1.0 Quantile RNSEN(-2) -1.0 -0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Quantile RNSEN(-3) -4 -3 -2 -1 0 1 2 0.0 0.2 0.4 0.6 0.8 1.0 Quantile RNSEP -2 -1 0 1 2 3 0.0 0.2 0.4 0.6 0.8 1.0 Quantile RNSEP(-1) -2 -1 0 1 2 0.0 0.2 0.4 0.6 0.8 1.0 Quantile RNSEP(-2) -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 Quantile RNSEP(-3)

**3.3 Summary **

The dissemination of implied volatility index by NSE has provided the academic researchers a new measure for exploring volatility. This chapter describes the information content of implied volatility index of India over a duration of nearly three years. This chapter intends to investigate the univariate time-series properties, and seasonal patterns exhibited by IVIX. The risk-return relationship is studied using three methodologies which include regression of changes in implied volatility index on: lead-lag, contemporaneous and contemporaneous absolute stock market returns; on positive and negative returns; and finally, the short-term return-volatility relationship is determined using the quantile regression method.

The time series properties of daily and weekly IVIX changes are found stationary. The autocorrelation function for the daily changes in IVIX is significantly negative and small, which reflects that it follows a mean reverting process. A significant Monday effect is observed in the IVIX series.

The implied volatility exhibits a negative and asymmetric temporal relation with stock market returns. This absolute value of coefficient of non-linear regression shows that there exists an asymmetric and non-linear relationship, best described as downward sloping reclined S-curve by Low (2004) but the degree of association is less for the Indian markets as compared to the American markets.

On an average, a short-term negative asymmetric return-volatility relation is also found which is shown by quantile regression method. The behavioural explanation holds true while explaining the return-volatility relation. However, when sharp upside moves are observed in the market, changes in IVIX and Nifty returns move independent of each other. On the other hand, when the market moves down, the relationship becomes insignificant only for the steepest or the worst declines.

Finally, it can be summarized that there is an asymmetric relationship between changes in implied volatility and index returns. The contemporaneous returns are found to be significantly negative and are the most important determinants of changes in current implied volatility. These results support the representativeness and affect heuristic of behavioural theories and reject the fundamental theories of leverage and volatility feedback. The main conclusion drawn from the above discussion is that the IVIX can act as a sentiment index or investor fear gauge for the Indian capital markets. It can therefore be used by practitioners both as a market timing tool and in portfolio management.