# The properties of India Vix

Abstract: Volatility indices reflect the general consensus of the expected short term volatility. India is one of the emerging markets with a very active trading in index options along with a volatility index hence it ought to be rich in information. This paper examines the distributional properties of volatility index of India, India Vix (Ivix) using quantile regression methodology. The results show that changes in Ivix are negatively related to stock market returns but there is only a feeble asymmetric response to positive and negative returns in stock returns. Daily variance forecasts obtained from Ivix contains important information about future market variance but is a biased forecast of the realized variance. a comparative analysis of the developed and emerging market indices shows that generally volatility in an emerging market like India is more than that observed in developed markets but during the peak of the global crisis the volatility in India is significantly less than that observed in the US, UK or Japan. Finally, the study finds that one day lagged volatility spillover is observed between the US market and Indian market while transmission in opposite direction was not observed.

Keywords: Implied volatility, Volatility indices, quantile regression, volatility spillover

JEL Classifications: G10, G14, G15, G17, C22

### Introduction

A volatility index is designed to reflect the expected short term future volatility. CBOE was the first exchange to introduce the volatility index VIX in 1993 since then it became synonymous with US stock market volatility. Just as increases in the Dow are cheered by the market, in contrast, increases in VIX are feared as an increase in volatility means increase in uncertainty. In fact some commentators nicknamed it is as the investors fear gauge. Financial media reporting of VIX almost paralleled Dow or the S & P 500 index. Subsequently other leading exchanges had introduced volatility indices notably the Deutsche Brse introduced VDAX in 1994 and VX1 and VX 6 by the French Options exchange in 1997. Table 1 below presents some of the major volatility indices currently used to measure the respective market volatilities. As can be noted most of the indices are for the US and European markets and India Vix is the only emerging market volatility index. By 2008 India is the first market in Asia-Pacific with a volatility index computed and disseminated by an organized exchange.

Though the idea of volatility index was first proposed by Gastineau (1977) it was the seminal work of Whaley (1993) that paved the way not only the introduction of a risk measure but also to facilitate volatility trading in an economical way. Initially VIX was calculated as a weighted average of the implied volatilities derived from eight at-the-money, nearby and second-nearby options on S & P 100 index (OEX option contract). The implied volatilities were computed by inverting the Black-Scholes option pricing model. These implied volatilities were interpolated in such a way that they represent the implied volatility of a hypothetical at the money OEX option with exactly 30 days to mature. VIX is quoted in percentage points just like the standard deviation of rates of return.

In September 2003 CBOE revamped the VIX calculation methodology and introduced the new VIX based on the S & P 500 index. In the new methodology[1] VIX is computed from all the available option prices that are out of the money and not from the implied volatilities. Another significant difference is the calculation is not based on any option pricing model and is based on the following formula given by Britten-Jones and Neuberger (2000).

where s is the VIX/100; F is the forward index level derived from index option prices; K0 is the first strike below the forward index level, F; Q(Ki) is the mid-price of the bid/ask spread for the option with strike Ki. ?Ki is the interval between strikes on either side of Ki, R is the risk-free interest rate to expiration and T is the time to maturity in minutes.

National Stock Exchange of India (NSE) introduced the country's first volatility index, India Vix in April 2008. Broadly the calculation methodology is similar to that of CBOE's current VIX calculation. NSE uses the near and mid month options bid and offer prices of the Nifty 50 index call and put options. The exchange made available the past data of India Vix from Nov 1 2007. Since the usefulness of the volatility index depends on its properties and the information contained to forecast realized volatility in this study an attempt is made to examine the same for India Vix (Ivix hereafter). The study also examines the relationship with stock market returns as it proves the utility of volatility futures/options in portfolio diversification. Finally the study investigates the volatility transmission between India, an emerging market and developed markets from three geographic regions North America (represented by the US), Europe (represented by the UK), and Asia (represented by Japan).

This study contributes to the existing volatility analysis literature in the following ways. First, it examines the properties of a volatility index from an emerging market that is constructed free of any pricing model. There is abundant work done in the context of developed markets and the Indian derivatives market didn't receive attention that it warrants. This despite Indian derivatives market is one of the leading markets of the world. This can be gauged from Table 2 that depicts the turnover of index options from world's major exchanges. Second, the study utilizes Quantile regression in our study which is superior, especially when the data exhibit fat tails, to OLS regression employed by earlier studies. Financial data are characterized by fat tails and hence using quantile regression to examine the empirical regularities observed elsewhere will give a new perspective. To the best of our knowledge there is no published work in this area using Quantile regression method. Third, this study looks at volatility transmission between developed markets and an emerging market using the volatility indices that are ex ante while past studies in this domain are not only few but are based on the returns and volatility data that are ex ante. Fourth, this study looks at whether emerging markets behave differently and become more volatile than developed markets in a crisis. Finally, the results of the study will be of interest to to traders and institutional investors who may consider trading OTC volatility futures/forwards based on India Vix.

### Data

This study uses Nifty index returns (denoted as RNifty), Ivix, VIX, VFTSE, and VXJ index series. Daily closing values[2] of the indices spanning from Nov 1 2007 till Dec 31 2009 have been used in this study. The time period of the study represents a full cycle with Nifty reaching a crest of 6287.5 points and followed by a decline to a trough of 2524.2 points.

### Statistical Properties

Figure 1 depicts a time series plot of the Nifty index and Ivix movements over the period Nov 2007 till Dec 2009. In conjunction with the descriptive statistics from Table 3 it can be noted that on an average Ivix hovered around 37.79%. The mean of Ivix is found to be approximately close to the annualized standard deviation of returns. Ivix reached the maximum value of 85.13% on Nov 17 2008 almost during the peak of the global financial crisis and a minimum of 22.69% on Dec 24 2009. During the period under consideration the median closing level is 36.04% and 50% of the time it ranged between 43.46% to 30.06% (a range of 13.4 percentage points) and 90% of the time it closed between 57.41% and 25.06% (a range of 32.35 percentage points). The average of the first difference of the Ivix (dIvix) is not statistically different from zero indicating the absence of any trend. The higher moments suggest that the distributions of Nifty returns and Ivix are leptokurtic. All the data series are non-normally distributed as can be noted from the normality tests - Jarque-Bera and Doornik-Hansens' test statistics both strongly reject the hypothesis of normal distribution. The results of unit-root tests indicate that the time-series of the Nifty and Ivix are non-stationary in levels but are stationary in first differences.

A look at the cross correlations between changes in Ivix and the Nifty returns reveals that there is a negative contemporaneous correlation (significantly different from zero) however the degree of correlation is rather weak compared with the extent of negative correlations observed in other markets. There is no significant correlation between changes in Ivix and lagged Nifty returns.

The study proceeds to investigate the existence of any seasonality or predictable trends in the Ivix data. Figure 2 shows the average Ivix on different days of the week. It appears that Ivix decreases on an average as the week progresses from Monday to Thursday and increases on the Fridays. As a preliminary check (results not reported) pair-wise T-tests of means were conducted and all the tests do not reject the null hypothesis of zero mean differences. Hence the observed trend is not statistically significant. Formally this is tested using day-of-the-week dummy variable for testing the seasonality. The dummy variable takes a value of 1 for a given weekday zero otherwise. The model includes dummy variables for all days except for Mondays and an intercept term is also included. Presence of seasonality will be reflected in a statistically significant slope coefficient. Each coefficient measures the average difference in Ivix with reference to Monday's Ivix level and the intercept measures the average Ivix level for Monday.

Where j is the day of the week with Tuesday represented as 2, Wednesday as 3 etc., and Djt is the dummy variable that assumes a value of 1 on day j and 0 otherwise. Results from the regression presented in Table 5 and the test statistics are computed using HAC corrected standard errors. The results indicate that the intercept term is significant and is equal to the mean volatility on Monday. But none of the dummy variables are statistically significant and hence the apparent seasonality in the Ivix is not statistically significant.

### Relationship between Implied volatility index and the Stock market returns

Asset pricing theories like CAPM assert that stock prices decline as expected risk (volatility) increases. Black (1976) termed it as leverage effect, according to which as stock prices fall (vis--vis bond prices) the equity value decreases leading to an increase in financial leverage. An increase in financial leverage leads to a concentration of business risk on the equity holders hence the increase in risk of the firm's equity. Subsequent works of Christie (1982) and French et al (1987) all document a similar negative relationship. Moreover the association is just not negative but also asymmetric in nature. Schwert (1990) demonstrate that the increase in expected volatility corresponding to a given fall in stock prices is more than the decrease in expected volatility corresponding to a similar size gain in prices. In the volatility indices literature Flemming et al (1995), Whaley (2000) provides evidence of negative and asymmetric relationship from the VIX and S & P 100 indices. Simon (2003) analyzes the response of the Nasdaq 100 Volatility Index (VXN) to Nasdaq 100 index (NDX), and documents a fall in VXN in response to large positive NDX returns, and rise in VXN in response to large negative NDX returns. Giot (2005) finds there is a negative relationship between contemporaneous changes in implied volatility indexes and the underlying stock indexes for both the S&P 100 and the Nasdaq 100 but finds a subdued asymmetric relationship between Nasdaq 100 and VXN. Subsequent studies by Skiadopoulos (2004) for the Greek market and Ting (2007) for the Korean market also provide similar evidence. However Dowling and Muthuswamy (2003) for the Australian market finds no asymmetric response while Frijns et al (2008), using data over a longer period, provide mixed evidence for the Australian market. Siriopoulos and Fassas (2009) also find absence of a statistically significant asymmetric relationship for Russel 2000 volatility index, VIX, VXN, VBEL and MVX.

This above minimization can be formulated either as a linear programming or as a GMM problem.

The negative and asymmetric relationship between volatility changes and stock market returns (Rnifty) is tested by a quantile regression of the daily changes of the volatility index against the Nifty daily returns. Following the methodology first employed by Flemming et al (1995) and later adapted by Frijns et al (2008) the following quantile regression[3] is estimated at each decile.

Since there was a contemporaneous negative correlation between stock returns and Ivix changes it is expected that 0 to be negative and statistically significant. The asymmetry between stock returns and changes in volatility will be captured by the sum of and. If there is an asymmetric relationship then the sum of 0 and will have to be negative for increases in market returns and vice versa for market declines. For this to hold the magnitude of 0 shall be less than the size of.

Table 6 presents the results from the quantile regression and it can be noted that the constant term is statistically significant for most of the quantiles. The Rnifty term is having the expected negative sign for all quantiles and is statistically significant by and large except for the first decile. So this establishes the negative relationship between stock market returns and the changes in volatility. The absolute Nifty returns (Nifty_abs) coefficient is statistically significant across most of the quantiles except for the third decile.

Making use of the regression coefficients in Table 6 the expected change in dIvix (attributable to market returns only) for 1% increase (decrease) in Rnifty is computed in Table 7. It is evident that the negative relationship is not reflected at the higher quantiles when the market moves up and in the reverse direction also it is not observed at quantiles below the median. The negative relationship between market returns and volatility changes is observed correctly and in both directions only at the fifth and sixth deciles. The asymmetric relationship as hypothesized was observed only for the sixth decile. This finding[4] is in contrast to what was observed in most other markets, for instance Fleming et al (1995), and Whaley (2008) using the traditional regression methods. However for Australian markets too evidence on asymmetric relationship is rather mixed. This might be a reflection of the weaker negative correlation between Nifty returns and dIvix presented in Table 4. For instance Fleming et al (1995) find a very high negative correlation of (0.677) between S & P 100 index and Vix while correlation between Nifty returns and dIvix is only -0.135. Another reason may be that Ivix is constructed using only at-the-money options and excluding out-of-the money options is making the Ivix less negatively correlated with Nifty returns. Therefore NSE may consider increasing the range of options especially to include the entire smile in to the calculation of Ivix. This weak negative correlation raises doubts on the diversification benefits of including Ivix in an investment portfolio.

### Relationship between Implied volatility index and the realized volatility

The relationship between implied volatility and realized volatility is one of the well researched[5] topics. Results from the earlier studies are rather mixed with some studies reporting that implied volatility is a good predictor for realized volatility and later studies provide evidence in contrary. However at present, the general consensus is implied volatility though biased, has more predictive power than past realized volatility.

Dowling and Muthuswamy (2003) using the data from Australian options market and the Black-Scholes option pricing model find that implied volatility index AVIX performs poorly compared to historical volatility. In contrast to their study Frijns et al (2008) find from the Australian data that implied volatility index contains important information both in sample and out of sample. Giot (2003) demonstrates that volatility forecasts based on the VIX/VXN indexes have the highest information content, both in the volatility forecasting and market risk assessment frameworks. Jiang and Tian (2005) implemented the model-free implied volatility given by Britten-Jones and Neuberger (2000) to SPX options and find it to be more efficient at forecasting future realized volatility. Corrado and Miller (2005) report that the CBOE implied volatility indexes VXO, VIX, and VXN dominate historical index volatility in providing forecasts of future price volatility for the S&P 100, S&P 500, and Nasdaq 100 stock indexes. Magheribi et al (2007) find that the KOSPI 200 volatility index contains useful information on future volatility of Korean stock market. Siriopoulos and Fassas (2009) examined the predictive power of twelve equity volatility indices and find implied volatility though biased, has more predictive power than past realized volatility.

Results[6] from Table 9 show a is not significantly different from zero and is statistically different from zero hence it may be inferred that Ivix contains important information for future realized variance. If it is an unbiased forecast then the constant term shall be equal to zero and slope coefficient will be equal to 1. To verify this Wald test is conducted with the null as a=0 and = 1. The Wald test statistic at 14.9 rejects the null hypothesis at 1% level of significance hence it can be inferred that though implied variance contains useful information it is not an unabiased forecast of realized variance. Next, the encompassing regression is estimated that generally comprises two or more volatility forecasts as the explanatory variables. The study assumes the one period lagged realized volatility as a competing forecasting approach and the following regression is estimated:

Rvari= a + 1Fvari + 2Rvari-1 + e

If 1 and 2 are both statistically significant it means both the approaches add to each other's and to obtain best forecast both are needed while significance of only one slope coefficient indicates that one approach dominates the other and the information contained in the second approach is incorporated by the dominating approach.

Table 10 shows the results for the encompassing regression and it may be observed that the superiority of the Ivix as a competing forecast is not categorical. The slope coefficient of Fvar is statistically different from zero and that of realized variance is not different from zero only at 5% or less. A Wald test of joint hypothesis a = 0, 1 = 1 and 2= 0 is rejected as the test statistic is 8.23 with a P-value of 0.00411. Since the coefficient of Fvar improved in the encompassing regression it can be concluded that forecasts of Rvar can be improved up on by using both Ivix and past realized variances.

### Volatility spillover Effects

Before the transmission of implied volatility across markets is investigated, first an attempt is made to study the performance of volatility indices during the global financial crisis and its aftermath. Figure 3 presents the comparative movements of Ivix, VFTSE, VIX and VXJ for the period Nov 2007 - Dec 2009. From the comparative statistics[7] in Table 12 it may be observed that on an average Ivix is more than any of the developed market indices reflection of the riskiness of an emerging market. Yet the standard deviation of Ivix is lower than any of the developed markets indicating that the developed markets volatility was quite spiky during the period under consideration.

To understand the behavior of volatility in developed and emerging markets during peak of crisis and the remaining period the data is divided in to two sub-periods. The first sub-period pertains to the peak of global crisis defined as from September 2008 to April 2009. Bartram and Bodnar (2009) notes that massive increases in volatility began in September 2008 hence the study considers it as the start[8] of the first sub-period. The other end point is determined using the historic volatility range of VIX as a guide. The median VIX level is computed for every month after September 2008 to check whether the median volatility falls in the 90% historic normal range of VIX. This range was found to be 11.22% and 30.28% by Whaley (2000). This range is updated by including subsequent observations of VIX from 1999 till end of 2006 accordingly the normal range is found as 11.4% and 31.32%. From April 2009 median VIX levels are back to the normal range. The second sub-period is considered as the normal period defined as the period remaining after excluding the first sub-period from the whole sample period.

An important point that emerges from the comparative movements is that Ivix is at a higher level than other indices for a good deal of time but from October 2008 till early April 2009 Ivix was considerably lower than the volatility indices of developed markets. Again for the rest of the period when the developed markets calmed down Ivix climbed over the other indices. In other words Ivix is lower than other indices during the first sub period and higher during the second sub period. The average level of Ivix is 47.44 around 10% points more compared with its own overall average. Not only the developed markets show marked increase in the average level of volatility indices during the crisis period but it exceeded the average level of Ivix as can be noted from Table 12. This is not surprising as the developed markets were at the centre of the whole crisis. But what is surprising is the average level of VXJ far exceeded that of any other market.

The exceedance of developed market's volatility level over Indian market's volatility level is examined by running the following regression:

Where IVdifft is the difference between the given volatility index (VIX, VFTSE, VXJ) and the Ivix, Dper is a dummy variable that assumes a value of 1 if the observation falls during Sep 2008 to Apr 2009 (crisis period) and zero otherwise. The constant term measures the average difference in volatility levels during the normal period and a significant slope coefficient indicates whether the exceedance/reduction in volatility level during the crisis period.

Table 13 presents the regression results with the coefficients and test statistics computed using HAC corrected standard errors. The results confirm the increase in volatility levels in developed markets during the crisis. The sharpest increase was observed in the Japanese market and the UK market witnessed a lower increase. It can be inferred that normally volatility levels are higher in an emerging market like India but during extreme conditions like the Global crisis the increase in volatility levels is not to the same order as that of developed markets and this feature will have implications in asset allocation and portfolio diversification.

A change in asset price volatility in one market leads to changes in price volatility in another market, apart from the change attributable to the local factors, it is said that volatility spilled over from first to the second market. Hamao et al (1990), King and Wadhwani (1990) Lin et al (1994), Bekart and Harvey (1997) all examined equity market volatility spillovers using ex post returns data. Gemmill and Kamiyama (1997) first examined volatility spillovers using the volatility indices that have an ex ante focus and find that there is a spillover from S & P options to FTSE and Nikkei options. Skiadopoulus (2004) find a contemporaneous spillover between the changes in VXO and the Greek options market. Siriopoulos and Fassas (2009) report similar evidence that globally VIX is the leading source of uncertainty and VSTOXX plays a similar role for the other European markets. This study examines the spillover effects between developed markets and India in the vector autoregression framework. The VAR model below examines whether a large change in one of the markets leads to a similar change in other markets.

The results of the VAR system are reported in the Tables 13 indicate that apart from the past changes in volatility indices changes in VIX is common factor effecting the next day's change in the volatility indices of all the countries. Only the overnight changes in VIX significantly impacts the next day volatility changes in India as can be gauged from the sizable coefficient of dVIX. This indicates that uncertainty from the US gets transmitted to Indian market reminding of the famous newspaper headline if US sneezes rest of the world catches cold. Similar transmission in the reverse direction from India to other markets is not observed as past values of changes in Ivix were not found to be significant in any of the other equations. Interestingly the Japanese market volatility neither influences Indian market volatility nor is influenced by Indian market volatility though both are major economies in Asia. Since the Japanese stock market closes just before the opening of the Indian market it is likely that contemporaneous changes in VXJ may influence change in Indian volatility. This is investigated with the following OLS regression:

### Conclusions

Volatility is quite central in finance and forecasting it had become one of the important pursuits of academics as well as practitioners. Exchange traded option prices reflect the market's expectation of the future volatility over the remaining life of the option. Hence option prices became a rich source of information about future volatility. Volatility indices on index options reflect the general consensus of the expected volatility of the overall market. This paper attempts to examine the distributional properties of the recently launched volatility index, titled as India Vix, derived from Nifty index options that trade on NSE. The study finds that changes in Ivix are negatively related to stock market returns but the asymmetric response of Ivix to changes in Nifty returns is fairly weak. Daily variance forecasts obtained from Ivix contains important information about future market variance but is a biased forecast of the realized variance. The analysis shows that generally volatility in an emerging market like India is more than that observed in developed markets but during the peak of the global crisis the volatility in India is significantly less than that observed in the US, UK or Japan. Finally, the study finds that uncertainty from the US flows to Indian market while transmission in opposite direction was not observed.

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- For an elaborate procedure of the calculation of new VIX refer CBOE (2003).
- All the data except for VXJ was obtained from the respective Exchanges' websites and VXJ data was obtained from the website of CSFI-VXJ homepage http://www-csfi.sigmath.es.osaka-u.ac.jp/en/activity/vxj.php
- Although lags of Nifty returns were included but none of them were found to be significant hence those results were not reported
- To check whether the results are sensitive to the specification of the model and to rule out that possibility alternate specifications used in Whaley (2000), Whaley (2008), and Siriopoulos and Fassas (2009) were also estimated and the results are qualitatively similar.
- For a comprehensive literature review on volatility forecasting see Poon and Granger (2003). Only post Poon and Granger (2003) and works related to volatility indices literature is reviewed here.
- For want of space only the results for the median were presented here. The results are qualitatively similar for the other quantiles.
- Since the data series is organized by synchronizing and adjusting for holidays in different markets hence the some of the summary statistics may not match with those of Ivix series presented in Table 3
- September 2008 is an eventual month, with the announcement of nationalization of Fannie Mae and Freddie Mac, filing for bankruptcy by Lehman Brothers, followed by another announcement of Bank of America that it would be purchasing Merrill Lynch and finally the massive bailout package of $700bn by the Federal Reserve.