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A Copula Approach to Volatility Index in India

Chinnaraja Chendur Pandian provides an overview of measuring volatility in India


Graph- Shutterstock
Photo: Shutterstock

Volatility asset class: portfolio diversifier during tail and catastrophic events

Portfolio diversification is a primary risk management tool to protect against the concentration of risk. Prior to the financial crisis of 2008, asset classes like equities, bonds and commodities were relatively less correlated. However during the 2008 crisis most of the asset values dropped in conjunction with the drop in equity values and the correlation among these assets rose significantly. As a result, many investors realised that portfolios which they believed to be well diversified based on historical data, were effectively not diversified at all.

In contrast, volatility indices in developed markets showed strong negative correlations during the crisis.  This led to an explosion of derivatives on volatility indices being used as effective tools for portfolio diversification to protect against tail or catastrophic events.

India VIX: Volatility index representing Indian market

A volatility index measures the market’s expected annualised volatility over the next 30 days as depicted by the strip of active out-of-the money option bid and offer quotes. The Chicago Board of Exchange (CBOE) designed the first volatility index (VIX) in 1993. It was first derived from the implied volatility of S&P 100 index options and later it was redefined based on the strategy of replicating variance swaps using options on the S&P 500 index. 

Recently Japan, Australia and Hong Kong launched derivatives on their respective volatility indices. India is not far behind either and has entered the market too. Since then, the Indian option market has grown rapidly ahead of many peers in Asia. The National Stock Exchange of India Ltd (NSE) topped the list of Exchanges with the most number of traded index options in 2013 (Table 1).

A Copula Approach to Volatility Index in India - Table1
Table 1: Top 5 exchanges by number of index options traded in 2013

Increasing option volumes based on NSE’s CNX Nifty, a well-diversified 50-stock benchmark index accounting for 23 sectors of the Indian economy, cemented the path for the volatility index in India. In 2008 India’s first volatility index was born and in 2014 futures on India VIX were introduced for the first time. India VIX uses CBOE’s computation methodology (fair price of a variance swap) with some adjustments for its own Indian market dynamics.

• Nifty future value used to identify the strip of out-of-the money options rather than determining the forward index level from the option book

Uses ‘Natural Cubic Spline’ to interpolate quotes for illiquid options

Three trading days left to expiry of the nearest month, India VIX ‘rolls’ to the next and further month

Contract specification of futures on India VIX is pretty unique with a weekly contract cycle and a minimum trade size of nearly $16K compared to $3.33K for futures on stock. Despite the high minimum contract size being five times the size of other derivative contracts traded in India, the traded volume so far is quite impressive. This was highly attributed to the national election events which unfolded during this period. Post-election the trading volume had dropped although more recently the volume has resumed due to increased uncertainty over the Iraq war. 

A Copula Approach to Volatility Index in India - Table2
Table 2: Futures on India VIX traded volume
A Copula Approach to Volatility Index in India - Table3
Table 3: Specification of Futures on India VIX

Co-movement behaviour of India VIX & Nifty:

To fully understand the use of India VIX, its dependencies with underlying option index CNX Nifty should be analysed.  Historically measuring and modelling dependencies has been centred on correlation. The correlation coefficient indicates the strength and direction of a linear relationship between two random variables. The best known correlation measure is the Pearson product-moment correlation coefficient. This is a reasonable measure of dependence when random variables are distributed as multivariate normal, but when this is not the case correlation gives spurious results.

The distribution of Nifty and India VIX is not normal as illustrated by the Normality plot (Figure 1a & 1b] and Jarque Bera test (table 4) where the Jarque Bera Stats are greater than the critical values.  This indicates that correlation may not be appropriate for measuring their dependency and its joint distribution is better understood by advanced methods like copula.

A Copula Approach to Volatility Index in India - Table4
Table 4:
A Copula Approach to Volatility Index in India - Figure1(a)
Figure 1(a)
A Copula Approach to Volatility Index in India - Figure1(b)
Figure 1(b)

Copula as dependence measure:

Using copula to build multivariate distributions is a flexible and powerful technique as it separates choice of dependence from marginal, on which no restrictions are placed. For two random variables we have the relationship:   

  F(X, Y) = C (G(x), H(y))

 The dependence relationship is entirely determined by the copula function ‘C’, while the scaling and the shape (for example, mean, standard deviation, skewness, and kurtosis) are entirely determined by the marginal distribution G(x) and H(y).

Best fitted copulas are constructed by considering the daily log returns of the volatility index and underlying option index over a sufficient time horizon, say  1 June 2011 to 30 May 2014.The steps involved in the process are as follows:

- Using the daily closing price the log returns for both Nifty and India VIX are calculated

- Fitting student t-distributions for both indices as this best fits the fat tail, which is the marginal distribution

- Fitting elliptical and Archimedean family of Copula for the joint movement returns of Nifty and India VIX

- Selecting the best copula fit using maximum likelihood estimation, Akaike and Bayesian information criteria

- Similarly extending the same exercise for S&P 500 (SPX) and VIX to do a comparison of the Indian market with the United States.

Interpretation of the fitted copula

Using the Nifty and India VIX, a Student-t copula appears to be the best one capturing the co-movement behaviour.  Table 5 shows that the Indian market has a low copula correlation and degree of freedom compared to the US market.

A Copula Approach to Volatility Index in India - Table5
Table5: Fitted t-copula parameters

One of the key properties of the volatility indices is strong negative correlation. In an ideal world, positive tail dependency should not be possible as it implies the probability of having a high (low) extreme value in one index given that a high (low) extreme value occurred in other index.

A Copula Approach to Volatility Index in India - Figure2(a)
Figure 2(a)
A Copula Approach to Volatility Index in India - Figure2(b)
Figure 2(b)
This is not the case for the Indian market. From the copula density function (Figure 2a & 2b) a significant amount of positive tail dependency is observed for the Indian market compared to the US market visually [lower wings towards (0, 0) and (1, 1) are elevated in figure 2a]. This is also strongly aided by the high tail dependency coefficient and copula density value at the lower percentile tail part (Table 6).
A Copula Approach to Volatility Index in India - Table6
Table 6:
Further, using the fitted t-copula function, 10,000 random numbers are generated and transformed into returns of respective indices. Scatter plots produced for the returns of the indices are shown in Figures 3a & 3b.
A Copula Approach to Volatility Index in India - Figure3(a)
Figure 3(a)
A Copula Approach to Volatility Index in India - Figure3(b)
Figure 3(b)

For the Indian market the plot is fanned out in the tails and a good amount of data lies in the first and third quadrant confirming positive tail dependency. This is not the case with the US where the scatter plot is less fanned at both ends and no extreme returns in the first and third quadrant. So the instances of observing an extreme positive or negative return concurrently in SPX and VIX is rare. Thus, derivatives on CBOE VIX represent a good asset for tail hedge against exposure to the US market.

The positive tail dependencies observed in the Indian market signify instances where a sharp drop (rise) in Nifty and India VIX may be observed concurrently which breaks down the diversification benefit during some tail events. This might be attributed to the method in which options are priced and the lack of depth in the market, as India VIX depicts the volatility build in the option prices. Once the positive tail dependency between India VIX and Nifty tends to zero, a derivative on India VIX may indeed represent a better asset class to protect against tail risk.

Chinnaraja Chendur Pandian currently works for Nomura, Global Risk