# The New Zealand implied volatility index.

1. IntroductionIn 1993, the Chicago Board Options Exchange (CBOE) was the first exchange to introduce an implied volatility index, called the VIX. Based on a weighted average of implied volatilities of the S&P100 Index call and put options, the VIX was designed to be a measure of broad market volatility. Following this development, many exchanges constructed similar implied volatility indices for risk hedging and portfolio management purposes. Examples are the German Derivatives Exchange (DTB), which introduced an implied volatility index (VDAX) in 1994, the French Futures Exchange (MATIF), which launched two implied volatility indices (VX1, VX6) in 1997, and most recently the NYSE Euronext, which introduced implied volatility indices on the Dutch AEX Index, the Belgian BEL 20 Index and the French CAC40 Index in September 2007.

The importance of these volatility indices is highlighted by two major developments. First, there is an increasing amount of literature suggesting that the implied volatility index is a measure of investor sentiment. Typically, implied volatilities rise in times of financial distress, and drop when markets boom. Therefore, the implied volatility index is often referred to as the "investor fear gauge" (Whaley, 2000). Giot (2002a, 2002b) further shows that these indices convey meaningful information about the future profitability and volatility of the stock market. Second, the development of worldwide derivative markets has led to the need for derivatives on market volatility. For the purpose of hedging market risk, the implied volatility index could be used as the underlying asset of volatility derivatives (Brenner and Galai, 1989; Brenner et al., 2006). An example is the CBOE, which launched futures and options on its implied volatility indices in March 2004 and February 2006, respectively.

In New Zealand, the derivatives market has only recently started to develop and until now no implied volatility index has been constructed. The aim of this paper is therefore to construct an implied volatility index, called the NZVIX. In constructing the NZVIX, this paper makes an important contribution. Since there are no derivative products traded on a New Zealand equity index, we propose an alternative approach to construct an implied volatility index using individual stock options. In New Zealand, we find that options are traded on four stocks, which are the largest constituents of the NZX 15 Index. We therefore construct the NZVIX by combining implied volatilities series of the existing four stock options, which are then used as a proxy for the implied volatility index of the NZX 15. This approach may prove to be useful for other markets where options are only traded on individual stocks and not on indices.

In addition to the construction of the index, we study some properties of the NZVIX. Firstly, because implied volatilities are considered to be forward-looking measures of market volatility (Fleming et al., 1995), we investigate whether the NZVIX has predictive power for NZX 15 market volatility. Secondly, we consider the relationship between the NZVIX and the NZX 15 Index returns. Past research has shown that implied volatility indices tend to rise when investors expect financial turmoil (e.g. Whaley, 2000). We therefore investigate whether there is a contemporaneous relationship between changes in the NZVIX and the NZX 15 Index returns, and whether changes in the NZVIX have any predictive power for index returns (Giot, 2002a). We find evidence implying that the NZVIX can predict market volatility and find a significant contemporaneous and weakly significant predictive relationship between the NZX 15 Index returns and changes in the NZVIX.

The remainder of this paper is organized as follows. The next section reviews the literature. This is followed by a short discussion of the products traded on the New Zealand Futures and Options Exchange (NZFOX) and the NZX 15 Index. Section 4 discusses the methodology of constructing the index and the data selection. Section 5 presents the empirical findings. The final section summarizes and concludes.

2. Literature Review

2.1 Construction of the Implied Volatility Index

One of the first papers on implied volatilities was by Latane and Rendleman (1976). They derived implied volatilities for 24 stock options trading on the CBOE and investigated their properties. After the CBOE introduced the S&P100 Index options in 1983, Cox and Rubinstein (1985) proposed to create a market implied volatility index by averaging near-the-money call options to obtain an at-the-money index with a constant time to maturity. This was a milestone in the study of implied volatilities and subsequent studies used a similar methodology in their analysis.

Whaley (1993) improved on this implied volatility index, by creating an at-the-money volatility index with a constant 30 calendar-day time to maturity. He achieved this by averaging the implied volatilities of eight S&P100 Index options, being both calls and puts with different times to maturity. This methodology was adopted by the CBOE in 1993 for constructing the VIX, and its underlying principle is still used today. In September 2003, the methodology to compute the VIX changed. The VIX started using S&P500 Index options, and the option pricing model was altered from the Black-Scholes option pricing model to an independent undisclosed model. The new VIX retains the name VIX, while the original index that uses S&P100 Index options is now referred to as the VXO.

Dotsis, Psychoyios and Skiadopoulos (2006) examine different methods to develop implied volatility indices for major European and American stock markets. Their findings show that the basic principles of the applied methods for the calculation of market implied volatility are the same, although different option valuation models are used in the construction of indices. All implied volatility indices are constructed for (hypothetical) at-the-money index options with a constant time to maturity. This is achieved by computing weighted averages of near-the-money puts and calls with different times to maturity. Moreover, because many countries have no implied volatility index, a number of studies have proposed methods for the construction of implied volatility indices for their markets. For example, Dowling and Muthuswamy (2005) introduce the Australian Volatility Index (AVIX) by using S&P/ASX200 Index options, and Skiadopoulos (2003) constructs a Greek Volatility Index (GVIX), based on FTSE/ASE-20 Index "options. These methodologies are very similar to that used for the construction of the original VIX.

2.2 Properties of Implied Volatility Index

Studies on implied volatility indices typically examine their properties in two dimensions. The first dimension is the relationship between the implied volatility index and stock market volatility (Dowling and Muthuswamy, 2005; Fleming et al., 1995; Giot, 2002b). The second dimension is the relationship between the implied volatility index and stock market returns (Fleming et al., 1995; Giot, 2002a; Skiadopoulos, 2003; Whaley, 2000).

A number of studies have shown that implied volatility can forecast market volatility. Latane and Rendleman (1976) and Chiras and Manaster (1978) found that implied volatility is a good predictor of market volatility. Feinstein (1989) demonstrated that implied volatility approximates the market expectation of average volatility over the remaining life of the option. This approximation is especially accurate for at-the-money and near-the-money options. Harvey and Whaley (1992) and Giot (2002b) also reported that the VIX provides meaningful and accurate information about the future S&P100 Index volatility, based on a regression of market volatility on two lags and the contemporaneous VIX. On the other hand, several studies have found that implied volatility cannot provide good forecasts of future market volatility. Canina and Figlewski (1993) and Figlewski (1997) explain why implied volatility cannot be a good predictor of future volatility. One of their reasons is that the proportion of large price movements is greater than those predicted by the log-normal diffusion process as assumed by Black-Scholes. Another reason is that traders may only hold options for a short time period rather than the whole life of the option. Therefore, it is difficult for the market makers to rely solely on the prediction of implied volatility indices. For Australia, Dowling and Muthuswamy (2005) found that the implied volatility index cannot forecast market volatility in Australia.

Early studies on the relationship between market volatility and stock market returns have found a strong negative relationship (Black, 1976; Christie, 1982). Poterba and Summers (1986) arrived at a similar conclusion by using a time-varying risk premium hypothesis. French, Schwert and Stambaugh (1987) confirmed these findings by examining the relationship between volatility and expected returns. Moreover, Schwert (1989, 1990) observed that the relationship between market returns and volatility is not symmetric. He found that an increase in stock prices has a small negative impact on volatility, while a decrease in stock prices has a greater positive impact on volatility. Fleming et al. (1995) reported that changes in the VIX have a significant negative relationship with the S&P100 Index by using a multivariate regression of daily index returns on two lags, two leads and contemporaneous VIX changes. Consistent with Schwert (1989, 1990), they also observed an asymmetric effect. The study by Dowling and Muthuswamy (2005) showed that the relationship between implied volatility changes and market returns is negative and significant but not asymmetric in the Australian market.

As the investor fear gauge, some research has focussed on the predictive power of implied volatilities. Since the VIX is typically found to be negatively related to contemporaneous stock market returns, Giot (2002a) suggests and shows that the VIX is able to predict market movements in the short-run. Skiadopoulos (2003), however, found that the implied volatility index in Greece cannot forecast equity index returns.

3. NZFOX Products and the NZX 15 Index

On 17 September 2003, the New Zealand Exchange (NZX) signed an agreement with the Sydney Futures Exchange (SFE) for the listing and trading of New Zealand equity derivative products. The NZX would provide a set of futures and options contracts by listing these products on the SFE. The deal enabled the NZX to develop, market and promote a range of futures and options products based on equity securities listed on the NZX under the name NZFOX (Vivian and Johnston, 2003). (1)

The NZFOX products currently include the NZX 15 Index futures (FOX15) and four New Zealand equity options on Contact Energy Ltd (CEN), Fletcher Building Ltd (FBU), Telecom Corporation of NZ Ltd (TEL) and the Warehouse Group Ltd (WHS). The FOX15 was originally listed on 31 August 2004 and the four stock option contracts were introduced on 23 August 2005. The four stock options are American-type options, with expiry months in March, June, September and December. Trading hours are from 9:45am to 4:55pm (New Zealand time).

The NZX 15 Index was introduced on 9 February 2004 and consists of the 15 largest and most liquid New Zealand securities listed on the NZSX. On average the NZX 15 Index accounts for more than 70% of the total market capitalization on the NZSX (Daucher, 2004). The constituent companies of the NZX 15 Index are weighted by free-float market capitalization with the weighting of any security capped at 30%. The NZSX reviews constituents every half year and rebalances the index. The underlying assets of four NZFOX share options represent the largest four constituents of the NZX 15. These four stocks typically represent more than half of the total capitalization of the NZX 15.

4. NZVIX Construction

4.1 Data

The construction of the implied volatility index requires a wide range of data from different sources for the sample period of this study, 2 September 2005 to 26 May 2006. Option data (strike prices, expiration dates and settlement prices) are obtained from TAQTIC. (2) Stock prices, equity index levels and risk-free rates are obtained from DataStream. The New Zealand interbank rate whose maturity is closest to the option expiration date is used as the risk-free rate. Information about the amount and ex-dividend date is obtained from the NZFOX website. Research has shown that volatility is much lower when the stock exchange is closed for trading than when it is open (Hull, 2005). Therefore, we use trading days instead of calendar days when calculating implied volatility.

Due to the infrequent trading of the NZFOX options, there are a significant number of days where option prices are not available. As a result, the data selection and the construction of the NZVIX are carried out on a weekly basis. Friday prices are used to calculate implied volatilities. If there is no option data available on Friday, the previously observed price is used. Since the New Zealand stock options trade on quarterly cycles, we compute the NZVIX with a time to maturity of three months. This differs from the construction of the VIX, which is based on a time to maturity of one month.

4.2 Construction of NZVIX

The methodology employed to construct the NZVIX differs from the VIX and other implied volatility indices. The VIX and other major market implied volatility indices are created from the option prices of equity indices. In New Zealand there are no options traded on equity indices. Options trade for four individual stocks (CEN, FBU, TEL and WHS). Therefore, we construct an implied volatility index for the New Zealand market by first computing implied volatility indices for each stock individually and then combining the implied volatilities of the four stock option prices. By doing this, we assume that the constructed implied volatility index from four stock options can be used as a measure for the implied volatility index of the NZX 15 Index option. The validity of this assumption will be assessed in a later section.

In constructing the implied volatility indices for each stock we follow the same approach as the construction of the VIX. Each individual stock's implied volatility index is computed as a weighted-average of the implied volatilities of eight near-the-money options consisting of four first-nearby (shorter than three months time to maturity) and four second-nearby options (longer than three months time to maturity). Each individual implied volatility is referred to as a component implied volatility. Since the options are American-type options and on stocks that pay dividends, we use the binomial tree approach proposed by Cox, Ross and Rubinstein (1979) to compute the component implied volatilities. This binomial tree is calibrated by changing the implied volatility to match the observed option price.

To circumvent problems of anomalous option trading during the period close to contract maturity, we ensure that the option series have at least eight days to expiration. For each maturity we include puts and calls with exercise price just below and just above the current stock price. We use both puts and calls to offset the upward (downward) bias in implied volatility on the call option against downward (upward) bias on the put option (Fleming et al., 1995).

The next stage is to average the eight component implied volatilities. Let [X.sub.L] be the strike price closest to but just below the current stock price, S, and [X.sub.U] be the strike price closest to but just above S. Also, let "1" ("2") and c (p) represent the first-nearby (second-nearby) option series and call (put) option, respectively. The eight component implied volatilities are shown in the following matrix:

Exercise First-nearby Price Call [X.sub.L] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [X.sub.U] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Exercise First-nearby Price Put [X.sub.L] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [X.sub.U] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Exercise Second-nearby Price Call [X.sub.L] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [X.sub.U] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Exercise Second-nearby Price Put [X.sub.L] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [X.sub.U] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To compute the implied volatility index for a single stock, we first average the call and put options' implied volatilities for each pair of options. For example, for the first-nearby option series with the exercise price of [X.sub.L], the average of the call and put options' implied volatilities is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

The other three averages, denoted [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], are computed in a similar fashion. Next, assuming that the volatility smile can be approximated by a piecewise linear function (see Dumas et al., 1994), we compute the implied volatility for an at-the-money option by linearly interpolating between the implied volatilities of the in-the-money and out-of-money options. We compute at-the-money implied volatilities for both expirations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

The last step is to create an implied volatility index with a constant time to maturity by combining the first-nearby and second-nearby at-the-money implied volatilities. Because we use trading days instead of calendar days to compute implied volatilities, we need to transform calendar days into trading days (see e.g. Whaley, 2000). Let [N.sub.CAL] be the number of calendar days to expiration and [N.sub.TRADE] the number of trading days to expiration. Then we use the following formula to convert calendar days to trading days:

[N.sub.TRADE] = [N.sub.CAL] - 2 x int( [N.sub.CAL] / 7). (4)

In our case, we intend to construct an implied volatility index for an artificially created option with a time to maturity of three months. Transforming calendar days into trading days leads to a time to maturity of 66 trading days (66 = 90 - 2 x int(90/7)). The 66-trading-day (or 90-calendar-day) implied volatility for an individual stock is computed as a weighted average of the first-nearby and the second-nearby at-the money options:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

where [[sigma].sub.i] is the implied volatility index of stock i, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the number of trading days until expiration of first-nearby contract, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the number of trading days until expiration of second-nearby contract. The number of trading days determines the weight given to the implied volatilities of the two options.

At this stage, four series of implied volatility series from four stock options have been computed. Next, we combine the four implied volatility series into a single implied volatility index, the NZVIX. The implied volatility series are combined using the formula:

NZVIX = [square root of ([K.summation over (i=1)] [w.sup.2.sub.i] [[sigma].sup.2.sub.i] + 2 [K-1.summation over (i=1)] [summation over (J>i)] [w.sub.i] [w.sub.j] [[rho].sub.ij] [[sigma].sub.i] [[sigma].sub.j])] (6)

where K is the number of stocks in the portfolio, [w.sub.i] is the weight of stock i in the portfolio and [[rho].sub.ij] is the correlation between four stocks' implied volatilities. The weight of each stock in the portfolio is based on the market capitalization in the NZX 15 Index. Information about the capitalization in the NZX 15 Index is obtained from the weekly diary reports of NZX Data.

Since equation (6) relates to a computation based on implied volatilities, we would also require the implied correlations to compute the implied volatility of the index. However, since options are only traded on single assets, we cannot observe or deduce these implied correlations. We therefore use historical correlations to compute the implied volatility index, even though this may introduce some imprecision into the NZVIX measure. The correlations of four stocks' returns are calculated based on daily returns from the past six weeks (30 observations). This particular number of observations was chosen as a trade-off between not using too much old data and still having enough observations to compute a meaningful statistic. Finally, we compute the NZVIX using equation (6), and use this measure as the implied volatility index for the NZX 15 Index.

5. Empirical Findings

This section examines some properties of the NZVIX. To justify that the implied volatility index of the portfolio is a realistic proxy for the implied volatility index of the NZX 15 Index, we start by examining the relationship between the portfolio and the NZX 15 Index. Next, we consider whether the NZVIX can be used as a forward-looking measure for market volatility as suggested by Fleming et al. (1995). Lastly, we investigate whether the NZVIX can be used as an "investor fear gauge", i.e. when markets are in turmoil (and implied volatilities increase) investors' fear depresses the stock market. We therefore examine whether there is a contemporaneous relationship between changes in the NZVIX and NZX 15 Index returns, as considered by Whaley (2000), and whether changes in the NZVIX have any predictive power for NZX 15 Index changes. Although finding a predictive relationship would contradict the semi-strong form market efficiency, such a predictive relationship has been found by Giot (2002a).

5.1 Relationship between the Portfolio of Stocks and the NZX 15

As mentioned before, the underlying stocks of the four options are the largest constituents of the NZX 15. These four stocks represent 53.14% of the total capitalization of NZX 15 Index. (3) To assess the relationship between the four stocks and the index, we construct a value-weighted portfolio of the four stocks. Figure 1 shows the relationship between the weekly returns of the portfolio and the NZX 15 Index for the period from 9 September 2005 until 26 May 2006. As can be seen, the movements of the portfolio of the four stocks are similar to the index movements, although the returns of portfolio are more volatile than the NZX 15 Index. (4) As a more formal analysis, we perform a regression of the portfolio returns on the return of the NZX 15 Index, i.e.

[FIGURE 1 OMITTED]

[R.sub.M ,t] = a + [beta][R.sub.P ,t], + [[epsilon].sub.t], (7)

where [R.sub.M,t] is the return on the NZX 15 Index and [R.sub.p,t] is the return on the portfolio. This regression yields the following results:

[R.sub.M ,t] = 0.0027 + 0.6453 [R.sub.P ,t] + [[epsilon].sub.t] [R.sup.2] = 0.844. (2.02) (13.64)

Based on the slope coefficient, the stocks in the portfolio have a higher market risk than the NZX 15 Index. (5) The highly significant t-stat of 13.64 indicates a strong relationship between the two return series, which is further highlighted by the high [R.sup.2]. The movement of the portfolio explains the movement in the index for 84.4%. The regression results of equation (7) therefore indicate that the returns of the NZX 15 Index are highly correlated with the returns of the portfolio.

Equation (7) also provides information about the relationship between the volatility of the portfolio and the volatility of the NZX 15 Index. Rewriting equation (7) in terms of variances we obtain:

[[sigma].sup.2.sub.M] = [[beta].sup.2][[sigma].sup.2.sub.P] + [[sigma].sup.2.sub.[epsilon]] (8)

where [[sigma].sup.2.sub.M] is the variance of the portfolio returns, [[sigma].sup.2.sub.P] is the variance of the market return and [[sigma].sup.2.sub.[epsilon]] is the variance of the error term. The regression [R.sup.2] of equation (7), defined as

[R.sup.2] = [[beta].sup.2][[sigma].sup.2.sub.P] / [[sigma].sup.2.sub.M], (9)

shows that 84.4% of the variance of the NZX 15 Index can be explained by the variance of the portfolio. We therefore argue that the implied volatility index derived from the four stock options can be used as a proxy for the implied volatility index of NZX 15. We have to keep in mind though that our derived NZVIX is based on a portfolio with higher market risk than the NZX 15 Index, which implies that the level of the NZVIX will be systematically higher than the level of the 'true' NZVIX.

5.2 NZVIX as a Predictor of Future Stock Market Volatility

Fleming et al. (1995) suggest that implied volatilities derived from index options provide a forward-looking measure for market volatility. Since options are priced based on investors' expectations about the volatility during the lifetime of the option, Fleming et al. suggest that implied volatility can be used as a predictor of observed or realized market volatility. We therefore want to test whether the observed market volatility can be explained by past implied volatility.

Since the NZVIX has been computed with an average time to maturity of 66 days (approximately 3 months), we compute a forward-looking realized volatility, by calculating the standard deviation over the subsequent 66 trading days after the computation of the NZVIX. In Table 1 we report some summary statistics for the NZVIX and the (66 day) forward-looking realized NZX 15 Index volatilities, expressed as a weekly figure. The range of the NZVIX is from 1.33% to 2.57%, with a mean of 1.83% and the standard deviation of 0.31%. The realized volatility has a mean of 1.66%, slightly lower than the NZVIX. This could among others, be due to the higher market [beta] of the portfolio used to construct the NZVIX. Both volatility series are non-normally distributed, both displaying a positive skewness. In addition the NZX 15 realized volatility also displays a negative excess kurtosis. The correlation between the NZVIX and the weekly volatility of the NZX 15 Index is 0.45, indicating a moderate positive relationship.

[FIGURE 2 OMITTED]

In Figure 2 we graph the NZX 15 realized volatility and the NZVIX. The positive relationship is roughly observed throughout the whole period, but the graph also shows that the NZVIX is more volatile than the NZX 15 Index volatility.

As a more rigorous analysis of the predictive power of the NZVIX, we conduct a regression of the NZX 15 realized volatility against the NZVIX,

[[sigma].sub.NZX,t] = [alpha] + [beta] [NZVIX.sub.t] + [[epsilon].sub.t]. (10)

where [[sigma].sub.NZX,t] is the NZX 15 realized volatility. This regression, however, does not indicate whether the NZVIX is a superior forecaster of implied volatility compared to a naive predicator of future volatility, such as past volatility. We therefore conduct two more regressions. First, we regress the NZX 15 forward-looking realized volatility on the 66-day past realized volatility,

[[sigma].sub.NZX,t] = [alpha] + [gamma][[sigma].sup.Past.sub.NZV ,t] + [[epsilon].sub.t]. (11)

where [[sigma].sup.Past.sub.NZX ,t] is the volatility of the NZX 15 Index over the past 66 trading days. Second, we run an encompassing regression to compare the performance of both forecasting measures, i.e.

[[sigma].sub.NZX,t] = [alpha] + [beta][NZVIX.sub.t] + [gamma][[sigma].sup.Past.sub.NZX ,t] + [[epsilon].sub.t]. (12)

To control for heteroskedasticity and autocorrelations in the residuals, we correct the standard errors using the procedure suggested by Newey and West (1987).

In Table 2 we present the results of regressions (10) to (12). In the first column in Table 2, we consider the NZVIX as a forecaster of future volatility. We observe a significant positive relationship between NZVIX and future volatility (a coefficient of 0.336 and t-stat of 3.45), with an adjusted [R.sup.2] of 0.185. These results indicate that the implied volatilities of the options contain useful information about the future volatility of the market as a whole. In the traditional sense, however, our expectations about an unbiased forecaster for future volatility should yield coefficients of [alpha] = 0 and [beta] = 1 in equation (10). There are several explanations as to why [alpha] and [beta] could be different from their expected values. Firstly, as regression (7) indicated, the portfolio of stocks on which the NZVIX is based has a higher market risk than the NZX15 Index. This causes the NZVIX to be more volatile than the volatility of the NZX 15 Index. (6) Secondly, the New Zealand options market is still an underdeveloped market which could add noise to the derived NZVIX. This noise increases the measurement error in the NZVIX, which downward biases the estimate for [beta] (see e.g. Greene, 2007). Lastly, it is not uncommon to find values for [alpha] and [beta] that differ from their expected values. Andersen et al. (2003), for example, consider the forecasting power for various measures of volatility, and find that in their out-of-sample tests [alpha] and [beta] are often different from their expected values. However, although unbiasedness is a desirable feature, it is by no means a necessary feature. As long as the bias is systematic, a biased estimator can be used for forecasting.

The second column of Table 2 reports the regression results for past 66-trading day volatility as a forecaster of future volatility. The results reveal an insignificant relationship, which implies that past volatility contains little information about the future volatility.

The last column of Table 2 reports the results of the encompassing regression using both NZVIX and past volatility. We find that only the coefficient of NZVIX is significant. This implies that NZVIX is a superior forecaster of future volatility compared to the measure of past volatility that we have used.

5.3 NZVIX and its Relation to Movements of the NZX 15

The previous analysis has shown that there is predictive power of the NZVIX for the future volatility of the NZX 15 Index. The next question we address is whether the NZVIX can be used as an investor fear gauge. Whaley (2000) has shown that the VIX in the US typically rises in times of financial turmoil and vice versa. To assess whether this relationship holds for the NZVIX and the NZX 15 Index, we consider the relationship between changes in the NZVIX and changes in the NZX 15 Index.

Table 3 displays summary statistics of changes in the NZVIX and the weekly returns of the NZX 15 Index. The sample mean for both series is almost zero, indicating that there is nearly no trend. Both series are not normally distributed; the changes in the NZVIX have a high excess kurtosis, while the NZX 15 returns are negatively skewed. The correlation between both series is -0.25, which indicates a weak inverse relationship.

Figure 3 illustrates the relationship between the changes in the NZVIX and the NZX 15 returns. From this graph, we observe a weak negative relationship. For example, the NZX 15 Index peaks at the end of December 2005, while the NZVIX is at a low. In 2006, this negative relationship also exists, such as in March, April and May. However, just as their correlation (-0.25) indicates, the negative relationship is weak.

To investigate the relationships between changes in the NZVIX and the NZX 15 returns more carefully, we consider a regression of weekly returns of the NZX 15 Index on (lagged) changes in the NZVIX, i.e.

[R.sub.M ,t] = [[alpha].sub.j] [[beta].sub.j] [DELTA][NZVIX.sub.t-j] + [[epsilon].sub.t], (13)

for different values of j. The purpose of running these regressions is to assess whether there is a contemporaneous relationship between changes in the NZVIX and the NZX 15 market returns, and to assess whether changes in the NZVIX are able to forecast the NZX 15 Index movements within three months. Regressions were performed for lags up to 12 weeks (three months) and standard errors are again HAC-corrected following Newey and West (1987). For the sake of brevity, we only report the most important results. The two relationships that might be most interesting are the regression of market returns on the contemporaneous value of the NZVIX and the regression of the market return on the one-week lagged NZVIX. It is these results that we report.

For the contemporaneous relationship, we obtain the following results:

[R.sub.M ,t] = 0.001-1.789 [DELTA][NZVIX.sub.t-j] + [[epsilon].sub.t] [R.sup.2] = 0.037. (0.42) (-2.37)

These findings suggest a significant negative contemporaneous relationship between the changes in the NZVIX and stock market returns. The negative relationship is in line with the CAPM which suggests that the expected market risk is priced and provides evidence for the time-varying risk premium hypothesis of French and Summers (1986). Our findings are in line with e.g. Whaley (2000), but contrast the findings of Dowling and Muthuswamy (2005), who found no significant relationship between the implied volatility index changes and weekly index returns in the Australian market.

The second regression considers the relationship between market returns and one-week lagged values of the NZVIX. The predictive power of implied volatility for stock market returns has been assessed and found by Giot (2002a). We obtain the following regression results:

[R.sub.M ,t] = 0.001-1.689 [DELTA][NZVIX.sub.t-1] + [[epsilon].sub.t] [R.sup.2] = 0.031. (0.25) (-1.85)

We find a weakly significant (at the 10% level) relationship between lagged values of the NZVIX and the NZX 15 Index. Although the relationship is weak, these findings are in line with Giot (2002a), who finds that implied volatilities can predict stock market returns in the short-run.

[FIGURE 3 OMITTED]

5.4 Limitations of the Current Study

Although the results presented here are promising, there are several limitations to this study. The foremost limitation is the short sample period that has been used. Stock options in New Zealand were introduced only at the end of August 2005, and since the introduction of the contracts, trading in these options has been infrequent. The current study is therefore limited to weekly data rather than daily data. Studies from the US and Australia show that weekly implied volatility indices in general show weaker relationships with market volatility and returns than do daily indices (see Fleming, Ostdiek and Whaley, 1995; Dowling and Muthuswamy, 2005). Therefore, further investigation at a later stage, when these options trade more frequently, is warranted.

The second limitation, also due to the limited trading in options, is the large bid-ask spread in option price quotes. These large spreads lead to a significant bid-ask bounce observed in the settlement prices of these options, and may introduce noise in the component implied volatilities. However, because the NZVIX is a weighted average of the four component implied volatilities, this problem is mitigated to some extent.

Thirdly, only implied volatilities from efficient option prices reflect the investor sentiment on the future stock market (Canina and Figlewski, 1993). The New Zealand market just launched the stock options for trading in August 2005. Thereby the option market in New Zealand is still at an initial stage and further development of this market may lead to improved efficiency.

6. Summary and Conclusions

The main objective of this paper is to construct an implied volatility index for the New Zealand stock market, which is called the NZVIX. Since there are no equity index options traded in New Zealand, we propose a new approach that uses stock options on the largest constituents of the stock index, the NZX 15 Index. Using a combination of the implied volatilities of the stocks, we construct the NZVIX. Using regression analysis, we show that our assumption of the stocks being representative for the index is reasonable. Our suggested approach could be useful for other markets where options are only traded on stocks and not on equity indices.

When assessing some of the properties of the NZVIX, we observe that the NZVIX has positive skewness and has a slightly higher mean than the weekly volatilities of the NZX 15, which is likely due to the underdiversification of the NZVIX relative to the NZX 15. When investigating the relationship between the NZVIX and future NZX 15 Index volatility, we find a significantly positive relationship. This finding is in line with previous research indicating the forecasting power of implied volatility indices for market volatility. We further find a significant negative relationship between changes in the NZVIX and changes in the NZX 15 Index. Finally, we find some weak evidence showing that the NZVIX has some predictive power for short-term stock price movements. However, given the limitations of this study, future research on larger samples, and if possible higher frequencies, is warranted.

Received on January 3, 2007; received in revised form on January 15, 2008; accepted on January 22, 2008.

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* We would like to thank the two anonymous referees and the editor for their helpful and constructive comments, and the Securities Industry Research Centre of Asia-Pacific Ltd (SIRCA) for providing the data.

(1) The New Zealand Futures and Options Exchange.

(2) TAQTIC is an e-database that belongs to SIRCA Ltd (Securities Industry Research Centre of Asia-Pacific Ltd),

(3) The weighting percentage was gathered on 23 August 2006. Historically, the total weight of these four stocks was around this percentage.

(4) Such higher volatility is expected because the portfolio of four stocks is considerably less diversified than the NZX 15 Index.

(5) To address market risk, one typically considers the reverse regression. This regression yields a beta of 1.30, indicating that the stocks in the portfolio have more market risk than the NZX 15 Index.

(6) We can control for the level of market risk is the NZVIX by dividing the slope coefficient of (10) (0.336) by the estimated beta of (7) (0.6453). This would yield a corrected slope coefficient of 0.520, which is still significantly different from [beta] = 1.

Bart Frijns (#) ** (+), Alireza Tourani-Rad (+), Yajie Zhang (+),

(#) Corresponding author. Department of Finance, Faculty of Business, Auckland University of Technology, Private Bag 92006, Auckland 1142, New Zealand, Tel.: +64 9 921 9999; Fax: +64 9 921 9629; Email: b.frijns@aut.ac.nz.

** Nijmegen School of Management, Radboud University Nijmegen, The Netherlands

(+) Department of Finance, Faculty of Business, Auckland University of Technology, New Zealand

Table 1. Summary statistics of the NZVIX and the (66 day) forward-looking realized volatility of NZX 15 Index NZX 15 NZVIX Volatility Mean 0.0183 0.0166 Median 0.0175 0.0159 Minimum 0.0133 0.0134 Maximum 0.0257 0.0204 Standard Deviation 0.0031 0.0023 Excess Kurtosis 0.0188 -1.461 Skewness 0.7938 0.3929 Table 2. Regressions of forward-looking realized NZX 15 Index volatilities NZVIX and Past Past NZVIX Volatility volatility [alpha] 0.010 (4.56) 0.012 (3.81) 0.010 (4.25) [beta] 0.336 (3.45) 0.323 (2.52) [gamma] 0.309 (1.33) 0.068 (0.30) [R.sup.2] (adj) 0.185 0.021 0.164 T-stats are presented in parentheses and are HAC-corrected following Newey and West (1987). Table 3: Summary statistics of the changes in the NZVIX and the weekly returns of NZX 15 Index Weekly Changes returns in NZVIX of NZX 15 Mean 0.0002 0.0009 Median 0.0004 0.0011 Minimum -0.0057 -0.0489 Maximum 0.0083 0.0433 Standard 0.0029 0.0204 Deviation Excess Kurtosis 1.025 0.0117 Skewness 0.0362 -0.2488

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Author: | Frijns, Bart; Tourani-Rad, Alireza; Zhang, Yajie |
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Publication: | New Zealand Economic Papers |

Article Type: | Report |

Geographic Code: | 8NEWZ |

Date: | Jun 1, 2008 |

Words: | 7064 |

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