The asymmetric relationship between volatility index and volatility-of-volatility index

ABSTRACT

We use the nonlinear autoregressive distributed lag model to assess the asymmetric relationship between the Chicago Board Options Exchange’s volatility index (VIX) and volatility-of-volatility index (VVIX) over the period January 2007 to March 2020. To control for potentially confounding factors, we include measures for economic policy uncertainty and the volatility risk premium. There are three key findings. First, we find that there is an asymmetric long run cointegrating positive relationship running from VVIX to VIX. In this long run relationship, VIX is more responsive to deceases in VVIX than increases in VVIX. Second, we also find an asymmetric short run relationship between VIX and VVIX. However, in contrast to the long run results, for the short run VIX is more responsive to increases in VVIX than decreases in VVIX. Third, consistent with other studies, we find that in the long run VIX rises with greater economic policy uncertainty but falls with increases in the volatility risk premium. We discuss the implications of our findings for practitioners in risk and portfolio management, derivative pricing, and trading.

KEYWORDS:

• Volatility-of-volatility index
• volatility index
• NARDL
• volatility risk premium
• asymmetry

JEL CLASSIFICATION:

• C32
• C53
• C58
• G13

Disclosure statement

No potential conflict of interest was reported by the authors.

Data availability statement

Data used in the study is publicly available. Citations for available data are included in the references section.

Notes

1 Fassas and Siriopoulos (2020) provide an extensive survey on existing volatility indices. They summarise empirical results regarding return-volatility relationship and the information content of volatility indices over realised volatility in a multi-country and multi-asset setting.

2 See Baker et al., (2016) for a discussion on economic policy uncertainty and Jurdo et al. (2015) for a discussion on the financial uncertainty indicator.

3 The Three Component Index is retrieved from Economic Policy Uncertainty (2020)

4 However, differently from Bollerslev et al. (2014), we use the standard deviation rather than variance to define the volatility because the implied volatilities such as VIX and VVIX are quoted as an annualised standard deviation of its underlying index.

5 Although Andersen et al. (2001) documented that high frequency data using intraday returns outperforms lower frequency data in the estimation of the realised variance, we use the daily returns to calculate the monthly volatility due to extreme volatility in the daily returns not been consistent with the focus of this study.

6 The mean and standard deviation from its rebased mean are 90 and 14.3% for VVIX and 19.6 and 47.9% for VIX over the period.

7 Employing Augmented Dickey-Fuller test with constant, we find that coefficient of the first lag of level variable is −0.348 for VVIX and −0.132 for VIX, indicating that VVIX has a stronger mean-reversion property.

8 Because the short run incorporates lags in changes in first differences and the impact of the error correction term, it is not possible to do a similar sort of analysis of mean reversion statistics.

9 The results in Table 4 also support Hibbert et al. (2008) and Badshah (2013) who argued that the return-volatility asymmetry is dominated by extreme changes in returns because changes in volatility are more extreme and volatile at higher levels of volatility.