Risk of investing in volatility products: A regime-switching approach

ABSTRACT

Volatility indexes provide a tool for investors to speculate and trade on market sentiment regarding future volatility. The risk of trading on volatility indexes can be measured by their second moments, namely, variance and correlation. This study considers the four representative volatility indexes published by the CBOE: stock market volatility index (VIX), crude oil volatility index (OVX), foreign exchange rate volatility index (EVZ), and gold price volatility index (GVZ). To examine their risk, we develop an extended multivariate Markov switching ARCH (MSARCH) model in which regime-switching variances, correlations, and variance-correlation relations are designed. Our empirical sample consists of the four volatility indexes from June 2008 to April 2020 for 612 weekly observations (Wednesday to Wednesday). For the conditional variances, we find evidence of regime-switching processes (switching between low and high volatility regimes) for the individual volatility index returns, with the exception of the GVZ. The estimated probability of the high volatility regime may be used to track economic distress and uncertainty shocks. These results provide evidence for volatility-of-volatility risk. For the conditional correlations, we find a regime-switching relation between variances and correlations. That is, the highest correlation appears when the paired volatility markets are simultaneously experiencing a state of high volatility. By contrast, when the paired volatility markets are encountering different volatility states, the correlation is weaker. These results indicate that the volatility-of-volatility risk is a factor affecting the dynamics of correlations between volatility indexes.

KEYWORDS:

• volatility index
• Markov-switching model
• GARCH/ARCH
• DCC

JEL CLASSIFICATION:

• C58
• G15

Acknowledgements

The research team would like to express our sincere appreciation to the two anonymous reviewers for their very helpful comments and suggestions. Any remaining mistakes are our own.

Data availability

All data are obtained from publicly available sources.

Notes

1 The deviation between our Equation (8)(8) $q_t=\tau +\pi \cdot q_{t-1}+\lambda \cdot e_{t-1}^{VIX}\cdot e_{t-1}^{OTH}/\sqrt{h_{t-1}^{VIX}\cdot h_{t-1}^{OTH}}$(8) and Engle’s (2002) Equation (23) is that the former simply shows the DCC for the bivariate data (i.e., the portfolio with two assets) and the latter uses a matrix to show the DCC setting for the multivariate data (i.e., the portfolio with n assets).

2 Theoretically, the correlation coefficient needs to range from -1.0 to 1.0. In Equation (9)(9) ${\rho }_t=\exp \left(q_t\right)/\left[\right(\exp \left(q_t\right)+1]$(9) , we employ the logistic function to control the magnitude of the correlation coefficient between zero and one. We even removed the restriction and considered a negative coefficient to rerun the MGARCH model. It should be noted that we are unable to obtain the estimation results since the Hessian matrix is singular and cannot be inverted. This result is consistent with the results shown in Table 2, a positive and significant correlation coefficient for each paired volatility indexes.

3 Gray (1996) proposed the Markov-switching GARCH model in which the Markov-switching mechanism and the GARCH model are combined. However, the model suffers from an ad hoc assumption, of which Hamilton and Susmel (1994) provide an in-depth discussion.

4 Our Equations (11)(11) $h_t^{VIX}=g_{s_t^{VIX}}^{VIX}\times [{\omega }^{VIX}+{\alpha }^{VIX}\cdot (e_{t-1}^{VIX}{)}^2]$(11) and (12)(12) $h_t^{OTH}=g_{s_t^{OTH}}^{OTH}\times [{\omega }^{OTH}+{\alpha }^{OTH}\cdot (e_{t-1}^{OTH}{)}^2]$(12) are consistent with Equation (4)(4) $H_t=\left[\begin{array}{cc}{h}_t^{VIX}& h_t^{VIX,OTH}\\ h_t^{VIX,OTH}& h_t^{OTH}\end{array}\right]$(4) of Ramchand and Susmel (1998). The difference in expression is that we move the parameter for the scale of volatility (i.e., g_st) from the left-hand side of the equation to the right-hand side.

5 One might consider generalising the MSARCH model with higher-order ARCH terms. Our empirical results (see Section 3) show that the estimates of the higher-order ARCH parameters are insignificant. Accordingly, this study did not consider the higher-order ARCH parameters for the sake of expediency.

6 To estimate the model, we use the software package, Broyden–Fletcher–Goldfarb–Shanno algebra in GAUSS, to search for the parameters that maximise the log-likelihood function. The program codes are available upon request.