On the Term Structure of VIX Futures’ Implied Convexity

Abstract

Before the global equity crash in October 1987, volatility could be reasonably approximated as a constant, consistent with Black-Scholes (1973) dynamics. Thereafter, a stylized feature of equity options markets is that both single-name and index options have exhibited consistent, regular deviations of volatility in both strike and maturity. The resulting volatility surface has been studied extensively (Rubinstein 1994, Jackwerth and Rubinstein 1996, Derman 1999, Cont and da Fonseca 2002, Gatheral 2006). Moreover, reduced-form representations of major equity indices’ volatility surfaces corresponding to “average” volatility (over strikes) accumulated through fixed maturities, for example, Cboe’s (formerly Chicago Board Options Exchange) Volatility Index (VIX), have been popularized as gauges of investor sentiment and risk-aversion. Likewise, there has been considerable interest in quantifying and interpreting the term structure of futures whose payoffs are tied to these indices (Zhu and Zhang 2007; Lu and Zhu 2009; Egloff et al. 2010). In the context of the risk-neutral distribution characterizing asset prices at contract maturity, these studies focus on futures’ expectations—their first moments; higher-order moments are less well-studied. Daigler et al. (2016) introduce implied convexity as a measure of variance, that is, the second moment. However, although many authors have studied the term structure of VIX futures’ expectations, to our knowledge, none has examined the term structure of their variances. This article extends the research of Daigler et al. in two important ways. First, it provides an alternative to their intermediate adjustments of the VIX near-term (VIN) and VIX far-term (VIF) component indices that is consistent with the assumptions underlying the calculation of all Cboe volatility indices. It is likewise consistent with volatility indices in foreign markets, for example, the Euro STOXX 50 Volatility (VSTOXX) index (Deutsche Börse Group 2022). Second, it characterizes the entire term structure of VIX futures’ second moments, rather than that of a single contract with a maturity of approximately one month. Additionally, we quantify the differences arising from various interpolation choices. We find that extrapolation based only on two maturities near thirty calendar days produces estimates of variance that differ considerably from interpolations based on all available expiries. Furthermore, the accuracy of extrapolation degrades as the absolute differences between a contract’s maturity and the maturities of the interpolants increase.