The Implied Convexity of VIX Futures

Abstract

With the advent of modern option theory and practice, volatility has become one of the most important financial variables. The CBOE Volatility Index (VIX) of implied volatility for the S&P 500 Index has spawned numerous similar volatility indexes for other asset classes and a profusion of VIX-related futures, options, and exchange-traded funds (ETFs). But even though variance is nicely behaved, rising linearly in proportion to the length of the holding period, the square root of variance volatility is nonlinear. Variance is easy to hedge with a linear forward contract, but volatility is not. A convexity correction for the curvature of the square root function is needed. This correction is a function of the volatility of volatility (i.e., of the VIX in this case), which is an object of considerable interest in its own right. In this article, Daigler, Dupoyet, and Patterson make use of the convexity correction relationship to extract implied convexity from the difference between the price of a variance swap and the square of implied volatility from options on VIX futures. Although the relationship is a little noisy in the data, implied convexity, which is the “vol of VIX,” behaves largely as one expects. The authors’ insight may offer a new and useful estimator for this important parameter.