The Implied Convexity of VIX Futures

An important component of theoretical CBOE Volatility Index (VIX) futures prices is a term correcting for the negative convexity of the square root function by subtracting from the forward-starting variance swap rate an estimate of the future volatility of VIX futures prices. In the same fashion that an index option's traditional implied volatility can be viewed as an aggregate market consensus of future realized volatility, this convexity value can be viewed as an aggregate market consensus of future volatility of volatility. This article examines the predictive properties and features of this convexity adjustment needed to value VIX futures prices by extracting it from the relationship between observed VIX futures prices and the corresponding spot option market prices used to compute the forward-starting variance swap rate. The authors find that implied convexity levels can indeed be used to forecast the future volatility of VIX futures prices, even though implied convexity consistently underestimates future realized VIX futures variance. They also show that implied convexity can at times violate strict theoretical conditions by being negative, although we are able to rule out arbitrage opportunities. Finally, they examine the properties of this implied convexity adjustment, both as a time series and with respect to various market volatility factors with which they find positive and statistically significant relations.