Asymptotic extrapolation of model-free implied variance: exploring structural underestimation in the VIX Index

We show that the VIX Index structurally underestimates model-free implied volatility because its implementation omits extrapolation of the volatility smile in the tails. We use the asymptotic behavior of the volatility surface to construct a correction term that is model-independent and only requires option prices at the two outermost strikes. We show how to apply this correction to the VIX Index ex-post as well as how to modify its implementation accordingly. Furthermore, we show that the degree of underestimation varies over time. For the S&P 500 Index and the DJIA Index the error is larger in periods of sustained low volatility. This cannot be observed for the Volatility-of-VIX Index.