Implicit expectiles and measures of implied volatility

We show how to compute the expectiles of the risk-neutral distribution from the prices of European call and put options. Empirical properties of these implicit expectiles are studied on a data-set of closing daily prices of FTSE MIB index options. We introduce the interexpectile difference Delta tau (X) := e(tau) (X) - e(1-tau) (X), for tau is an element of (1/2, 1], and suggest that it is a natural measure of the variability of the risk-neutral distribution. We investigate its theoretical and empirical properties and compare it with the VIX index computed by CBOE. We also discuss a theoretical comparison with implicit VaR and CVaR introduced in Barone Adesi [J. Risk Financ. Manage., 2016, 9].