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 dataset of closing daily prices of FTSE MIB index options. We introduce the interexpectile difference Δ_τ(X) := e_τ (X) - e_1-τ(X), for τ ∈ (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 and with implicit VaR and CVaR introduced in Barone Adesi (2016)