An Axiomatization of $Lambda$-Quantiles

We give an axiomatic foundation to Λ -quantiles, a family of generalized quantiles introduced in [M. Frittelli, M. Maggis, and I. Peri, Math. Finance, 24 (2014), pp. 442-463] under the name Lambda Value at Risk. Under mild assumptions, we show that these functionals are characterized by a property that we call "locality," which means that any change in the distribution of the probability mass that arises entirely above or below the value of the Λ -quantile does not modify its value. We make comparisons with a related axiomatization of the usual quantiles given by Chambers in [Math. Finance, 19 (2009), pp. 335-342], based on the stronger property of "ordinal covariance," meaning that quantiles are covariant with respect to increasing transformations. Further, we present a systematic treatment of the properties of Λ -quantiles, refining some of the results of Frittelli, Maggis, and Peri and [M. Burzoni, I. Peri, and C. M. Ruffo, Quant. Finance, 17 (2017), pp. 1735- 1743] and showing that in the case of a nonincreasing Λ the properties of Λ -quantiles closely resemble those of the usual quantiles.