Robust Lambda-quantiles and extremal distributions

TL;DR

This paper develops a framework for robust $ ext{Lambda}$-quantiles under partial distribution information, deriving explicit formulas for extremal distributions and applying them to portfolio optimization under uncertainty.

Contribution

It introduces a novel approach to compute robust $ ext{Lambda}$-quantiles using extremal distributions for various uncertainty sets, extending classical quantile robustness.

Findings

Explicit formulas for robust $ ext{Lambda}$-quantiles under moment, Wasserstein, and marginal constraints.

Application of the framework to optimal portfolio selection under model uncertainty.

Demonstration of the approach's effectiveness in practical risk management scenarios.

Abstract

In this paper, we investigate the robust models for $\Lambda$-quantiles with partial information regarding the loss distribution, where $\Lambda$-quantiles extend the classical quantiles by replacing the fixed probability level with a probability/loss function $\Lambda$. We find that, under some assumptions, the robust $\Lambda$-quantiles equal the $\Lambda$-quantiles of the extremal distributions. This finding allows us to obtain the robust $\Lambda$-quantiles by applying the results of robust quantiles in the literature. Our results are applied to uncertainty sets characterized by three different constraints respectively: moment constraints, probability distance constraints via the Wasserstein metric, and marginal constraints in risk aggregation. We obtain some explicit expressions for robust $\Lambda$-quantiles by deriving the extremal distributions for each uncertainty set. These results are applied to optimal portfolio selection under model uncertainty.