Numerical methods for lambda quantiles: robust evaluation and portfolio optimization

Lambda quantiles, originally introduced as lambda value at risk, generalise the classical value at risk by allowing for a variable confidence level. In this article, we examine their numerical properties. We begin by developing a globally convergent Newton-based algorithm for evaluating lambda quantiles, named Λ-Newton-Bis. We prove convergence under mild assumptions, including cases with discontinuities, and local convergence under stronger local regularity conditions. Additionally, we consider an optimal portfolio allocation problem using lambda quantiles, proposing two alternative solution approaches based on the Λ-Newton-Bis algorithm. Several examples compare the performance of the two approaches and confirm the numerical efficiency of Λ-Newton-Bis.