Paper: arXiv 2406.12999

Authors: Marcelo Righi, Fernanda Müller

Abstract

In this paper, we refine and generalize closed forms for worst-case law invariant convex risk measures with uncertainty sets based on: i) closed balls under $p$-norms and Wasserstein distance; and ii) moment constraints involving mean and variance. We also characterize the argmax of the worst-case problem in both settings. From such general results, we illustrate our framework by developing explicit closed forms for concrete examples of convex risk measures. Furthermore, we use extensive numerical simulations in order to assess the impact of robustness on capital determination and portfolio optimization.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 5.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical generalization of worst-case convex risk measures over p-norms, Wasserstein distance, and moment constraints, complemented by explicit closed-form examples. While the theoretical derivations are deep and comprehensive, the empirical component relies on numerical simulations rather than extensive real-world historical market backtests, slightly limiting its empirical rigor score.

Research Flowchart

  flowchart TD
    A[Goal: Refine & generalize worst-case law invariant convex risk measures] --> B[Input: Uncertainty Sets <br> p-norms, Wasserstein distance & Moment constraints]
    B --> C[Method: Derive analytical closed forms & characterize argmax worst-case]
    C --> D[Examples: Apply framework to concrete convex risk measures]
    C --> E[Simulations: Multi-Asset portfolio optimization & capital determination]
    D --> F[Outcome: Explicit robust risk formulas & quantitative impact analysis]
    E --> F