Paper: arXiv 2312.05973
Authors: Michael Kupper, Max Nendel, Alessandro Sgarabottolo
Abstract
In this paper, we study convex risk measures with weak optimal transport penalties. In a first step, we show that these risk measures allow for an explicit representation via a nonlinear transform of the loss function. In a second step, we discuss computational aspects related to the nonlinear transform as well as approximations of the risk measures using, for example, neural networks. Our setup comprises a variety of examples, such as classical optimal transport penalties, parametric families of models, uncertainty on path spaces, moment constrains, and martingale constraints. In a last step, we show how to use the theoretical results for the numerical computation of worst-case losses in an insurance context and no-arbitrage prices of European contingent claims after quoted maturities in a model-free setting.
Complexity vs Empirical Score
- Math Complexity: 8.5/10
- Empirical Rigor: 3.5/10
- Quadrant: Lab Rats — theoretically deep, empirically untested
Why this score: The paper presents a rigorous theoretical framework using weak optimal transport for convex risk measures, complete with advanced mathematical proofs and neural network approximations. However, it leans heavily on theory and synthetic numerical illustrations, lacking deep empirical backtesting on extensive real-world historical market datasets.
Research Flowchart
flowchart TD
A["Research Goal: Construct & compute convex risk measures using weak optimal transport (WOT) penalties"] --> B["Inputs: Loss functions, probability measures, constraints (Moments, Martingales)"]
B --> C["Theoretical Derivation: Explicit representation via a nonlinear transform of loss functions"]
C --> D["Computational Framework: Approximations using Neural Networks & algorithms for nonlinear transforms"]
D --> E["Outcomes: Numerical calculation of insurance worst-case losses & model-free no-arbitrage derivative pricing"]