Adversarial Training for Deep Hedging in Nonstationary Markets

Paper: arXiv 2610.07162

Authors: Philipp J. Schneider, Lukas Looser, Antoine Garin, Shuhan Liu, Daniel Kuhn

Abstract

Deep hedging learns trading policies from historical or simulated market trajectories, yet under nonstationarity these training paths may not represent future market conditions. We propose WRAP (Wasserstein-Reweighting Adversarial Perturbation), a drift-aware adversarial training framework derived from a two-budget distributionally robust optimization (DRO) formulation. The formulation is anchored to a weighted empirical reference distribution whose fixed baseline weights are chosen to balance sampling uncertainty against temporal drift. Around this reference distribution, the ambiguity set addresses two complementary forms of distributional misspecification by allowing an adversary to reweight the observed trajectories subject to a $ϕ$-divergence constraint and perturb their paths subject to an optimal-transport (OT) constraint. We derive a joint first-order expansion in which the leading-order increase over the nominal expected loss decomposes into a reweighting contribution determined by the dispersion of hedging losses across trajectories and a transport contribution determined by the sensitivity of the loss to path perturbations. This expansion yields an explicit finite-dimensional adversarial attack that replaces the distributional inner supremum with a tractable first-order approximation. Across stationary and nonstationary Heston dynamics and a generalized affine diffusion (GAD), the experiments show complementary benefits from reweighting and transport, with joint adversarial training providing the largest gains under nonstationarity.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a mathematically sophisticated framework for deep hedging under nonstationarity, combining DRO with Wasserstein and phi-divergence constraints. Its empirical section, while using simulated data, demonstrates rigorous testing across various market dynamics. The novelty lies in the joint first-order expansion and the WRAP framework for adversarial training.

Research Flowchart

  flowchart TD
    A[Research Goal: Deep Hedging in Nonstationary Markets] --> B{Problem: Nonstationary Market Conditions};
    B --> C[Methodology: WRAP Framework (Wasserstein-Reweighting Adversarial Perturbation)];
    C --> D{Inputs: Historical/Simulated Market Trajectories};
    D --> E[Computational Process: Two-Budget DRO, Optimal Transport, Phi-Divergence, First-Order Expansion];
    E --> F[Outcomes: Improved Hedging Policies, Largest Gains under Nonstationarity];

Cited References (64)

1. \href https://arxiv.org/abs/2510.18566 Don’t look back in anger: Wasserstein distributionally robust optimization with nonstationary data (2025) — link
2. \href https://papers.nips.cc/paper_files/paper/2025/hash/bee8c2bc757f6bbc3efd7cf1b979f0c9-Abstract-Conference.html Distributional adversarial attacks and training in deep hedging (2025)
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5. \href https://doi.org/10.1017/S0962492924000084 Distributionally robust optimization (2025)
6. \href https://doi.org/10.1007/s00440-014-0583-7 On the rate of convergence in Wasserstein distance of the empirical measure (2015)
7. \href https://doi.org/10.1007/978-3-540-71050-9 Optimal Transport: Old and New (2009)
8. \href https://doi.org/10.1098/rspa.2021.0176 Sensitivity analysis of Wasserstein distributionally robust optimization problems (2021)

0 of 64 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.