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First order Martingale model risk and semi-static hedging

Nathan Sauldubois, Nizar Touzi

TL;DR

This work analyzes first-order model-risk sensitivities in a distributionally robust framework for functionals on the Wasserstein space under martingale constraints and/or fixed first marginals. It integrates semi-static hedging instruments to reduce the worst-case impact of model misspecification and derives explicit first-order hedging strategies by solving convex/variational problems in weighted Sobolev spaces; the results are established for both adapted and standard Wasserstein deviations and extended to optimal stopping. Key contributions include differentiability at the origin of the robust hedging value under various deviation sets, explicit characterizations of optimal hedges (denoted by $h_{\rm ad,M}$ and $h_{\rm M}$), and, in one dimension with $p=2$, a Fredholm-integral equation characterizing the martingale hedge. The numerical illustrations demonstrate that standard Wasserstein deviations yield higher sensitivities than adapted deviations, underscoring the practical significance of distance choice, and show that the proposed hedges meaningfully reduce first-order model risk in forward-start European and American options. Overall, the paper provides a rigorous, implementable framework for robust hedging under model uncertainty with dynamic and marginal constraints, offering both theoretical insights and practical hedging rules for financial applications.

Abstract

We investigate model risk distributionally robust sensitivities for functionals on the Wasserstein space when the underlying model is constrained to the martingale class and/or is subject to constraints on the first marginal law. Our results extend the findings of Bartl, Drapeau, Obloj \& Wiesel \cite{bartl2021sensitivity} and Bartl \& Wiesel \cite{bartlsensitivityadapted} by introducing the minimization of the distributionally robust problem with respect to semi-static hedging strategies. We provide explicit characterizations of the model risk (first order) optimal semi-static hedging strategies. The distributional robustness is analyzed both in terms of the adapted Wasserstein metric and the more relevant standard Wasserstein metric.

First order Martingale model risk and semi-static hedging

TL;DR

This work analyzes first-order model-risk sensitivities in a distributionally robust framework for functionals on the Wasserstein space under martingale constraints and/or fixed first marginals. It integrates semi-static hedging instruments to reduce the worst-case impact of model misspecification and derives explicit first-order hedging strategies by solving convex/variational problems in weighted Sobolev spaces; the results are established for both adapted and standard Wasserstein deviations and extended to optimal stopping. Key contributions include differentiability at the origin of the robust hedging value under various deviation sets, explicit characterizations of optimal hedges (denoted by and ), and, in one dimension with , a Fredholm-integral equation characterizing the martingale hedge. The numerical illustrations demonstrate that standard Wasserstein deviations yield higher sensitivities than adapted deviations, underscoring the practical significance of distance choice, and show that the proposed hedges meaningfully reduce first-order model risk in forward-start European and American options. Overall, the paper provides a rigorous, implementable framework for robust hedging under model uncertainty with dynamic and marginal constraints, offering both theoretical insights and practical hedging rules for financial applications.

Abstract

We investigate model risk distributionally robust sensitivities for functionals on the Wasserstein space when the underlying model is constrained to the martingale class and/or is subject to constraints on the first marginal law. Our results extend the findings of Bartl, Drapeau, Obloj \& Wiesel \cite{bartl2021sensitivity} and Bartl \& Wiesel \cite{bartlsensitivityadapted} by introducing the minimization of the distributionally robust problem with respect to semi-static hedging strategies. We provide explicit characterizations of the model risk (first order) optimal semi-static hedging strategies. The distributional robustness is analyzed both in terms of the adapted Wasserstein metric and the more relevant standard Wasserstein metric.

Paper Structure

This paper contains 20 sections, 14 theorems, 149 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Notations and Definitions
  3. Main Results
  4. Adapted Wasserstein Model Deviation
  5. Wasserstein Model Deviation
  6. One-dimensional First-Marginal Constraint
  7. Optimal Stopping Problem
  8. Example for Specific $g$
  9. Stochastic Optimization Problem
  10. Stochastic Games
  11. Numerical Illustrations
  12. Forward-start European option model risk sensitivities
  13. American put model risk sensitivities
  14. Proofs
  15. Reduction to the linearized problem
  16. ...and 5 more sections

Key Result

Proposition 3.3

Under Assumption ass:on g, $\overline{G}^{\rm M}_{\rm ad}$ and $\underline{G}^{\rm M}_{\rm ad}$ are differentiable at the origin and and $J, \partial_x^{\rm ad}$ defined by Equation eqdef:caus_grad,J. Moreover, $U^{\rm M}_{{\rm ad}}$ is convex and coercive (in the sense of Definition ch01def:coercivity); hence, the optimization problem eq:deriv mart adapted admits a solution $h_{\rm ad, M}$ chara

Figures (10)

  • Figure 1: Sensitivities in the Black-Scholes model, $g ( \mu ) = E^{\mu} [ ( X_2 - X_1 )^+ ]$.
  • Figure 2: Relative sensitivities in the Black-Scholes model for $g \left( \mu \right) = E^{\mu} [ ( X_2 - X_1 )^+ ]$.
  • Figure 3: Worst-case scenarios in the Black-Scholes model for $g(\mu) = \mathbb{E}^{\mu} [ ( X_2 - X_1 )^+ ]$, $\sigma = 0.4$ and $r = 0.5$ (Black-Scholes distribution in red).
  • Figure 4: Optimal Hedges for $g(\mu) = \mathbb{E}^{\mu} [ ( X_2 - X_1 )^+ ]$, $\sigma = 0.4$
  • Figure 5: Sensitivities in the Black-Scholes model for the American put option $g(\mu)=\inf_{\tau \in \rm ST} \mathbb{E}^{\mu} [(e^{- \rho \tau} K - X_\tau)^+]$.
  • ...and 5 more figures

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 3.2
  • Proposition 3.3
  • Remark 3.4
  • Proposition 3.5
  • Proposition 3.7
  • Remark 3.8
  • Remark 3.10
  • ...and 15 more