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Model Risk Static-Hedging a Constrained Distributionally Robust Optimization approach

Nathan Sauldubois

TL;DR

The paper develops a rigorous framework for model risk via constrained distributionally robust optimization (DRO) under marginal and martingale constraints, focusing on static hedging with second-period vanilla payoffs. It introduces a novel Implicit Function Theorem–based method to analyze constrained DRO in the two-period setting, yielding first-order sensitivity formulas that reflect the structure of adapted Wasserstein distances and constraint operators. The authors derive explicit optimal hedging components and sensitivity expressions for both marginal and martingale constraints, and extend results to semi-static hedging and American option stopping problems, with numerical illustrations comparing Black–Scholes and Bachelier models. Overall, the work provides a versatile toolkit for quantifying and hedging model risk under realistic market-calibration constraints, highlighting how second-marginal constraints can significantly reduce sensitivity and alter hedging strategies.

Abstract

We investigate model risk and distributionally robust optimization (DRO) under marginal and martingale constraints. Building on our previous work, we address the previously open case of static hedging with second-period maturity vanilla options and hedging strategies involving a vanilla payoff. We also extend recent sensitivity results to settings where admissible models must satisfy a martingale coupling constraint. Our approach relies on a weak implicit function theorem argument to construct families of measures satisfying the prescribed constraints. We derive closed-form sensitivity formulas and characterize the corresponding hedging strategies when the underlying process is real-valued.

Model Risk Static-Hedging a Constrained Distributionally Robust Optimization approach

TL;DR

The paper develops a rigorous framework for model risk via constrained distributionally robust optimization (DRO) under marginal and martingale constraints, focusing on static hedging with second-period vanilla payoffs. It introduces a novel Implicit Function Theorem–based method to analyze constrained DRO in the two-period setting, yielding first-order sensitivity formulas that reflect the structure of adapted Wasserstein distances and constraint operators. The authors derive explicit optimal hedging components and sensitivity expressions for both marginal and martingale constraints, and extend results to semi-static hedging and American option stopping problems, with numerical illustrations comparing Black–Scholes and Bachelier models. Overall, the work provides a versatile toolkit for quantifying and hedging model risk under realistic market-calibration constraints, highlighting how second-marginal constraints can significantly reduce sensitivity and alter hedging strategies.

Abstract

We investigate model risk and distributionally robust optimization (DRO) under marginal and martingale constraints. Building on our previous work, we address the previously open case of static hedging with second-period maturity vanilla options and hedging strategies involving a vanilla payoff. We also extend recent sensitivity results to settings where admissible models must satisfy a martingale coupling constraint. Our approach relies on a weak implicit function theorem argument to construct families of measures satisfying the prescribed constraints. We derive closed-form sensitivity formulas and characterize the corresponding hedging strategies when the underlying process is real-valued.
Paper Structure (25 sections, 27 theorems, 235 equations, 14 figures)

This paper contains 25 sections, 27 theorems, 235 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Model Risk through Distributionally Robust Optimization.
  3. Model Risk Hedging through Constrained DRO.
  4. Our contribution: A new method to study constrained DRO.
  5. Notations and Definitions
  6. Main Results
  7. Finite Moment and Conditional Law Constraint
  8. Static Hedging/Marginal Constraints in One Dimension
  9. Semi-Static Hedging Strategy
  10. Optimal Stopping Problem
  11. Numerical illustration
  12. Comparison of Sensitivities
  13. Comparison of Hedging Strategies
  14. Proofs
  15. A Sufficient Condition
  16. ...and 10 more sections

Key Result

Proposition 3.3

Fix $\mathbf{d} \in\{ \mathbb{W}_p, \mathbb{W}_p^{\rm ad} \}$. Let $\varphi$ satisfy Assumption ass:regul and estim on g, $\psi$ and $\mu$ satisfy Assumption ass:comp and growth adapted wass and $\Theta$ a compactly supported $C^1$ function in $\mathbb{L}^p_{\mathbf{d}} (\mu )$, which is defined by where $\mathcal{L}^{\mathbf{d}}_{ \varphi, \psi}$ is defined by eqdef:operatogenconstraints and

Figures (14)

  • Figure 1: Impact of the marginal constraint on sensitivities for the Black-Scholes model.
  • Figure 2: Impact of the marginal constraint on martingale sensitivities for the Black-Scholes model.
  • Figure 3: Sensitivities and Vega in the Black-Scholes model.
  • Figure 4: Impact of the marginal constraint on sensitivities for the Bachelier model.
  • Figure 5: Impact of the marginal constraint on martingale sensitivities for the Bachelier model.
  • ...and 9 more figures

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Remark 3.5
  • Proposition 3.6
  • ...and 29 more