Paper: arXiv 2610.06495

Authors: Lokman A Abbas-Turki, Chassagneux Jean-Fran{\c c}ois, Jean-Philippe Lemor, Gr{é}goire Loeper, Simon Sananes

Abstract

We develop a posteriori primal-dual bounds for numerical approximations of European option prices in the Uncertain Volatility Model. Given a smooth candidate approximation of the value function, a feedback control induced by its Hessian yields a primal lower bound. On the dual side, we derive a representation based on a matrix-valued Gamma field, which provides an upper bound through a nonnegative Hamiltonian penalty. We further show that, when the dual field is generated by the candidate itself, this upper bound admits an equivalent representation in terms of the residual of the associated Black-Scholes-Barenblatt equation. We then study discrete-time approximations of these primal and dual quantities and quantify the corresponding discretization errors. Finally, we investigate their numerical evaluation for candidates obtained by stochastic policy-gradient and physics-informed neural-network methods, and compare the resulting estimates with a martingale dual approach from the stochastic-control literature. The numerical experiments highlight the importance of derivative accuracy, and in particular of second-order information, for obtaining tight a posteriori bounds.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a highly mathematical and novel approach to error bounding in the Uncertain Volatility Model, demonstrating strong theoretical rigor. While it includes numerical experiments, the primary focus is on the derivation of the bounds rather than extensive empirical validation, hence a moderate empirical rigor score.

Research Flowchart

  flowchart TD
    A[Research Goal: Develop a posteriori primal-dual error bounds for UVM option prices] --> B{Methodology: Primal-Dual Bounding Framework};
    B --> C[Primal Lower Bound: Hessian-induced feedback control];
    B --> D[Dual Upper Bound: Gamma field / Black-Scholes-Barenblatt Residual];
    D --> E{Computational Process: Discrete-time approximation & Numerical Evaluation};
    E --> F[Inputs: Candidate value functions from SPG & PINN];
    F --> G[Key Findings: Importance of derivative accuracy (2nd order) for tight bounds];
    G --> H[Outcomes: Quantified discretization errors & comparison with martingale dual];