A posteriori error bounds for the Uncertain Volatility Model
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.
Related
- Copula Active Subspaces II: Error Decomposition, A Posteriori Estimation, and Sharpness of the Bounds
- Stochastic Policy Gradient Methods in the Uncertain Volatility Model
- Residual-based a posteriori error estimate for a parareal approximation of the cardiac monodomain problem
- A Priori and A Posteriori Error Estimates for a Crank Nicolson Type Scheme of an Elliptic Problem with Dynamical Boundary Conditions
- A posteriori error estimates for variational inequalities of the second kind in an abstract framework
BibTeX
@misc{abbasturki2026posteriori,
title = {{A posteriori error bounds for the Uncertain Volatility Model}},
author = {Lokman A Abbas-Turki and Chassagneux Jean-François and Jean-Philippe Lemor and Grégoire Loeper and Simon Sananes},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06495},
eprint = {2610.06495},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05776673v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A posteriori error bounds for the Uncertain Volatility Model
AU - Lokman A Abbas-Turki
AU - Chassagneux Jean-François
AU - Jean-Philippe Lemor
AU - Grégoire Loeper
AU - Simon Sananes
PY - 2026
DA - 2026/10/03
PB - HAL
DO - 10.48550/arxiv.2610.06495
UR - https://hal.science/hal-05776673v1
AB - 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.
ER -