A posteriori error bounds for the Uncertain Volatility Model

Lokman Abbas-Turki (), Chassagneux Jean-François, Jean-Philippe Lemor (), Grégoire Loeper () and Simon Sananes ()
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Lokman Abbas-Turki: LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité
Chassagneux Jean-François: ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - Groupe ENSAE-ENSAI - Groupe des Écoles Nationales d'Économie et Statistique - IP Paris - Institut Polytechnique de Paris
Jean-Philippe Lemor: BNP-Paribas
Grégoire Loeper: BNP-Paribas
Simon Sananes: BNP-Paribas, LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité

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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.

Keywords: Uncertain volatility model; a posteriori error bounds; robust option pricing; stochastic control; primal-dual bounds; Hamilton-Jacobi-Bellman equations (search for similar items in EconPapers)
Date: 2026-10-03
Note: View the original document on HAL open archive server: https://hal.science/hal-05776673v1
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