发表机构
Université Paris Cité; Sorbonne Université; Laboratoire de Probabilités, Statistique et Modélisation (LPSM, UMR CNRS 8001); ENSAE-CREST; Institut Polytechnique de Paris; BNP Paribas Global Markets（巴黎西岱大学; 索邦大学; 概率、统计与建模实验室（CNRS联合研究单位8001）; 国立高等经济统计学院-CREST; 巴黎理工学院; 法国巴黎银行全球市场）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为不确定波动率模型中的欧式期权定价发展后验原始-对偶误差界，利用Hessian反馈控制和对偶Gamma场分别获得下界和上界，并通过数值实验验证了二阶信息对紧致界的关键作用。
AI 中文摘要
我们为不确定波动率模型中的欧式期权价格的数值近似发展了后验原始-对偶界。给定价值函数的光滑候选近似，由其Hessian矩阵诱导的反馈控制产生原始下界。在对偶方面，我们推导了一个基于矩阵值Gamma场的表示，该表示通过非负Hamiltonian惩罚提供上界。我们进一步证明，当对偶场由候选本身生成时，该上界允许以相关的Black-Scholes-Barenblatt方程的残差形式进行等价表示。然后，我们研究这些原始和对偶量的离散时间近似，并量化相应的离散化误差。最后，我们探讨了对于通过随机策略梯度和物理信息神经网络方法获得的候选值的数值评估，并将所得估计与随机控制文献中的鞅对偶方法进行比较。数值实验强调了导数精度，特别是二阶信息，对于获得紧的后验界的重要性。
英文摘要
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.
