Neural network approximation for superhedging prices. (12th September 2022)

• Record Type: Journal Article
• Authors: Biagini, Francesca; Gonon, Lukas; Reitsam, Thomas
• Is Part Of: Mathematical finance. Volume 33:Number 1(2023)
• Page Start: 146
• Page End: 184
• Publication Date: 2022-09-12
• Subjects: deep learning -- quantile hedging -- superhedging; Business mathematics -- Periodicals; 332
• DOI: 10.1111/mafi.12363
• Languages: English
• ISSNs: 0960-1627

Abstract

Abstract: This article examines neural network‐based approximations for the superhedging price process of a contingent claim in a discrete time market model. First we prove that the α‐quantile hedging price converges to the superhedging price at time 0 for α tending to 1, and show that the α‐quantile hedging price can be approximated by a neural network‐based price. This provides a neural network‐based approximation for the superhedging price at time 0 and also the superhedging strategy up to maturity. To obtain the superhedging price process for t > 0 $t>0$, by using the Doob decomposition, it is sufficient to determine the process of consumption. We show that it can be approximated by the essential supremum over a set of neural networks. Finally, we present numerical results.