Neural network approximation for superhedging prices

TL;DR

This paper develops neural network methods to approximate superhedging prices and strategies in discrete-time markets, proving convergence of quantile hedging prices and providing practical numerical implementations.

Contribution

It introduces neural network approximations for superhedging prices and strategies, including convergence results and a method to approximate the process of consumption.

Findings

Neural network-based approximation of superhedging prices converges as quantile level approaches 1.

The method effectively approximates superhedging strategies up to maturity.

Numerical results demonstrate the practical applicability of the approach.

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 $\alpha$-quantile hedging price converges to the superhedging price at time $0$ for $\alpha$ tending to $1$, and show that the $\alpha$-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 $t>0$, 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.