A comparison principle for PDEs arising in approximate hedging problems: application to Bermudan options

TL;DR

This paper establishes a comparison principle for PDEs related to approximate hedging problems, enabling better analysis of Bermudan options' quantile hedging prices in a non-linear setting.

Contribution

It proves a comparison theorem for a class of PDEs with discontinuous operators by rewriting them with continuous operators, advancing the mathematical understanding of hedging problems.

Findings

Comparison theorem for PDEs with discontinuous operators

Application to quantile hedging of Bermudan options

Enhanced mathematical tools for non-linear hedging analysis

Abstract

In a Markovian framework, we consider the problem of finding the minimal initial value of a controlled process allowing to reach a stochastic target with a given level of expected loss. This question arises typically in approximate hedging problems. The solution to this problem has been characterised by Bouchard, Elie and Touzi in [1] and is known to solve an Hamilton-Jacobi-Bellman PDE with discontinuous operator. In this paper, we prove a comparison theorem for the corresponding PDE by showing first that it can be rewritten using a continuous operator, in some cases. As an application, we then study the quantile hedging price of Bermudan options in the non-linear case, pursuing the study initiated in [2]. [1] Bruno Bouchard, Romuald Elie, and Nizar Touzi. Stochastic target problems with controlled loss. SIAM Journal on Control and Optimization, 48(5):3123-3150,2009. [2] Bruno Bouchard, Romuald Elie, Antony R\'eveillac, et al. Bsdes with weak terminal condition. The Annals of Probability, 43(2):572-604,2015.