A numerical scheme for the quantile hedging problem

TL;DR

This paper introduces a new numerical scheme combining PCPT and finite difference methods to approximate the quantile hedging price in non-linear markets, with proven convergence and practical efficiency demonstrated through financial examples.

Contribution

It presents a novel numerical approach for quantile hedging in non-linear markets, integrating PCPT with finite difference schemes and providing convergence proofs.

Findings

The scheme converges reliably for non-linear market models.

Numerical experiments show the method's efficiency in practical financial scenarios.

The approach effectively handles market imperfections in quantile hedging.

Abstract

We consider the numerical approximation of the quantile hedging price in a non-linear market. In a Markovian framework, we propose a numerical method based on a Piecewise Constant Policy Timestepping (PCPT) scheme coupled with a monotone finite difference approximation. We prove the convergence of our algorithm combining BSDE arguments with the Barles & Jakobsen and Barles & Souganidis approaches for non-linear equations. In a numerical section, we illustrate the efficiency of our scheme by considering a financial example in a market with imperfections.