An optimal transport approach for the multiple quantile hedging problem

TL;DR

This paper reformulates the multiple quantile hedging problem as an optimal transport problem, enabling the use of duality theory and numerical algorithms to compute hedging prices in non-linear markets.

Contribution

It introduces a novel optimal transport framework for the multiple quantile hedging problem and develops a dual formulation with a numerical solution approach.

Findings

Reformulation as Monge and Kantorovich optimal transport problems

Proof of no duality gap in the linear case

Development of a numerical method using SGA algorithms

Abstract

We consider the multiple quantile hedging problem, which is a class of partial hedging problems containing as special examples the quantile hedging problem (F{\"o}llmer \& Leukert 1999) and the PnL matching problem (introduced in Bouchard \& Vu 2012). In complete non-linear markets, we show that the problem can be reformulated as a kind of Monge optimal transport problem. Using this observation, we introduce a Kantorovitch version of the problem and prove that the value of both problems coincide. In the linear case, we thus obtain that the multiple quantile hedging problem can be seen as a semi-discrete optimal transport problem, for which we further introduce the dual problem. We then prove that there is no duality gap, allowing us to design a numerical method based on SGA algorithms to compute the multiple quantile hedging price.