Optimal Hedging in Incomplete Markets

TL;DR

This paper develops a framework for optimal hedging in incomplete markets modeled by a Lévy-Itô process, using a least squares criterion to determine the best hedge portfolio relative to a benchmark process.

Contribution

It introduces a method to find optimal hedge portfolios in Lévy-Itô markets based on a least squares error criterion, considering the market's incompleteness.

Findings

Existence of an optimal hedge portfolio under the specified criteria.

Framework applicable to markets driven by Brownian motion and Poisson measures.

Optimal hedging strategy minimizes expected squared error over a given time frame.

Abstract

We consider the problem of optimal hedging in an incomplete market with an established pricing kernel. In such a market, prices are uniquely determined, but perfect hedges are usually not available. We work in the rather general setting of a L\'evy-Ito market, where assets are driven jointly by an $n$-dimensional Brownian motion and an independent Poisson random measure on an $n$-dimensional state space. Given a position in need of hedging and the instruments available as hedges, we demonstrate the existence of an optimal hedge portfolio, where optimality is defined by use of an expected least squared-error criterion over a specified time frame, and where the numeraire with respect to which the hedge is optimized is taken to be the benchmark process associated with the designated pricing kernel.