Mean-Variance Hedging on uncertain time horizon in a market with a jump

TL;DR

This paper addresses mean-variance hedging with a random horizon that includes a jump time, formulating it as a stochastic control problem and solving it using backward stochastic differential equations with jumps.

Contribution

The work introduces a novel approach to mean-variance hedging with a jump time horizon by linking it to BSDEs with jumps and providing a verification theorem for optimal strategies.

Findings

Derived the optimal hedging strategy using BSDEs with jumps.

Proved existence of solutions to the BSDE system via filtration enlargement.

Connected the hedging problem to a stochastic control framework.

Abstract

In this work, we study the problem of mean-variance hedging with a random horizon T ^ tau, where T is a deterministic constant and is a jump time of the underlying asset price process. We rst formulate this problem as a stochastic control problem and relate it to a system of BSDEs with jumps. We then provide a veri cation theorem which gives the optimal strategy for the mean-variance hedging using the solution of the previous system of BSDEs. Finally, we prove that this system of BSDEs admits a solution via a decomposition approach coming from ltration enlargement theory.