Hedging of Defaultable Contingent Claims using BSDE with uncertain time horizon

TL;DR

This paper develops a mathematical framework for solving backward stochastic differential equations with random, non-stopping time horizons, enabling better hedging strategies for defaultable claims and life insurance contracts.

Contribution

It proves existence and uniqueness of solutions for BSDEs with non-stopping time horizons and applies this to financial hedging of defaultable claims.

Findings

Established conditions for existence and uniqueness of BSDE solutions with uncertain terminal times.

Derived explicit hedging strategies for defaultable claims and life insurance contracts.

Extended the mathematical theory of BSDEs to include non-stopping random terminal times.

Abstract

This article focuses on the mathematical problem of existence and uniqueness of BSDE with a random terminal time which is a general random variable but not a stopping time, as it has been usually the case in the previous literature of BSDE with random terminal time. The main motivation of this work is a financial or actuarial problem of hedging of defaultable contingent claims or life insurance contracts, for which the terminal time is a default time or a death time, which are not stopping times. We have to use progressive enlargement of the Brownian filtration, and to solve the obtained BSDE under this enlarged filtration. This work gives a solution to the mathematical problem and proves the existence and uniqueness of solutions of such BSDE under certain general conditions. This approach is applied to the financial problem of hedging of defaultable contingent claims, and an expression of the hedging strategy is given for a defaultable contingent claim or a life insurance contract.