Optimization problem and mean variance hedging on defaultable claims

TL;DR

This paper addresses the challenge of pricing and hedging claims dependent on the default times of two firms, using only bonds of one firm, by applying indifference pricing and mean variance hedging methods.

Contribution

It introduces a novel approach to hedge claims linked to multiple firms' defaults using only available bonds, employing HJB equations and BSDEs.

Findings

Derived explicit formulas for indifference prices in a Markov setting.

Developed a mean variance hedging framework using backward stochastic differential equations.

Demonstrated the effectiveness of the methods through theoretical analysis.

Abstract

We study the pricing and the hedging of claim {\psi} which depends on the default times of two firms A and B. In fact, we assume that, in the market, we can not buy or sell any defaultable bond of the firm B but we can only trade defaultable bond of the firm A. Our aim is then to find the best price and hedging of {\psi} using only bond of the firm A. Hence, we solve this problem in two cases: firstly in a Markov framework using indifference price and solving a system of Hamilton-Jacobi-Bellman equations, secondly, in a more general framework, using the mean variance hedging approach and solving backward stochastic differential equations (BSDE).