Credit risk premia and quadratic BSDEs with a single jump

TL;DR

This paper develops a framework using quadratic BSDEs with jumps to determine credit risk premia for defaultable claims, filling a theoretical gap in existence and uniqueness results for such equations.

Contribution

It introduces new existence and uniqueness results for quadratic BSDEs with jumps, enabling their use in credit risk valuation models.

Findings

Established conditions for quadratic BSDEs with jumps.

Provided a constructive algorithm for solving these BSDEs.

Applied results to credit risk premia valuation.

Abstract

This paper is concerned with the determination of credit risk premia of defaultable contingent claims by means of indifference valuation principles. Assuming exponential utility preferences we derive representations of indifference premia of credit risk in terms of solutions of Backward Stochastic Differential Equations (BSDE). The class of BSDEs needed for that representation allows for quadratic growth generators and jumps at random times. Since the existence and uniqueness theory for this class of BSDEs has not yet been developed to the required generality, the first part of the paper is devoted to fill that gap. By using a simple constructive algorithm, and known results on continuous quadratic BSDEs, we provide sufficient conditions for the existence and uniqueness of quadratic BSDEs with discontinuities at random times.