Modelling and hedging of default risk

Monique Jeanblanc and Marek Rutkowski

INTRODUCTION

Our aim in this chapter is to present the fundamental methods and results relating to the valuation and hedging of credit derivatives (defaultable claims) within the structural and reduced-form approaches. In contrast to some other works that address this issue by invoking a suitable version of the martingale representation theorem, we directly analyse the possibility of replication of a given contingent claim by means of a trading strategy based on default-free and defaultable securities. It should be stressed that not only is the exact replication of the value process of a defaultable claim valid prior to the default time, but it also includes the jump of this value at time of default, if it occurs prior to or at the maturity date of this claim. We believe that such an approach, motivated by Vaillant (2001), is far more intuitive and leads directly to explicit forms for replicating strategies. Moreover, the issue of a judicious choice of tradeable securities appears in a natural way.

Although the approach may be applied to the general set-up of random interest rates and stochastic intensity, we deliberately focus on the case of deterministic interest rates and