In this paper we examine the problem of partially hedging a given credit risk exposure. We derive hedges which satisfy certain optimality criteria: For a given investment into the hedge they minimize the remaining risk, or vice versa. This is motivated by the fact that it is a core business of financial intermediaries to carry risks, and that therefore they do not want to hedge their risks completely. In contrast to the usual mean-variance criterion, our hedging strategies try to minimize either the shortfall probability (SP) or the expected shortfall (ES). In complete markets this allows the investor to save money by hedging only part of the claim, while taking a certain (minimal) risk that the hedge does not cover the claim completely. In incomplete markets, a perfect hedge is not always available, and this methodology introduces a new way to find a hedging strategy which minimizes the shortfall risk. We apply this to a credit risk model, where default occurs at the first jump time of a Poisson process. The write-down after default is stochastic and independent of the time of default. In this stylized model we compare hedging strategies for defaultable bonds and credit default swaps which minimize either the SP (Quantile Hedging) or the ES. We consider first a complete market where the martingale measure is unique and derive explicit results. Hedging strategies for both objectives are compared. In the incomplete markets setting, we consider two situations: In the first, we assume that the default risk premium is unknown from the beginning, and therefore we have to select the worst-case martingale measure from the set of possible martingale measures. In the second, the market is complete at the beginning, but at a future time point the default risk parameter will change randomly, for example because of a rating change, and this makes the market incomplete. Strategies for both situations are developped.
