# SFB 303 Discussion Paper No. B - 457

**Author**: Lotz, Christopher

**Title**: Optimal Shortfall Hedging of Credit Risk

**Abstract**: 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.

**Keywords**: VaR, Credit Risk, Quantile Hedging, Incomplete Markets

**JEL-Classification-Number**: G12,G13,G33

**Creation-Date**: September 1999

**URL**: ../1999/b/bonnsfb457.pdf
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14.09.1999, Webmaster