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In this paper we discuss a credit risk model with a pure jump L\'evy processfor the asset value and an unobservable random barrier. The default time is thefirst time when the asset value falls below the barrier. Using theindistinguishability of the intensity process and the likelihood process, weprove the existence of the intensity process of the default time and find itsexplicit representation in terms of the distance between the asset value andits running minimal value. We apply the result to find the instantaneous creditspread process and illustrate it with a numerical example.

Intensity Process for a Pure Jump L\'evy Structural Model with Incomplete Information