Fourier inversion algorithms for generalized CreditRisk+ models and an extension to incorporate market risk

A popular model to describe credit risk in practice is CreditRisk+ and in this paper a Fourier inversion to obtain the distribution of the credit loss is proposed. A deeper analysis of the Fourier transformation showed that there are at least two methods to obtain the distribution although the corresponding characteristic function is not integrable. The CreditRisk+ model will be extended such, that general dependent sector variables can be taken into consideration, for example dependent lognormal sector variables. Then the transfer to a continuous time model will be performed and the sector variables become processes, more precisely geometric Brownian motions. To have a time continuous credit risk model is an important step to combine this model with market risk. Additionally a portfolio model will be presented where the changes of the spreads are driven by the sector variables. Using a linear expansion of the market risk, the distribution of this portfolio can be determined. In the special case that there is no credit risk, this model yields the well known Delta normal approach for market risk, hence a link between credit risk and market risk has been established