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Financial modelling with multivariate local jump processes

Final Activity Report Summary - FINMODJUMP (Financial modelling with multivariate local jump processes)

Credit Default Swaps (CDSs) have become during the last decade a very important financial derivative that allow investors to transfer the credit risk of obligors to other investors which are willing to take on this risk in exchange for a periodically paid premium. At the event of a default the seller of default protection agrees to cover the losses of the investor with exposure to the obligor. To be able to calculate the fair CDS premium that the seller and buyer can agree on we need a way of calculating the default probability of the reference obligor.

We have used a firm value model approach where a default event occurs when the asset value of the firm crosses a deterministic low barrier. This barrier corresponds to the recovery value of the firm's debt. The default probability, that is, the probability that the firm value hits the barrier is of vital importance when pricing Credit Default Swaps (CDS) and other financial derivatives where the default risk has to be taken into account.

We have worked with stochastic models that include jumps, in contrast to the standard firm value models which are assuming that the firm's value is evolving over time according to a continuous path, that is, the firm value cannot change dramatically through jumps. The default of a firm typically comes as a chock and we therefore believe that a model for credit risk should contain jumps.

In the project different jump models have been studied and for each class of models numerical schemes have been developed to be able to calculate the default probability of the firm. As mentioned above, the default probabilities are essential in the pricing of the CDSs with reference to the firm. Furthermore, the project has resulted in the methods to price options on CDSs and more advanced forms of CDSs called Constant Maturity Credit Default Swaps.