Let's Jump Together - Pricing of Credit Derivatives: From Index Swaptions to CPPIs

Abstract

This paper describes a dynamic multivariate jump driven model in a credit setting. We set up a dynamic Levy model, more precisely a Multivariate Variance Gamma (VG) model, for a series of correlated spreads. The parameters of the model come from a two step calibration procedure. First, a joint calibration on swaptions on the spreads is performed and second, a correlation matching procedure is applied. For the first calibration step, we make use of equity-like pricing formulas for payer and receiver swaptions, based on the characteristic function and the Fast Fourier Transform (FFT) method. In the second calibration step, we fix the correlation in the model to match the prescribed (in casu historically observed) correlation. This can be done fast since a closed form expression is readily available. The resulting jump driven dynamic model generates correlated spreads very fast. This model can be used to price a whole range of exotic structures. We illustrate this by pricing the currently popular credit Constant Proportion Portfolio Insurance (CPPI) structures. Because of the built in jump dynamics a better assessment of gap risk is possible.