Closed Form Solutions for Contingent CDS on Cross Currency Swaps

Abstract

A practical implementation of a CVA calculation on a portfolio of different types of derivatives with one counterpart can be complicated and needs to be validated in various ways. One very simple possibility is provided by the present paper: When the portfolio is made up only of CCY swaps (which differ only by the notional amounts and are otherwise identical) and the driving model is the one used here then one can use a closed form solution to predict the CVA with very high accuracy. This can be used for tests that an implementation should pass. For that purpose, the simplest possible model that includes vols and correlations for interest rates in both currencies, the
FX rate and the default intensity is chosen. It allows for a closed form solution for the PV of a contingent credit default swap (CCDS) that pays in default the outstanding mark to market price of a cross currency swap provided the latter is positive. The paper also provides conditions which determine the directions of the sensitivities of this PV with respect to changes in the correlations.