Perturbation Methods in Credit Derivatives

CHAPTER 12Rates‐Credit‐FX Hybrid Modelling

12.1 PREVIOUS WORK

We now seek to extend the pricing kernel derived in the previous chapter to incorporate credit risk. In other words we wish to consider multi‐currency cash flows which are either contingent on survival of a given debt issuer (or counterparty) or else are default contingent (protection or loss payment). To this end we will consider a multi‐factor model with four underlying assets. Essentially we shall be extending the three‐factor modelling of Chapter 11 with two interest rates and an FX rate to include credit risk in the manner set out in Chapters 8 and 10, allowing the FX rate to jump at default.

Our model is closely related to that considered by Itkin et al. [2019] except in that they take the interest rates to be governed by CIR square‐root volatility processes rather than Hull–White and they also allow of a jump in the foreign interest rate at default. They consider the basis spread between CDS spreads in the two currencies and find a significant near‐linear dependence on the FX jump but a much weaker non‐linear dependence (a few bps) on the interest rate jump, which furthermore diminishes as the FX jump size increases. The credit‐FX correlation is found to have a strong linear impact and the others very little.

On that basis we expect, for such “linear” products as quanto CDS, that the difference in prices compared with the use of the simpler model presented in Chapter 10 will not be great. We none the less ...