Perturbation Methods in Credit Derivatives

CHAPTER 11Multi‐Currency Modelling

Having looked at all models of interest based on two stochastic underlyings in Chapters 7 to 10, we move on in our investigation to consider pricing problems with three or more underlyings, starting with the simplest case where we have two interest rates and a spot underlying.¹ Specifically, we consider stochastic interest rates for foreign and domestic currencies and a stochastic exchange rate between these two currencies.

11.1 PREVIOUS WORK

The problem of finding analytic pricing kernels and/or option pricing formulae in the case of two stochastic interest rates with a stochastic FX rate appears to have received less attention than the rather simpler problem in relation to a hybrid equity‐rates model which we visited in Chapter 7. We follow closely the method of derivation used there based on the ideas expounded in Chapter 6.

A model closely related to the one we consider here was proposed by Jarrow and Yildirim [2003], albeit in the context of inflation derivatives pricing, rather than FX options. In their case the “nominal” interest rate is simply the domestic interest rate, while the “real” interest rate corresponds to the foreign interest rate and the consumer price index (CPI) to the FX rate. Jarrow and Yildirim [2003] produced exact closed‐form pricing formulae for vanilla options on the inflation index under the assumption of an HJM framework for the interest rates (Heath et al., 1992). An extensive discussion of the model and related ...