ABSTRACT
The cross-economy version of shifted BGM links foreign and domestic shifted BGM models via the forward FX exchange rate. In the Gaussian HJM framework the link is possible with deterministic volatilities for domestic and foreign instantaneous forwards, and also the FX rate; that is, deterministic volatilities are totally compatible with lognormal models for the prices of domestic and foreign bonds and the spot and forward FX rates. But, as shown by Schlogl [112], in cross-economy BGM some among the do-
mestic and foreign interest rate volatilities, and FX forward volatilities must be stochastic. Nevertheless, as with swaption volatilities in domestic BGM, with appropriate choices it is possible to obtain approximations for stochastic volatilities in cross-economy BGM that are good enough to return by simulation fairly accurate values for the implied volatilities to which the model is calibrated. To set the scene, in the following Section-14.1 we first work through rele-
vant ideas in a cross-economy HJM framework, before considering the BGM equivalent. Our notation for the foreign economy will be to superfix f to domestic variables to denote the equivalent foreign variable; for example, if PT and B (t, T ) are respectively the T -forward measure and zero coupon in the domestic economy, then PfT and Bf (t, T ) are the equivalent in the foreign economy.
