On Cross-Currency Models with Stochastic Volatility and Correlated Interest Rates

Abstract

Abstract We construct multi-currency models with stochastic volatility (SV) and correlated stochastic interest rates with a full matrix of correlations. We first deal with a foreign exchange (FX) model of Heston-type, in which the domestic and foreign interest rates are generated by the short-rate process of Hull–White (Hull, J. and White, A. [1990] Pricing interest-rate derivative securities, Review of Financial Studies, 3, pp. 573–592). We then extend the framework by modelling the interest rate by an SV displaced-diffusion (DD) Libor Market Model (Andersen, L. B. G. and Andreasen, J. [2000] Volatility skews and extensions of the libor market model, Applied Mathematics Finance, 1[7], pp. 1–32), which can model an interest rate smile. We provide semi-closed form approximations which lead to efficient calibration of the multi-currency models. Finally, we add a correlated stock to the framework and discuss the construction, model calibration and pricing of equity–FX–interest rate hybrid pay-offs.