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

Lech Grzelak and Cornelis Oosterlee

Applied Mathematical Finance, 2012, vol. 19, issue 1, 1-35

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.

Date: 2012
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Working Paper: On cross-currency models with stochastic volatility and correlated interest rates (2010)
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DOI: 10.1080/1350486X.2011.570492

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