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

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.

Key Words:

• foreign exchange (FX)
• stochastic volatility
• Heston model
• stochastic interest rates
• interest rate smile
• forward characteristic function
• hybrids
• efficient calibration

Acknowledgement

The authors thank Sacha van Weeren and Natalia Borovykh from Rabobank International for their fruitful discussions and helpful comments.

Notes

1 According to Duffie et al. (2000) the n-dimensional system of SDEs, is of the affline form iffor i, j = 1,… ,n, with r(X(t)) being an interest rate component.

2 Since the moments of the square root process under the T-forward measure are difficult to determine for , we have set or, in other words, the expectation is calculated under measure .

3 The model parameters do not satisfy the Feller condition,

4 As it is insightful to relate the covariance matrix with the necessary model approximations, the correlation structure is introduced here by means of instantaneous correlation of the scalar diffusions.

5 In Grzelak and Oosterlee (2010), the proof for this statement is given when a single yield curve is considered.

6 As in the standard Black–Scholes analysis for the log-transform gives