Article2011

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

Lech Aleksander Grzelak · Cornelis W. Oosterlee

48 citations0/0 files storedApplied Mathematical Finance

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Stochastic processes and financial applications Financial Risk and Volatility Modeling Risk Management in Financial Firms

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.

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References

1993A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options 1985A Theory of the Term Structure of Interest Rates 1992Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation 2000Transform Analysis and Asset Pricing for Affine Jump-diffusions 1990Pricing Interest-Rate-Derivative Securities 2001Interest Rate Models Theory and Practice 2004Stochastic Calculus for Finance II: Continuous-Time Models 1997Martingale Methods in Financial Modelling

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