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Pricing Cross-Currency Derivatives in a Libor Market Model

The Libor market model, developed in a series of papers (Brace et al., 1997; Miltersen et al., 1997; Jamshidian, 1997), is one of the most used interest rate model by both practitioners and academics. The model has some appealing features such as the use of an observable underlying, the forward Libor rate, and the assumption that the underlying follows a geometric Brownian motion. Actually, the latter has considerable implication with respect to the implementation and calibration of the model since it meets market practice which relies on the (Black, 1976) model to price the most important over-the-counter interest rate derivatives, i.e. caps and swaptions. The Libor market model has been extended to a multicurrency framework by Schlögl (2002) and Mikkelsen (2002).

The authors have relied on the dynamics of the forward foreign exchange rate to derive the dynamics of the foreign forward Libor rates under a forward domestic measure. This method suffers from the fact that the obtained dynamics does not involve explicitly the covariance matrix between domestic and foreign instantaneous forward Libor volatilities. In addition, their framework does not take into account some well documented facts on the behavior of both interest rates as well as foreign exchange rates. Specifically, Das (2002) and Johannes (2004), for instance, show that interest rates are subject to jumps. Similarly, Jorion (1988) reached the same conclusion in the case of exchange rates.

Therefore, including jumps in the dynamics of the forward Libor and foreign exchange rates may be, therefore, a more appropriate description of the reality. Adding jumps to the dynamics of the underlying can also be argued on the ground of option pricing since it provides a more accurate setting to recover the smile typically observed in the interest rate option markets. This lead, for example, in the single currency case, Glasserman and Kou (2003) to develop a Libor market model with a compound Poisson process and in more general setting Eberlein and Özkan (2005) to rely on the Lévy process to describe the dynamics of the forward Libor rates.

This paper extends the multicurrency Libor market model to a jump-diffusion framework. The interest rate processes as well as the foreign exchange rate process are assumed to follow, in addition to the diffusion component, multivariate Poisson processes. A more general setting can be found, in the multicurrency case, in Eberlein and Koval (2006) where all the rates involved follow Levy processes. However, like Schlögl (2002), Eberlein and Koval (2006) rely on modeling the forward foreign exchange rate to depict the different changes of probability measures. The approach followed in this article considers the dynamics of the spot, rather than the forward, foreign exchange rate to establish the appropriate probability measure change.

The procedure can be described as follows. The foreign Libor rate dynamics is first written, using the appropriate change of probability measure, under the domestic risk-neutral measure and then switched to the domestic forward measure. This approach provides the advantage that it involves in an explicit way the covariance matrix between foreign exchange rate volatilities and foreign instantaneous forward Libor rate volatilities, on the one hand, and the covariance matrix between the instantaneous volatilities of foreign and domestic forward Libor rates, on the other hand. The rest of the paper is organized as follows. In the next section, we extend to a jump-diffusion setting the multicurrency Libor model. Section (3) prices, under specific assumptions, various cross-currency derivatives. Finally, section (4) concludes.

Pricing Cross-Currency Derivatives in a Libor Market Model