# An Asymptotic FX Option Formula in the Cross Currency Libor Market Model

## Atsushi Kawai & Peter Jäckel introduce analytic approximation formulae for FX options in the Libor market model (LMM). The method to derive the formulae is an asymptotic expansion technique introduced in Kawai

In this article, we introduce analytic approximation formulae for FX options in the Libor market model (LMM). The method to derive the formulae is an asymptotic expansion technique introduced in Kawai [Kaw03].

We first apply the method to the lognormal LMM and lognormal FX model. Then, the method is applied to the displaced diffusion LMM and the displaced diffusion FX model. Some numerical examples show that the derived formulae are sufficiently accurate for practical applications.

The Libor market model developed by Brace, Gatarek, and Musiela [BGM97], Jamshidian [Jam97], Miltersen, Sandmann, and Sondermann [MSS97] is one of the most popular interest rate models among both academics and practitioners. It is interesting to use the model not only for pure interest rate products but also for long-dated equity/FX products or hybrid products.

In this article, we focus on modeling cross currency FX markets using the Libor market model, that is, the cross currency hybrid LMM. The dynamics of the model are a straightforward extension of the standard LMM formulation, and considerations regarding the specific choice of FX state variable (spot, forward, rolling spot) are discussed in the literature [Sch02]. Whilst a variety of accurate and efficient approximations for vanilla swaption pricing, and thus model calibration are available [JR00, HW00, Kaw02, Kaw03], very little has been published with respect to vanilla FX option approximations in cross-currency FX/interest rate models. The most notable exceptions are the very recent works by Osajima [Osa06] for a Gaussian forward rate stochastic volatility setup, and [AM06a, AM06b] for a cross-currency Libor market model without explicit skew on FX and interest rates. Since European FX options are the most important hedge instruments for the cross-currency exposure of FX/interest rate contracts, and since the pricing of vanilla options using Monte Carlo simulations for calibration purposes can be rather cumbersome, analytical approximations for FX plain vanilla option prices are highly desirable.

Using an asymptotic expansion method introduced in Kawai [Kaw03], we derive an analytic approximation formula for European FX options in the lognormal1 cross currency hybrid LMM. Then, we extend the method to the displaced diffusion cross currency hybrid LMM.