Duru-Kleinert Asymptotic Expansions for Long-Term Foreign Exchange and Swaptions Implied Volatility Smile

In this paper, we develop asymptotic formulas for long-dated Foreign Exchange (FX) and swaptions implied volatilities. We extend the method exposed in Decamps and De Schepper (2009b) to a generic model with time-dependent parameters. Imposing a condition on the skew, we derive averaging formulas for the parameters. The method is applied to the pricing of FX options when the domestic and foreign interest rate curves are driven by Gaussian short-term rate models and to the pricing of swaptions in the Libor market model.