Efficient Long-Dated Swaption Volatility Approximation in the Forward-LIBOR Model

We provide an efficient swaption volatility approximation for longer maturities and tenors, under the lognormal forward-LIBOR model. In particular, we approximate the swaption volatility with a mean update of the spanning forward rates. Since the joint distribution of the forward rates is not known under a typical pricing measure, we resort to numerical discretisation techniques. More specifically, we approximate the mean forward rates with a multi-dimensional weak order 2.0 Ito-Taylor scheme. The higher order terms allow us to more accurately capture the state dependence in the drift terms and compute conditional expectations with second-order accuracy. We test our approximation with a quasi-Monte Carlo study and find it to be substantially more effective when compared to existing approximations for longer maturities and tenors, particularly for calibration purposes