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

We provide efficient swaption volatility approximations for longer maturities and tenors under the lognormal forward-LIBOR model (LFM). 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 discretization techniques. More specifically, we approximate the mean forward rates with a multi-dimensional weak order 2.0 Itō–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 approximations for longer maturities and tenors using a quasi-Monte Carlo (QMC) study and find them to be substantially more effective when compared to the existing approximations, particularly for calibration purposes.