Long-dated interest rate modelling

Abstract: In this thesis, we consider problems in long-dated interest rate modelling, with particular focus on the forward-LIBOR market model (LFM) and its displaced diffusion extension, known as the displaced lognormal forward-LIBOR model (DLFM). We provide swaption volatility approximations for longer maturities and tenors that strike a balance between efficiency and accuracy when compared to the Rebonato, Hull-White and Kawai methods. Our approximations depend on a mean-update of the underlying jointly-distributed forward-LIBOR rates for which we use a multi-dimensional weak order 2.0 It¯o-Taylor scheme. The higher order scheme more accurately captures the state dependence present in the LFM dynamics. Furthermore, we develop an algorithm under the DLFM to approximate moments of the jointly-distributed forward-LIBOR rates based on a lognormal assumption. This algorithm is an iterative procedure that consists of a multi-dimensional full weak order 2.0 It¯o-Taylor expansion and a second-order Delta method that reincorporates state information that is normally frozen at a previous state. This combination more effectively captures the state dependence present in the dynamics of the DLFM. Lastly, we introduce a diagnostic tool in the form of a quantile approximation under the DLFM to analyse the behaviour of the jointly distributed long-dated forward-LIBOR rates. This is useful since the class of forward-LIBOR market models can produce unrealistically high forward rates under certain volatility parametrisations, particularly for maturities beyond liquid market calibration instruments. We further explore a methodology to keep these rates within realistic limits. All our approximations are extensively benchmarked using quasi-Monte Carlo (QMC) simulations and shown to improve upon previous approaches.Ph.D. (Mathematical Statistics