In this study, two interest rate models are analysed in context of counterparty credit risk. The goal of the study is to find a model that performs well on historical simulation for the PFE and EPE. The two models analysed are the Dynamic Nelson-Siegel model and the Displaced Diffusion model.
In the Dynamic Nelson-Siegel model, a Nelson-Siegel curve is fitted against the historical yield curves. The fit gives an historical series of the parameter values of the Nelson-Siegel curve, which are modelled via a stochastic process to obtain future yield curve predictions. In historical backtesting, the classic model using AR(1) processes for the parameters performs inadequate. Analysis on the underlying assumptions of the model show that the mean-reverting behaviour that is modeled is the cause. In addition the data is likely to feature heteroskedastic behaviour, which is not incorporated by the model. An adjusted model in which one parameter is modeled with a random walk with drift performs well on longer maturity rates, however shorter maturity rates are not modeled satisfactory.
The Displaced Diffusion model uses a lognormal diffusion process that is shifted to model Libor rates. As it is a Libor market model, all libor rates are modelled seperately using correlated Brownian motions. The shift parameter allows negative rates to be modeled, and is initially assumed constant. The backtesting results are mixed; some observed libor rates are modeled inadequately and some cannot be rejected to come from the Displaced Diffusion model and thus are modelled correctly. When backtesting the PFE, the results are good at the short term. At the 2-year window, PFE estimates are not always conservative but the number of excesses are of medium severity when compared to the probabilities used in the green-orange-red system dictated by the Basel committee for VaR backtesting.