Interest rate products form a large segment of over-the-counter derivatives. When the interest rate became negative, for the first time, in July 2009, interest rate models needed to adjust. Where first a log-normal model, as the Brace Gatarek Musiela (BGM) model, might have seemed logical for interest-rate products, as they were bounded by zero, now a normally distributed model, as the Hull-White model, could be considered more practical. To our knowledge, no comparison of the Hull-White model and the Brace Gatarek Musiela model has been made on the Expected Positive Exposure (EPE) (and thus Credit Valuation Adjustment (CVA)) of a swap portfolio. Therefore, this thesis compares the Hull-White model with the BGM model on the EPE of a swap portfolio. First, we show how both models can be simulated with Monte Carlo simulation and calibrated to caplets, after which we validate the used simulation. Finally, both models are compared on the convergence, computation time and EPE. It was found that the Hull-White model had a faster convergence and computation time than the BGM model for our implementation. Moreover, it was shown that the Hull-White model and BGM model have significantly different swap EPEs, except for far in-the-money (ITM) swaps and single payment swaps. Therefore, the models used for the EPE of a swap portfolio have a model risk.

The aim of this thesis is to model fully collateralized exposures in the presence of the Margin Period of Risk, i.e., the time between the last successful collateral call to the time where the amount of the loss crystallizes. We start with introducing a closed-form expression to model fully collateralized exposures for fixed versus floating interest rate swaps. Then the Brownian bridge method is introduced and applied, which allows for improvements computational wise. Finally, fully collateralized exposures are modeled in the presence of the Margin Period of Risk, with the Least Squares Monte Carlo method in the case of one European call option under Black-Scholes assumptions.