A Short Note on the Exact Stochastic Simulation Scheme of the Hull-White Model and Its Implementation

Abstract

In this short note we derive an exact simulation scheme for the joint distribution of (r(t),N(t)), where r denotes the short rate following a Hull-White model and $N$ denotes the numeraire.To sample the correct joint distribution of (r(t),N(t)) our scheme requires a two-factor Brownian driver. In other words: the time-discrete Hull-White model is a two factor model.We investigate the performance of this scheme compared to the classical schemes for correlation dependent products like the LIBOR in arrears and Bermudan swaptions. We show that the exact sampling of the joint distribution is important in some applications, e.g., the valuation of Bermudan options using American Monte-Carlo method. Traditional approximation of the numeraire may generate biases if the model uses larger simulation time-steps.Large time-step simulation of short rate models and the American Monte-Carlo method is of interest in the calculation of portfolio effects (xVAs), where usually many risk factors have to be simulated and computational resources need to be saved.