Validate Correlation of an ESG: Treasury Yields across

Within an internal model the Economic Scenario Generator (ESG) is an important component. In order to get a regulatory approval of an internal model it is required that the implemented models (must be) passed a rigorous validation process, see Ceiops [2009]. In this paper we focus on the particular problem to judge the contribution of correlations between interest rate risks across countries in the ESG. To that end we apply two strategies: an analytical and a statistical one. The analytical approach yields necessary conditions in terms of upper and lower bounds for correlations within the chosen model. A system of stochastic differential equations is used to describe several economies simultaneously. In this framework we derive a lower and upper bound of the correlation of the treasury yields between two economies by solving the associated ordinary differential inequalities. In order to deepen our understanding about the correlation structure we consider three modeling types of correlations of historical datasets. We first derive the realized correlations as outlined by Andersen et al. [2003] for the historical treasury yields of two economies. Furthermore we include Engle’s parsimonious multivariate GARCH models – known as Dynamical Conditional Correlation (DCC) model, see Engle [2009] – and we derive conditional correlations out of our ESG. We then exploit a nice relationship outlined by Andersen et al. [2003], which relates the realized correlation and conditional correlations in oder to compare the three model by their ability to capture the stylized facts of the underlying processes. In this respect the long memory of the correlation processes is of particular importance. We give a series of statistical analysis that highlight the adequacy of the model.