In this thesis, we study a structural default model of an interconnected banking system with mutual liabilities. We develop novel numerical and analytical methods for the computation of important characteristics of the model and assess their impact on counterparty credit risk. First, we study the case when the banks' asset values follow a pure diffusion process and develop analytical and semi-analytical methods for default probabilities, which are the first of their kind in the three-dimensional case. Next, we consider a jump-diffusion model and develop new splitting finite difference methods for the resulting partial integro-differential equations, which we test numerically in the case of two banks. Due to the curse of dimensionality, these methods are not extendable practically to a large number of banks. Therefore, we consider a mean-field approximation of the model in this case, which leads to a McKean-Vlasov equation where the drift term depends on the absorption rate on the boundary. We develop and analyse a novel simulation method and an alternative approach based on reduction of the corresponding nonlinear, nonlocal PDE to a coupled system of Volterra integral equations.