Parameter estimation of default portfolios using the Merton model and phase transition

Abstract

We discuss the parameter estimation of the probability of default (PD), the correlation between the obligors, and a phase transition. In our previous work, we studied the problem using a beta-binomial distribution. A non-equilibrium phase transition with an order parameter occurs when the temporal correlation decays by a power law. In this study, we adopt the Merton model, which uses an asset correlation as the default correlation, and find that a phase transition occurs when the temporal correlation decays by the power law. When the power index is less than one, the PD estimator converges slowly. Thus, it is difficult to estimate the PD with limited historical data. Conversely, when the power index is greater than one, the convergence speed is inversely proportional to the number of samples. We investigate the empirical default data history of several rating agencies. The estimated power index is in the slow-convergence range when we use long history data. This suggests that the PD can have a long memory and that it is difficult to estimate parameters due to slow convergence.