Superstatistical Fluctuations in Time Series of Leverage Returns

Abstract

Fat tails of q-Gaussian distributions of daily log-leverage-returns of 520 North American industrial firms reported by Katz and Tian (2013) imply a significantly higher credit risk at short time-horizons and/or large initial distances to the default barrier than forecasted by traditional structural models. Superstatistics can explain the origin of this behavior and allows for a straightforward generalization of conventional models of default. Here, we verify manifestation of these distributions with Kolmogorov-Smirnov test and analyze to what extent emergence of q-Gaussians can be described by superstatistics. To this end, we compare mean values of the Tsallis entropic parameter q obtained by two independent methods: i) direct fitting of q-Gaussians to distributions of log-leverage-returns and ii) derived from shape parameters of Gamma distributions fitted to histograms of inversed realized variances of these returns. For a vast majority of companies, we observe the striking consistency of the average values of q, converging to 1.47, obtained by both methods. This finding supports the applicability of superstatistical hypothesis, which assumes that q-Gaussians result from locally normal fluctuations with Gamma-distributed precision. Notably, for a small group of firms (~8%) having distributions with divergent second moment, the mean values of q derived by both methods are very different. It is likely that in this group of issuers, fluctuations of daily log-leverage-returns are so violent that the superstatistical model assumptions are not applicable. These findings are important for practical implementations of credit risk valuations and pricing of credit securities.