Fluctuating volatility and credit risk: Q-gaussian model of default

Structural models of default (such as Merton’s model) typically assume that the volatility of the issuer’s assets is time-invariant. This approach fails to recognize intermittent bursts of the volatility of the stock price and, hence, the company’s market value of assets. In this paper, we derive the expedient analytical formula for the term structure of the probability of default (PD), which captures uncertainty related to the slowly fluctuating volatility of the firm’s assets. The minimal version of the model contains only two parameters: the modified “distance to default” and the Tsallis entropic parameter q. For , related to the normal statistics, our model replicates results of traditional structural models. For q > 1, corresponding to complex systems with non-additive entropy, it forecasts much higher one-year PD for investment grade issuers than valuations based on the assumption of a constant volatility. Qualitatively, it reflects the simple intuition: the volatility values along the realized path may be much higher than the spot volatility of the log-asset returns at the valuation time. We report an excellent fitting of the derived 2-parametric formula to the 20-years’ term-structure of the cumulative default rates in all categories of credit rating. We find the strong correlation between large values of q > 5/3, corresponding to the divergent second moment of the probability density function (PDF) of log-asset returns and actual defaults among 645 North American industrial firms, 2006 -2012. Our findings are essential for credit risk analysis as well as (under the risk-neutral measure) valuations of credit spreads.