Default risk modeling beyond the first-passage approximation: Extended Black-Cox model

Abstract

We develop a generalization of the Black-Cox structural model of default risk. The extended model captures uncertainty related to firm's ability to avoid default even if company's liabilities momentarily exceeding its assets. Diffusion in a linear potential with the radiation boundary condition is used to mimic a company's default process. The exact solution of the corresponding Fokker-Planck equation allows for derivation of analytical expressions for the cumulative probability of default and the relevant hazard rate. Obtained closed formulas fit well the historical data on global corporate defaults and demonstrate the split behavior of credit spreads for bonds of companies in different categories of speculative-grade ratings with varying time to maturity. Introduction of the finite rate of default at the boundary improves valuation of credit risk for short time horizons, which is the key advantage of the proposed model. We also consider the influence of uncertainty in the initial distance to the default barrier on the outcome of the model and demonstrate that this additional source of incomplete information may be responsible for nonzero credit spreads for bonds with very short time to maturity.

A Random Matrix Approach to Credit Risk

“…The recovery rates are typically modeled independently, for example, by a reduced–form model, see [15] , [16] , or are even assumed to be constant, see, eg, [6] . Recent approaches aim at improving first passage models by including the chance of full recovery, even if a company's market value is below the threshold, see [17] , and estimating correlations between default probabilities of industry sectors, see [18] .…”

Default risk modeling with position-dependent killing

“…where is some normalized function, , which reflects market's indeterminacy in the initial value of x (see, e.g., Refs. [30,33,39] This expression was first derived by probabilistic methods in the work of Duffie and…”

“…where is some normalized function, , which reflects market's indeterminacy in the initial value of x (see, e.g., Refs. [30,33,39]). To simplify notations hereafter we are not showing the limits of integration over x from 0 to infinity.…”

“…This conclusion does not depend on the strength of the killing measure or choice of the boundary condition. Forecasts of contact models can be improved, however, if one takes into account the indeterminacy in the distance to the default barrier [30,33,39]. Following the CreditGrades model [30], suppose that the initial value of x is normally distributed with the mean and the variance : .…”

“…We establish the limits of validity of models that exploit the assumption of the time-independent hazard rate. Comparison between obtained formulae and historical data on average global corporate defaults [37] clearly demonstrates i) the applicability of the model-independent perturbation method for companies with high credit quality; ii) reveals excellent fitting capabilities of the extended Black-Cox (EBC) model [ 39 ] in all categories of credit ratings; iii) allows for differentiation between "microscopic" models of bankruptcy for companies in the speculative grade categories.…”