Structural Credit Risk using Time-changed Brownian Motions: A tale of two models
by Tom R. Hurd of McMaster University, and
September 13, 2011
Abstract: We consider a structural credit risk framework introduced in  where the log-leverage ratio of the firm is a LÚvy process in the form of a time-changed Brownian motion (TCBM) where the time-change process has identically distributed independent increments. In models of this type, "vanilla" credit derivative pricing formulas are in closed form in terms of an explicit one-dimensional Fourier transform. Our primary aim is to investigate whether two very simple specifications of the time change process, namely the variance gamma (VG) model and the exponential jump model (EXP), can lead to good fits to CDS data for a representative firm with an interesting credit history, Ford Motor Co. Statistical inference in this class of intrinsically non-Gaussian hidden state models presents some new challenges which are a second main focus of the paper. We consider several variations of nonlinear filtering and statistical inference for TCBM models applied to a 4.5 year time series of credit default swap (CDS) prices on Ford, with the goal of finding a fast, accurate, robust scheme. The main conclusion is that the two TCBM models significantly outperform the classic Black-Cox model. Secondly, we show that a new inference method called the "linearized measurement scheme" is much faster than a standard numerical integration scheme (up to 100 times faster), and yields equivalent performance. The statistical methodology proposed in this paper can be effectively implemented for many other variations of TCBMs and applied to a wide range of firms, and opens the door to far-ranging explorations of a new class of structural credit risk models.
Keywords: Credit risk, structural model, first passage problem, LÚvy process, fast Fourier transform, credit default spread, maximum likelihood estimation.
Previously titled: Statistical Inference for Time-changed Brownian Motion Credit Risk Models