On the formulation of credit barrier model using radial basis functions

Abstract

Albanese et al in 2003 and Avellaneda and Zhu in 2001 develop the framework of credit barrier model. They provide special solutions to the model in case of simple stochastic structure. The technical aspect of the model remains wide open for general stochastic structure that is crucial when the model is required to calibrate with aggregate amount of empirical data. This paper provides a technical solution to this problem with the use of radial basis functions (RBF). This paper discusses the numerical implementation of the credit barrier model using the RBF method. It also demonstrates that the RBF method is numerically tractable in this problem and allows in the model richer stochastic structure capable of capturing relevant market information.

Keywords:

• credit migration
• Kolmogorov equation
• radial basis function

Acknowledgements

This paper is an extension of our previous work with P. Baup. We thank him for initiating the research.

Notes

1 Note the use of the integral ∫₀^∞ds G(s, t) e^Λs=(1−νΛ)^−t/ν.

2 Denote φ′(||x−x_j||)=−2κ(x−x_j) φ (||x−x_j||) and φ′′(||x−x_j||)=−2κ[1−2κ(x−x_j)²] φ (||x−x_j||) as the first and second derivative of φ (||x−x_j||)=exp(−κ||x−x_j||²), respectively, with respect to x.

3 Suppose dx=(c₀+c₁x+c₂x²) dt+σ√x dz. We can rescale x=σ²y to give dy=[(c₀/σ²)+c₁y+(c₂σ²)y²] dt+√y dz.

4 In the discrete scheme, we assume β′(t) is not too large such that the contribution to ∂/∂t coming from the change in b(t) in the summation can presumably be neglected.