Levy Subordinator Model: A Two Parameter Model of Default Dependency

Subordinators are Levy processes with non-decreasing sample paths. They are natural processes to model default dependency. They help ensure that the loss process is non-decreasing leading to a promising class of dynamic models. The simplest subordinator is the Levy subordinator, a maximally skewed stable process with index of stability 1/2. Interestingly, this simplest subordinator turns out to be the appropriate choice as the basic process in modeling default dependency. It involves just two parameters to assess dependency risk, a measure of correlation and that of the likelihood of a catastrophe. Its attractive feature is that it admits a closed form expression for its distribution function. This helps in automatic calibration to individual hazard rate curves and efficient pricing with Fast Fourier Transform techniques. It is structured similar to the one-factor Gaussian copula model and can easily be implemented within the framework of the existing computational infrastructure. As it turns out, the Gaussian copula model can itself be recast into this framework highlighting its limitations. The model can also be investigated numerically with a Monte Carlo simulation algorithm. As is now well appreciated, random recovery is helpful in better pricing of the senior tranches and the model admits a tractable framework of random recovery. The model is investigated numerically and the implied base correlations are presented over a wide range of its parameters. The investigation also demonstrates its ability to generate reasonable hedge ratios.