Modeling and understanding the dynamics of credit risk are
critical for credit derivative markets from both pricing and
investment properties. We consider the approach of using
the distance-to-default to measure the credit quality of
a firm, and model its random behavior in time by a Levy process.
We use the model to investigate two closely related issues: the
default term structure implied from the market, and credit
rating transitions estimated from historical data. The first
is based on a risk-neutral probability measure and the
second is based on the real world probability measure, and our
model serves as a bridge to connect these two aspects.
The Fokker-Planck equation for the survival probability
density function provides a powerful tool to study the
properties of the Markov chain, and to describe
the evolution of quantities such as credit spread and default
probability. The model calibration is achieved through solving
the partial integro-differential equation (PIDE) in regions
separated by barriers, with rating transitions and defaults
represented by barrier crossings. Using finite difference
approximations, we are able to match exactly the default
probabilities for all ratings, and through
numerical optimization, generate transition matrices quite close
to those estimated from historical data. Our results show that
the processes in different regions are characterized by drifts
and volatilities that can be interpreted and connected with
realistic economic considerations.