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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.