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The credit migration process contains important information about the dynamics of a firm's credit quality, therefore, it has a significant impact on its relevant credit derivatives. We present a jump diffusion approach to model the credit rating transitions which leads to a partial integro-differential equation (PIDE) formulation, with defaults and rating changes characterized by barrier crossings. Efficient and reliable numerical solutions are developed for the variable coefficient equation that result in good agreement with historical and market data, across all credit ratings. A simple adjustment in the credit index drift converts the model to be used in the risk-neutral setting, which makes it a valuable tool in credit derivative pricing.

The main theme of this volume is credit risk and credit derivatives. Recent developments in financial markets show that appropriate modeling and quantification of credit risk is fundamental in the context of modern complex structured financial products. Moreover, there is a need for further developments in our understanding of this important area. In particular modeling defaults and their correlation has been a real challenge in recent years, and still is. This problem is even more relevant after the so-called subprime crisis that hit in the summer of 2007. This makes the volume very timely and hopefully useful for researchers in the area of credit risk and credit derivatives.