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Credit Dynamics in a First Passage Time Model with Jumps

by Natalie Packham of the Frankfurt School of Finance & Management,
Lutz Schl÷gl of Nomura International Plc, and
Wolfgang M. Schmidt of the Frankfurt School of Finance & Management

September 2009

Abstract: The payoff of many credit derivatives depends on the level of credit spreads. In particular, the payoff of credit derivatives with a leverage component is sensitive to jumps in the underlying credit spreads. In the framework of first passage time models we extend the model introduced in [Overbeck and Schmidt, 2005] to address these issues. In the extended a model, a credit quality process is driven by an It˘ integral with respect to a Brownian motion with stochastic volatility. Using a representation of the credit quality process as a time-changed Brownian motion, we derive formulas for conditional default probabilities and credit spreads. An example for a volatility process is the square root of a LÚvy-driven Ornstein-Uhlenbeck process. We show that jumps in the volatility translate into jumps in credit spreads. We examine the dynamics of the OS-model and the extended model and provide examples.

JEL Classification: G12, G13, G24, C69.

AMS Classification: 60G35, 60H05, 91B28.

Keywords: gap risk, credit spreads, credit dynamics, first passage time models, LÚvy processes, general Ornstein-Uhlenbeck processes.

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