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Credit Gap Risk in a First Passage Time Model with Jumps

by Natalie Packham of the Frankfurt School of Finance & Management,
Lutz Schloegl of Nomura International Plc, and
Wolfgang Schmidt of the Frankfurt School of Finance & Management

November 2009

Abstract: The payoff of many credit derivatives depends on the level of credit spreads. In particular, credit derivatives with a leverage component are subject to gap risk, a risk associated with the occurrence of jumps in the underlying credit default swaps. In the framework of first passage time models, we consider a model that addresses these issues. The principal idea is to model a credit quality process as an It˘ integral with respect to a Brownian motion with a stochastic volatility. Using a representation of the credit quality process as a time-changed Brownian motion, one can 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. The model can be implemented efficiently using a technique called Panjer recursion. Calibration to a wide range of dynamics is supported. We illustrate the effectiveness of the model by valuing a leveraged credit-linked note.

JEL Classification: G12, G13, G24, C69.

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

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