### Abstract

Joint default modeling for a set of firms is crucial in the context of pricing credit derivatives. We consider here a model for defaults among multiple firms based on Vasicek or Ornstein- Uhlenbeck models for the hazard rates of the underlying companies or "names." We analyze the impact of volatility time scales on the default distribution for the set of firms. We also consider the associated impact on a particular credit derivative contract, the so-called collateralized debt obligation (CDO). We demonstrate how correlated fluctuations in the parameters of the firm hazard rates affect the loss distribution and prices associated with the CDOs. The effect of stochastic parameter fluctuations is to change the shape of the loss distribution and cannot be captured by using averaged parameters in the original model. Our analysis assumes a separation of time scales and leads to a singular-regular perturbation problem [J.-P. Fouque, G. Papanicolaou, and R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, UK, 2000; J.-P. Fouque et al., Multiscale Model. Simul., 2 (2003), pp. 22-42]. This framework allows us to compute perturbation approximations that can be used for effective pricing of CDOs.

Original language | English (US) |
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Pages (from-to) | 1956-1978 |

Number of pages | 23 |

Journal | Multiscale Modeling and Simulation |

Volume | 7 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2009 |

### All Science Journal Classification (ASJC) codes

- Chemistry(all)
- Modeling and Simulation
- Ecological Modeling
- Physics and Astronomy(all)
- Computer Science Applications

### Keywords

- Asymptotic analysis
- Defaults
- Stochastic volatility

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## Cite this

*Multiscale Modeling and Simulation*,

*7*(4), 1956-1978. https://doi.org/10.1137/080726719