A set-valued Markov chain approach to credit default

Abstract

We propose a novel credit default model that takes into account the impact of macroeconomic factors and intergroup contagion on the defaults of obligors. We use a set-valued Markov chain to model the default process, which includes all defaulted obligors in the group. We obtain analytic characterizations for the default process and derive pricing formulas in explicit forms for synthetic collateralized debt obligations (CDOs). Furthermore, we use market data to calibrate the model and conduct numerical studies on the tranche spreads of CDOs. We find evidence to support that systematic default risk coupled with default contagion could have the leading component of the total default risk.

Keywords:

• Credit risk
• Collateral default obligation (CDO)
• Markov chain
• Jump diffusion
• Tranche spread

Acknowledgments

We are extremely grateful to the managing editor and two anonymous referees for their constructive comments. We also thank Tahir Choulli, Zhengyu Cui, Xianhua Peng and Chao Shi for insightful discussions and comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The filtration $\mathbb{F}$ is rich enough to accommodate all the stochastic processes introduced in the paper, i.e. those processes are well defined and adapted to the filtration $\mathbb{F}$. We may restrict to a smaller filtration (a subset of $\mathbb{F}$) in applications (e.g. in Definitions 2.3 and 2.4). Please see Giesecke et al. (2011) for a similar probabilistic setup.

2 In practice, $R_i$ is often set to 40% for all i, see ISDA standard CDS converter specification at http://www.cdsmodel.com/cdsmodel/assets/cds-model/docs.

3 Throughout this paper, for a stochastic process, if the subscript is reserved for special meaning, then we write time variable t in parenthesis, see, e.g. ${\lambda }_i\left(t\right)$ and ${\lambda }_L\left(t\right)$; otherwise, we may write time variable t in subscript or parenthesis exchangeably, see, e.g. $L_t$ and $L\left(t\right)$.

4 This remark is modified from the report of an anonymous referee.

5 Giesecke et al. (2011) consider similar feasible region in their studies. Changing the feasible region Θ will only slightly affect calibration.

6 The model parameters in table 4 are calibrated under the criterion (20(20) $\underset{\mathit{x}\in \mathrm{\Theta }}{min}\sum _{i=1}^6{\left(\frac{{Tranchei}^{model}-\overline{{Tranchei}^{market}}}{\overline{{Tranchei}^{market}}}\right)}^2,$(20) ), and hence, there is no guarantee which tranche will have the best fitting performance. The fittings for the equity and index tranches are not as accurate as for other tranches either in Seo and Wachter (2018).

7 Independent Poisson processes do not jump simultaneously.

Additional information

Funding

The research of Jun Deng is supported by the National Natural Science Foundation of China [grant number 11501105]. The research of Bin Zou is supported by a Start-up grant from the University of Connecticut.