A contagion process with self-exciting jumps in credit risk applications

Abstract

The modeling of the probability of joint default or total number of defaults among the firms is one of the crucial problems to mitigate the credit risk since the default correlations significantly affect the portfolio loss distribution and hence play a significant role in allocating capital for solvency purposes. In this article, we derive a closed-form expression for the default probability of a single firm and probability of the total number of defaults by time $t$ in a homogeneous portfolio. We use a contagion process to model the arrival of credit events causing the default and develop a framework that allows firms to have resistance against default unlike the standard intensity-based models. We assume the point process driving the credit events is composed of a systematic and an idiosyncratic component, whose intensities are independently specified by a mean-reverting affine jump-diffusion process with self-exciting jumps. The proposed framework is competent of capturing the feedback effect. We further demonstrate how the proposed framework can be used to price synthetic collateralized debt obligation (CDO). Finally, we present the sensitivity analysis to demonstrate the effect of different parameters governing the contagion effect on the spread of tranches and the expected loss of the CDO.

Keywords:

• Joint default risk
• contagion process
• affine jump-diffusion
• Abel equation of second kind
• collateralized debt obligations

Acknowledgments

The first version of this article was completed when the first author was pursuing his Ph.D. at Indian Institute of Technology Delhi, India. Authors are thankful to the editor and the anonymous reviewer for their valuable suggestions and comments which helped improve the paper to a great extent.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Here, we follow the convention $0^0=1$.

2 Unlike the standard reduced-form models where the first jump of the point process makes the firm default, we here assume that a jump in the point process can make the firm default with probability d. One can think of the probability d as the resistance level of the firm against default. Furthermore, the firm survives upon the arrival of bad news with probability 1−d. Therefore, the probability of survival is $\begin{array}{rl}P({\tau }_i>T)& =\sum _{n=0}^{\mathrm{\infty }}P({\tau }_i>T\mid {\tilde{N}}_i(T)=n)P\left({\tilde{N}}_i\right(T)=n)\\ & =\sum _{n=0}^{\mathrm{\infty }}(1-d_i{)}^{n}P({\tilde{N}}_i\left(T\right)=n)=E[(1-d_i{)}^{{\tilde{N}}_i\left(T\right)}].\end{array}$

3 We know that, in theory, the proposed intensity process can become negative with very small probability due to the term $\sigma \mathrm{d}W\left(t\right)$. Nevertheless, the use of such process for default intensity is popular among both academicians and practitioners due to its analytical tractability (For reference, see [15,22,23,27]). Further, one can take $\sigma =0$, if required, in order to avoid the possibility of negative values.

4 Note in Equation (7(7) $P\left(D\right(t)=j\mid N(t)=n)=N{C}_j\left(P_i\right(t,n\left){)}^j\right(1-P_i(t,n){)}^{N-j}$(7) ) that N is the number of firms in the portfolio (a constant) and n is the number of jumps (can take values $0,1,2,\dots$) in the common point process governing the default intensities of the firms.

Additional information

Funding

The first author is thankful to Council of Scientific and Industrial Research (CSIR), India (award number 09/086 (1207)/2014-EMR-I) for the financial grant.