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01.11.2015 | Research Article

Credit risk and contagion via self-exciting default intensity

verfasst von: Robert J. Elliott, Jia Shen

Erschienen in: Annals of Finance | Ausgabe 3-4/2015

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Abstract

Recent empirical evidences indicate that default rates are influenced not only by the observable or latent risk factors, but also depend on the history of past defaults. Motivated by this empirical finding, we consider in this paper a reduced-form, intensity-based credit risk model, which allows for both frailty and default contagion, using a so-called “self-exciting” intensity, in the sense that the default intensity varies not only with the risk factors, but also depends on the previous default history of all the firms. With “self-exciting” default intensity, we are able to obtain closed-form expressions for the pricing of credit derivative securities in our model. The estimation of parameters using the EM algorithm is considered as well.

Vorheriger Artikel Robustness of equilibrium in the Kyle model of informed speculation

Nächster Artikel Optimization of relative arbitrage

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In Azizpour et al. (2014), an observable factor is also included in the model to represent the macro-economic risk factor. Our model and filtering-based approach can be easily extended to allow for observable risk factors, as observable factors will not change the filtering equations essentially. For simplicity, we only model the latent frailty and contagion effects through the hidden, unobservable Markov state process, as our main focus is to filter out the current state with the “self-exciting” default intensity, which is a novel contribution. We also obtain the robust filter with the “self-exciting” default intensity.

In Azizpour et al. (2014), the frailty factor is modeled as a mean-reverting CIR square-root diffusion process.

In Giesecke and Schwenkler (2014), the authors consider a filtered likelihood estimate of parameters for marked point processes. Using numerical analysis, they compare estimates obtained with their approach and with the EM algorithm, and find that for the majority of the parameters, their estimates are more accurate and less susceptible to the initial values.

For \(\lambda ^i_t\) of the special form \(\lambda ^i(t,X_{t-},N_{(\cdot )})=\phi \left( \int _{-\infty }^th(t-s,X_{t-})dN_s\right) \), appropriate conditions for integrability are given by researchers. A key example of interest is when \(\lambda ^i(t,X_{t-},N_{(\cdot )})=\langle \alpha ,X_{t-}\rangle +\langle \beta ,X_{t-}\rangle \int _0^t e^{-\langle \gamma ,X_{t-}\rangle (t-s)}dY^i_s\), where \(Y^i_t=\sum _{j\ne i}N^i_t=\sum _{j\ne i}{\mathbf {1}}_{\{t\ge \tau _i\}}\), and \(\alpha ,\beta ,\gamma \) are vectors of parameters to be determined. This is a variant of the Hawkes’ process.

For example, a variant of the Hawkes’ process is \(\lambda ^i(t,X_{t-},N_{(\cdot )})=\langle \alpha ,X_{t-}\rangle +\langle \beta ,X_{t-}\rangle \int _0^t e^{-\langle \gamma ,X_{t-}\rangle (t-s)}dY^i_s\), where \(Y^i_t=\sum _{j\ne i}N^i_t=\sum _{j\ne i}{\mathbf {1}}_{\{t\ge \tau _i\}}\), and \(\alpha ,\beta ,\gamma \) are vectors of parameters to be determined. This is a parametric specification where \(\lambda ^i(t,X_{t-},N_{(\cdot )})\) is self-exciting, and not depending on \(N^i_{[0,t-]}\).

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Titel: Credit risk and contagion via self-exciting default intensity
verfasst von: Robert J. Elliott
Jia Shen
Publikationsdatum: 01.11.2015
Verlag: Springer Berlin Heidelberg
Erschienen in: Annals of Finance / Ausgabe 3-4/2015
Print ISSN: 1614-2446
Elektronische ISSN: 1614-2454
DOI: https://doi.org/10.1007/s10436-015-0259-z

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