Top

Annals of Finance

Published in:

01-11-2015 | Research Article

Credit risk and contagion via self-exciting default intensity

Authors: Robert J. Elliott, Jia Shen

Published in: Annals of Finance | Issue 3-4/2015

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

previous article Robustness of equilibrium in the Kyle model of informed speculation

next article Optimization of relative arbitrage

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

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 \(N^i_{[0,t-]}\).

Azizpour, S., Giesecke, K., Schwenkler, G.: Exploring the Sources of Default Clustering. Working paper, Stanford University (2014)

Bielecki, T.R., Rutkowski, M.: Credit Risk: Modeling, Valuation and Hedging. Springer, Berlin (2002)

Black, F., Cox, J.C.: Valuing corporate securities: some effects of bond indenture provisions. J. Finance 31(2), 351–367 (1976)

Collin-Dufresne, P., Goldstein, R., Helwege, J.: Is Credit Event Risk Priced? Modeling Contagion Via the Updating of Beliefs. Preprint, Carnegie Mellon University (2003)

Cvitanić, J., Ma, J., Zhang, J.: Laws of large numbers for self-inciting correlated defaults. Stoch Process Appl 122, 2781–2810 (2011)CrossRef

Das, S.R., Duffie, D., Kapadia, N., Saita, L.: Common failings: how corporate defaults are correlated. J. Finance 62(1), 93–117 (2007)

Duffie, D., Lando, D.: Term structure of credit spreads with incomplete accounting information. Econometrica 69, 633–664 (2001)CrossRef

Duffie, D., Singleton, K.: An econometric model of the term structure of interest-rate swap yields. J Financ 52(4), 1287–1321 (1997)CrossRef

Duffie, D., Singleton, K.: Modeling term structures of defaultable bonds. Rev Financ Stud 12, 687–720 (1999)CrossRef

Duffie, D., Singleton, K.: Credit Risk: Pricing, Measurement and Management. Princeton University Press, Princeton (2003)

Duffie, D., Eckner, A., Horel, G., Saita, L.: Frailty correlated default. J Financ 64, 2089–2123 (2009)CrossRef

Elliott, R.J., Aggoun, L., Moore, J.B.: Hidden Markov models: Estimation and control. Applications of Mathematics (New York), 29, Springer, New York, xii+361 pp (1995)

Frey, R., Runggaldier, W.: Pricing credit derivatives under incomplete information: a nonlinear filtering approach. Financ Stochast 14, 495–526 (2010)CrossRef

Frey, R., Schmidt, T.: Pricing and hedging of credit derivatives via the innovations approach to nonlinear filtering. Financ Stochast 16, 105–133 (2012)CrossRef

Giesecke, K., Longstaff, F., Schaefer, S., Strebulaev, I.: Corporate bond default risk: a 150-year perspective. J Financ Econ 102, 233–250 (2011)CrossRef

Giesecke, K., Longstaff, F., Schaefer, S., Strebulaev, I.: Macroeconomic effects of corporate default crises: a long-term perspective. J Financ Econ 111, 297–310 (2014)CrossRef

Giesecke, K., Schwenkler, G.: Filtered likelihood for point processes. Working paper. Stanford University (2014)

Giesecke, K., Spiliopoulos, K., Sowers, R.: Default clustering in large portfolios: typical events. Ann Appl Probab 23(1), 348–385 (2013)CrossRef

Hajek, B., Wong, E.: Stochastic Processes in Engineering Systems. Springer, Berlin (1985)

Jarrow, R.A., Turnbull, S.M.: Pricing derivatives on financial securities subject to credit risk. J Financ 50, 53–86 (1995)CrossRef

Jarrow, R.A., Lando, D., Turnbull, S.M.: A Markov model for the term structure of credit risk spreads. Rev Financ Stud 10(2), 481–523 (1997)CrossRef

Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, Berlin (1988)CrossRef

Merton, R.C.: On the pricing of corporate debt the risk structure of interest rates. J. Finance 29(2), 449–470 (1974)

Schönbucher, P.: The term structure of defaultable bond prices. Rev Deriv Res 2, 161–192 (1998)

Schönbucher, P.: Information Driven Default Contagion. Working Paper, ETH, Zurich (2003a)

Schönbucher, P.: Credit Derivatives Pricing Models: Models, Pricing and Implementation. Wiley, Chichester (2003b)

Title: Credit risk and contagion via self-exciting default intensity
Authors: Robert J. Elliott
Jia Shen
Publication date: 01-11-2015
Publisher: Springer Berlin Heidelberg
Published in: Annals of Finance / Issue 3-4/2015
Print ISSN: 1614-2446
Electronic ISSN: 1614-2454
DOI: https://doi.org/10.1007/s10436-015-0259-z

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 3-4/2015

Evidence on exercise pricing in CEO option grants in two countries

Optimization of relative arbitrage

Financial innovation and risk: the role of information

Optimal investment in multidimensional Markov-modulated affine models

Robustness of equilibrium in the Kyle model of informed speculation

Bounds for path-dependent options