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A dynamic contagion process

Part of: Stochastic processes Markov processes Mathematical finance

Published online by Cambridge University Press:  01 July 2016

Angelos Dassios  and
Hongbiao Zhao
Angelos Dassios*
Affiliation:
London School of Economics
Hongbiao Zhao*
Affiliation:
London School of Economics
*
∗ Postal address: Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK.
∗ Postal address: Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK.
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Abstract

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We introduce a new point process, the dynamic contagion process, by generalising the Hawkes process and the Cox process with shot noise intensity. Our process includes both self-excited and externally excited jumps, which could be used to model the dynamic contagion impact from endogenous and exogenous factors of the underlying system. We have systematically analysed the theoretical distributional properties of this new process, based on the piecewise-deterministic Markov process theory developed in Davis (1984), and the extension of the martingale methodology used in Dassios and Jang (2003). The analytic expressions of the Laplace transform of the intensity process and the probability generating function of the point process have been derived. An explicit example of specified jumps with exponential distributions is also given. The object of this study is to produce a general mathematical framework for modelling the dependence structure of arriving events with dynamic contagion, which has the potential to be applicable to a variety of problems in economics, finance, and insurance. We provide an application of this process to credit risk, and a simulation algorithm for further industrial implementation and statistical analysis.

Keywords

Dynamic contagion processCox process with shot noise intensitypiecewise-deterministic Markov processcluster point processself-exciting point processHawkes process

MSC classification

Primary: 60J75: Jump processes
Secondary: 60G55: Point processes 60G44: Martingales with continuous parameter 91G40: Credit risk
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 3 , September 2011 , pp. 814 - 846
Copyright
Copyright © Applied Probability Trust 2011 

References

Azizpour, S. and Giesecke, K. (2008). Self-exciting corporate defaults: contagion vs. frailty. Working paper, Stanford University.Google Scholar
Bordenave, C. and Torrisi, G. L. (2007). Large deviations of Poisson cluster processes. Stoch. Models 23, 593–625.CrossRefGoogle Scholar
Brémaud, P. and Massoulié, L. (1996). Stability of nonlinear Hawkes processes. Ann. Prob. 24, 1563–1588.Google Scholar
Brémaud, P. and Massoulié, L. (2002). Power spectra of general shot noises and Hawkes processes with a random excitation. Adv. Appl. Prob. 34, 205–222.Google Scholar
Chavez-Demoulin, V., Davison, A. C. and McNeil, A. J. (2005). Estimating value-at-risk: a point process approach. Quant. Finance 5, 227–234.CrossRefGoogle Scholar
Costa, O. L. V. (1990). Stationary distributions for piecewise-deterministic Markov processes. J. Appl. Prob. 27, 60–73.Google Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, 2nd edn. Springer, New York.Google Scholar
Dassios, A. and Jang, J.-W. (2003). Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity. Finance Stoch. 7, 73–95.Google Scholar
Dassios, A. and Jang, H.-W. (2005). Kalman-Bucy filtering for linear systems driven by the Cox process with shot noise intensity and its application to the pricing of reinsurance contracts. J. Appl. Prob. 42, 93–107.CrossRefGoogle Scholar
Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. J. R. Statist. Soc. B 46, 353–388.Google Scholar
Davis, M. H. A. (1993). Markov Models and Optimization. Chapman and Hall, London.Google Scholar
Duffie, D. and Gârleanu, N. (2001). Risk and valuation of collateralized debt obligations. Financial Analysts J. 57, 41–59.Google Scholar
Duffie, D. and Singleton, K. J. (1999). Modeling term structures of defaultable bonds. Rev. Financial Studies 12, 687–720.CrossRefGoogle Scholar
Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Prob. 13, 984–1053.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York.Google Scholar
Errais, E., Giesecke, K. and Goldberg, L. (2009). Affine point processes and portfolio credit risk. Working paper, Stanford University.Google Scholar
Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90.CrossRefGoogle Scholar
Hawkes, A. G. and Oakes, D. (1974). A cluster representation of a self-exciting process. J. Appl. Prob. 11, 493–503.CrossRefGoogle Scholar
Jarrow, R. A. and Yu, F. (2001). Counterparty risk and the pricing of defaultable securities. J. Finance 56, 1765–1799.Google Scholar
Jarrow, R. A., Lando, D. and Turnbull, S. M. (1997). A Markov model for the term structure of credit risk spreads. Rev. Financial Stud. 10, 481–523.CrossRefGoogle Scholar
Lando, D. (1998). On Cox processes and credit risky securities. Derivatives Res. 2, 99–120.CrossRefGoogle Scholar
Longstaff, F. A. and Rajan, A. (2008). An empirical analysis of the pricing of collateralized debt obligations. J. Finance 63, 529–563.Google Scholar
Massoulié, L. (1998). Stability results for a general class of interacting point processes dynamics, and applications. Stoch. Process. Appl. 75, 1–30.CrossRefGoogle Scholar
Oakes, D. (1975). The Markovian self-exciting process. J. Appl. Prob. 12, 69–77.Google Scholar
Stabile, G. and Torrisi, G. L. (2010). Risk processes with non-stationary Hawkes claims arrivals. Methodology Comput. Appl. Prob. 12, 415–429.Google Scholar