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A reduced-form model for correlated defaults with regime-switching shot noise intensities. (English) Zbl 1343.60117

Summary: In this paper, we consider a two-dimensional reduced form contagion model with regime-switching interacting default intensities. The model assumes that the intensities of the default times are driven by a macro-economy described by a homogenous Markov chain and that the default of one firm may trigger a positive jump, associated with the state of the Markov chain, in the default intensity of the other firm. The intensities before the default of the other firm are modeled by a two-dimensional regime-switching shot noise process with common shocks. By using the idea of “change of measure” and some closed-form formulas for the joint conditional Laplace transforms of the regime-switching shot noise processes and the integrated regime-switching shot noise processes, we derive the two-dimensional conditional and unconditional joint distributions of the default times. Based on these results, we can express the single-name credit default swap (CDS) spread, the first and the second-to-default CDS spreads on two underlyings in terms of fundamental matrix solutions of linear, matrix-valued, ordinary differential equations.

MSC:

60J28 Applications of continuous-time Markov processes on discrete state spaces
60J27 Continuous-time Markov processes on discrete state spaces
60H30 Applications of stochastic analysis (to PDEs, etc.)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
91G40 Credit risk
91G80 Financial applications of other theories
60G46 Martingales and classical analysis
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