We propose a reduced-form model for credit risk in a multivariate setting where the intensities of defaults are driven by affine jump diffusions processes. An important example of application for the model is a portfolio of bonds issued by many obligors, such as a Collateralized Debt Obligation (CDO). By modeling the intensities of defaults of each obligor, one can produce default scenarios that are necessary to simulate the cashflows of the collateral portfolio and hence to estimate the risk associated to any tranche of a CDO.
We assume that defaults may occur as a consequence of three independent types of credit events: the idiosyncratic one, depending only on the particular situation of the obligor, the sectoral one, affecting all the obligors belonging to a given group, the general one, affecting all the obligors. The intensities of the independent credit events are affine jump diffusion processes and the default intensity process of each obligor is a linear combination of them.
The idea of using affine processes to model the intensity of defaults has been frequently used in the financial literature. In particular, our approach is closely related to Duffie and Gârleanu [7] who focus on the analysis of the effect of correlation on the values of CDOs with different cash flow structures. Variations of the Duffie and Gârleanu model were proposed by Mortensen [18], Eckner [12] who consider a unique common factor, but allow for sensibility coefficients depending on the obligors. All these works impose some restrictions on the parameters in order to simplify computations to get affine default intensities. Chapovsky et al. [2] model all default intensities with a unique affine common factor and deterministic idiosyncratic components. In this paper we consider a generalization of such approaches, defining a a framework where single name default intensities have different sensibilities to the three types of credit events.
The model we consider belongs to the general class of doubly stochastic processes of default. The basic idea of such models is that, after conditioning for the intensities, defaults are independent, therefore, default correlation is solely determined by the correlation of the intensities. It has often been debated if the double stochastic hypothesis is able to produce sufficient default correlation. The main critique to the effectiveness of this approach is that conditional independence makes the connection between defaults too indirect. An empirical study by Das et al. [4] tests a double stochastic model on data of U.S. corporations from 1979 to 2004, showing that it does not explain all default clustering observed.
In particular, it has been argued (see for instance Schönbucher [19]) that the introduction of joint jumps in the default intensities is necessary to produce significant levels of correlation. Duffie and Gârleanu [7] studied the impact of correlation on valuation of CDO’s tranches. Chapovsky et al. [2] showed that a simple, one factor model can be calibrated efficiently on market prices of synthetic CDO’s and reproduce the observed correlation skew. Positive results in the same direction are obtained by Mortensen [18] and Eckner [12], who calibrate to CDS, credit indices and credit tranche spreads, and prove the capability of their models to generate correlation consistent with market-implied levels. In our general setting we obtain an even more flexible correlation structure. Moreover, a nice feature of our class of models is that default correlations can be explicitly computed. We can therefore compare the impact of the diffusion and of the jump components on default correlation. We will give evidence that an appropriate choice of parameters can produce any level of default correlation.
The standard approach to simulate default scenarios is to discretize the SDEs of all the intensity processes involved. Such a straightforward methodology is affected by a discretization error, that may be controlled by decreasing the length of the time interval, with a greater computational cost. We will follow an alternative idea producing an exact simulation of the default scenarios. The simulation is “exact” in the sense that it produces the times of default and the identity of the defaulters from the exact probability distributions, without resorting to approximation or discretization. Moreover, it improves the computational efficiency because it does not require the simulation of the intensity processes of all the obligors. Such a methodology was originally proposed by Duffie and Singleton [10] for general intensity processes. Here we develop a toolbox for the specific case of affine processes and compare the performances of the method with the standard approach.
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