Chapter 3: Default intensities implied by CDO spreads: inversion formula and
3.4 Local intensity function implied by credit portfolio loss models
3.4.4 Gaussian and Student-t copula models
Because of its tractability, the one-factor Gaussian copula model has been the financial industry benchmark despite some well-known drawbacks. The Student-t copula model, which embeds the Gaussian copula model as a limit when the degree of freedom goes to infinity, is widely used as well. More precisely, given a family of marginal default time distributions (Fi, i= 1, ..., n), the joint
distribution of the default times τi is modeled by first defining latent factors Xi =S ρZ0+ p 1−ρ2Z i , withS =
1 for Gaussian copula r
ν
V for Student-t copula
whereZ0, Zi ∼ N(0,1), V ∼χ2ν are independent variables, and then defining the default times by
τi=Fi−1(FXi(Xi)),
whereFXi(.) denotes the distribution ofXi. We refer readers to [70] for details.
To study the local intensity function implied by these two bottom-up models, we first calibrate them using the base-correlation method [75]. Then, we study the local intensity function corre- sponding to different base correlations. Notice that these two models are static, which means that there is no default intensity defined in this framework: we are in effect representing the expected tranche notionals in these models in terms of an equivalent local intensity function, which then enables to compare these static models with dynamic models presented above.
Figure 3.8 shows the local intensity functions implied by the two copula models with base correlations corresponding to tranche [6%,9%]. Observe that the main difference between the two local intensity functions is that for the Gaussian copula model, there is a sharp increase for short times when the number of defaults is larger than 100. Other than that, both local intensity functions appear to have a smooth dome shape. This relatively restricted form of local intensity function can explain why a single correlation cannot fit the full set of CDO tranche spreads and why we usually observe a base-correlation skew. Also, since the Student-t copula embeds the Gaussian copula as a limit, it is not surprising that the local intensity functions implied by the two models are similar in general shape. This also suggests that the additional degree of freedom in the Student-t copula is still not able to generate a flexible enough local intensity function to match the full set of CDO market data.
3.4.5 Affine jump-diffusion model
Many “bottom-up” reduced form models [41, 45, 83, 92] based on diffusive or jump-diffusion dy- namics for default intensities have been proposed for pricing portfolio credit derivatives. Most of these models are built in the “doubly stochastic” framework by specifying the default intensities for each name in the portfolio. A prominent example, which lends itself to implementation, is the model proposed by Duffie and Gˆarleanu [41] where the default intensities follow correlated affine jump-diffusion processes. We consider the extension of this model considered in Mortensen [83] here. The default intensity for nameiis represented as
λi,t=Xti+aiXt0,
wherea1, ..., an are parameters, and (Xti), i= 0, ..., n, are independent affine jump-diffusions with
dXti = κi(bi−Xti)dt+σi q
XtidWti+dJti,
where (Wti), i= 0, ..., n, are independent Brownian motions and (Jti), i= 0, ..., n, are independent compound Poisson processes with exponentially distributed jumps. This general specification is theoretically appealing, but the calibration to 125 individual CDS spreads and 6 tranche spreads of a CDO involves the solution to a nonlinear optimization problem in dimension 881: 125 factor loadings, 126 initial risk factor values and 630 parameters for the risk factor dynamics. Eckner [45] proposes a parsimonious version of this model, which we will adopt here.
Interestingly, Figure 3.8 shows that the local intensity function implied by the affine jump- diffusion model [45] is similar to the one implied by the one-factor Student-t copula model. This is a surprising result because the two modeling frameworks are fundamentally different: one is dynamic, while the other one is static.
3.5
Conclusion
We have proposed a simple and efficient calibration method for recovering the default intensity of a portfolio from CDO spreads. Our method is based on two ingredients: a nonparametric method based on quadratic programming for recovering expected tranche notionals from CDO spreads, and an inversion formula for computing the local intensity function from the expected tranche notionals. This method is shown to be much more stable, with respect to changes in inputs, than the commonly used nonlinear least squares method based on parametric models (see, e.g. [60]). Contrarily to the base-correlation method, our method yields an arbitrage-free model.
Comparing our calibration algorithm to a parametric calibration method [60] and to a nonpara- metric entropy minimization method [28] using iTraxx Europe index CDO spreads, we observe that these different calibration methods lead to quite different values of default intensity while main- taining a good match to the observations: this illustrates clearly the ill-posedness of the calibration problem. We also find that model-dependent quantities such as forward starting tranche spreads and jump-to-default ratios are quite sensitive to the calibration method used, even within the same model class.
On the other hand, comparing the local intensity functions implied by different credit portfolio models reveals that apparently different models, such as static Student-t copula models and reduced- form affine jump-diffusion models, lead to similar marginal loss distributions and tranche spreads. Thus, market prices alone are insufficient to discriminate between these model classes.
These results emphasize the importance of model uncertainty when addressing the pricing and hedging of portfolio credit derivatives and call for more research in this direction.