Two factors and two-name credit
Default dependence created by a common regime, and where it enters the expansion.
Two default intensities follow independent Ornstein–Uhlenbeck processes, $dx_j = \kappa_j(\theta_{j,y} - x_j)\,dt + \sigma_{j,y}\,dW_j$, whose mean levels and volatilities switch with a common two-state regime. The same reduction prices any positive combination $c_1x_1 + c_2x_2$, which covers two-factor Gaussian short rates as well. Survival of each name and of both names is
$$S_{c}(t) = \mathbb{E}\Big[e^{-\int_0^t (c_1 x_1 + c_2 x_2)}\Big] = e^{-c_1B_1x_1 - c_2B_2x_2}\,a_{c}(t),$$with $c = (1,0)$, $(0,1)$ and $(1,1)$. Here $\kappa = (1, 1.5)$, $\theta_1 = (0.06, 0.01)$, $\theta_2 = (0.05, 0.015)$, $\sigma = (0.02, 0.005)$ for both names, $x_0 = (0.03, 0.03)$ and $t = 3$.
The table gives the correlation of the two default indicators, numerically and after each order. The averaged model has independent names, so order 0 gives zero. The dependence appears at order $1/\lambda$. For a symmetric chain the first-order term is
$$\log\frac{S_{(1,1)}}{S_{(1,0)}\,S_{(0,1)}} = \frac{1}{\lambda}\int_0^t \tilde g_1(r)\,\tilde g_2(r)\,dr + O(\lambda^{-2}),$$where $\tilde g_j = -\kappa_j\tilde\theta_j B_j + \tfrac12\tilde s_j B_j^2$ is each name's regime half-difference. It is positive when the regimes move both names the same way.
| switching rate | numerical | order 0 | order 1 | order 2 | order 3 | order 4 |
|---|---|---|---|---|---|---|
| 2 | 0.003086 | 0 | 0.003360 | 0.003130 | 0.003066 | 0.003092 |
| 4 | 0.001718 | 0 | 0.001785 | 0.001725 | 0.001717 | 0.001718 |
| 8 | 0.000903 | 0 | 0.000919 | 0.000904 | 0.000903 | 0.000903 |
The reduction was checked against Monte Carlo of the switching model in verify_models.py.