Intensity basket

rl.SwitchingIntensityBasket(models): several default intensities, each a SwitchingVasicek or SwitchingCoxIngersollRoss, driven by one chain with independent diffusions.

The model

\[d\lambda^k_t = a_k\,(b^k_{y_t} - \lambda^k_t)\,dt + \sigma^k_{y_t}\,dW^k_t, \qquad k = 1, \dots, K,\]

with one regime \(y_t\) shared by all names. The common regime is the only source of dependence.

Reduction

Given the regime path the names are independent, so the joint survival probability is the bond of the sum of the intensities:

\[\mathbb{P}(\text{no default by } T \mid y_0 = i) = \mathbb{E}\big[e^{-\int_0^T\sum_k\lambda^k}\big] = e^{-\sum_kB_k(T)\,\lambda^k_0}\,a_i(T), \qquad g_i = \sum_k g^k_i .\]

The forcings add and the prefactors multiply. The product of the marginal survivals has the forcings solved separately, and the difference between the two is the dependence the regime creates.

Closed form

For two Vasicek names with a common reversion speed the half-differences add, \(\tilde g = \tilde g^1 + \tilde g^2\), and the Green–Kubo term of the joint survival contains the cross term \(\varepsilon\int\tilde g^1\tilde g^2\), which the product of the marginals lacks. To first order

\[\log\frac{Q_{12}(T)}{Q_1(T)\,Q_2(T)} = \varepsilon\int_0^T\tilde g^1\tilde g^2 + O(\varepsilon^2), \qquad \int_0^T\tilde g^1\tilde g^2 = a^2\tilde b^1\tilde b^2I_2 - \tfrac12a\big(\tilde b^1\tilde s^2 + \tilde b^2\tilde s^1\big)I_3 + \tfrac14\tilde s^1\tilde s^2I_4 .\]

It is positive when the same regime raises both intensities and negative when it raises one and lowers the other.

Python

x0, a, sig, lam, T = 0.02, 0.5, 0.003, 5.0, 5.0
chain = rl.RegimeChain.twoState(lam, lam)
name = lambda levels: rl.SwitchingVasicek(chain, x0, a, levels, sig)
E, B, I1, I2, I3, I4 = powers_of_B(a, T)

for label, b1, b2 in (("same regime raises both", [0.06, 0.01], [0.06, 0.01]),
                      ("regime raises one, lowers the other", [0.06, 0.01], [0.01, 0.06])):
    closed = (1 / lam) * a * a * half(b1)[1] * half(b2)[1] * I2          # sigma does not switch here
    n1, n2 = name(b1), name(b2)
    joint = library_bond(rl.SwitchingIntensityBasket([n1, n2]), T)
    library = np.log(joint / (library_bond(n1, T) * library_bond(n2, T)))
    print(f"{label}:")
    print(f"  closed form, first order   {closed:+.6f}")
    print(f"  library, numerical         {library:+.6f}")
    print(f"  default correlation        {rl.SwitchingIntensityBasket([n1, n2]).defaultCorrelation(T):+.4f}")

same regime raises both:
  closed form, first order   +0.000290
  library, numerical         +0.000280
  default correlation        +0.0017
regime raises one, lowers the other:
  closed form, first order   -0.000290
  library, numerical         -0.000279
  default correlation        -0.0018

rl.FirstToDefaultSwap on the basket is a single-name credit default swap on the summed intensity.