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Quantitative Finance · المسرد

ما معنى Student-t copula؟

Definition 15.10 Rates, Credit, XVA and Risk · الفصل 15 — Portfolio Credit

In the Student-t copula with ν\nu degrees of freedom the latent variables are Xi=W(ρ Z+1−ρ εi)X_i = \sqrt W(\sqrt\rho\,Z+\sqrt{1-\rho}\,\varepsilon_i) with W=ν/χν2W = \nu/\chi^2_\nu common to all names; a name defaults if Xi<tν−1(pi)X_i < t_\nu^{-1}(p_i). The common scale WW makes the names default together in bad draws: the copula has tail dependence, which the Gaussian lacks.

Probability that the five-year pool loss exceeds x (log scale), same marginals and asset correlation 25%, simulated with 400 000 draws of the factors. The Student-t copula puts several times more probability on large losses. Data: the chapter’s tutorial.
Figure 15.4. Probability that the five-year pool loss exceeds xx (log scale), same marginals and asset correlation 25%, simulated with 400 000 draws of the factors. The Student-t copula puts several times more probability on large losses. Data: the chapter’s tutorial.

أمثلة

Example 15.11 (Tails)

At an asset correlation of 25%, the probability that the pool loses more than 12% in five years is 2.24% under the Gaussian copula and 5.23% under the Student-t copula with four degrees of freedom; for a loss above 24%, 0.13% against 1.13%, 8.6 times more (Figure 15.4). Senior tranches priced with the Gaussian copula at one correlation look cheap for the same reason the skew exists: the market prices more tail than the Gaussian gives.

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