Table 1 The Archimedean copulas
From: Stressed portfolio optimization with semiparametric method
Types | Copula function | Copula multivariate function |
|---|---|---|
Clayton | \(\phi \left(t\right)=\frac{1}{\theta }\left({t}^{-\theta }-1\right)\) | \(C \left({u}_{1} {,u}_{2} ,\dots ,{ u}_{m}\right)={max({{{(u}_{1}}^{-\theta }+\dots +{{u}_{m}}^{-\theta }+m-1)}^{-\frac{1}{\theta }}, 0)}\) |
Frank | \(\phi \left(t\right)=-ln\left(\frac{{exp}\left(-\theta t\right)-1}{{exp}\left(-t\right)-1}\right)\) | \(C \left({u}_{1} {,u}_{2} ,\dots ,{ u}_{m}\right)=-ln\left(1+\frac{({exp}\left(-\theta {u}_{1}\right)-1)\dots ({exp}\left(-\theta {u}_{m}\right)-1)}{{exp}\left(-\theta \right)-1}\right)\) |
Gumbel | \(\phi \left(t\right)={(-lnt)}^{\theta }\) | \(C \left({u}_{1} {,u}_{2} ,\dots ,{ u}_{m}\right)={exp}\left(-{\left({\left(-ln{u}_{1}\right)}^{\theta }+\dots +{(-ln{u}_{m})}^{\theta }\right)}^{\frac{1}{\theta }}\right)\) |