214 Frequently Asked Questions In Quantitative Finance
the statistics, the mean, standard deviation, distribution,
etc., of the present value of resulting cashflows.
Here are some examples of bivariate copula functions.
They are readily extended to the multivariate case.
Bivariate Normal:
C(u,v)=N 2
(
N− 11 (u),N 1 −^1 (v),ρ
)
, − 1 ≤ρ≤1,
whereN 2 is the bivariate Normal cumulative distribution
function, andN 1 −^1 is the inverse of the univariate Normal
cumulative distribution function.
Frank:
C(u,v)=
1
α
ln
(
1 +
(eαu−1)(eαv−1)
eα− 1
)
, −∞<α<∞.
Fr ́echet–Hoeffding upper bound:
C(u,v)=min(u,v).
Gumbel–Hougaard:
C(u,v)=exp
(
−
(
(−lnu)θ+(−lnv)θ
) 1 /θ)
,1≤θ<∞.
This copula is good for representing extreme value dis-
tributions.
Product:
C(u,v)=uv
One of the simple properties to examine with each of
these copulas, and which may help you decide which is
best for your purposes, is thetail index. Examine
λ(u)=
C(u,u)
u
.
