Modelling Dependence with Copulas - IFOR

9 Modelling Extremal Events in Practice
is used, where τ stands for Kendall’s tau.
However, this choice of copula might prove dangerous. Since the value of the
contract depends on simultaneous exceedences, a copula such as the Gaussian would
result in a price too low if the true dependence structure has the property of upper
tail dependence and the thresholds were high enough. The seller of the contract
would then tend to undervalue the contract. A safer approach from the sellers point
of view would be to use a Gumbel copula. From the expression for the bivariate
Gumbel survival copula it follows that for the bivariate case the probability of
payout is given by
P[N =2] = Ĉ(F 1(k 1 ), F 2 (k 2 ))
(
= F 1 (k 1 )+F 2 (k 2 ) − 1+exp −[(− ln F 1 (k 1 )) θ +(− ln F 2 (k 2 )) θ ] 1/θ) .
For higher dimensions we use the multivariate extension of the Gumbel family
presented in chapter 5.5. For the trivariate case we have
P[N =3] = Ĉ(F 1(k 1 ), F 2 (k 2 )F 3 (k 3 ))
= F 1 (k 1 )+F 2 (k 2 )+F 3 (k 3 ) − 2
+C θ2 (F 1 (k 1 ),F 2 (k 2 )) + C θ1 (F 1 (k 1 ),F 3 (k 3 )) + C θ1 (F 2 (k 2 ),F 3 (k 3 ))
−C θ1,θ 2
(F 1 (k 1 ),F 2 (k 2 ),F 3 (k 3 )),
where
and
C θ1,θ 2
(u 1 ,u 2 ,u 3 )=exp{−
)
C θi (u, v) =exp
(−[(− ln u) θi +(− ln v) θi ] 1/θi
(
[(− ln u 1 ) θ2 +(− ln u 2 ) θ2 ] θ1/θ2 +(− ln u 3 ) θ1 ) 1/θ1}.
To parametrizise the Gumbel copulas the relation
1
θ i =
, i =1,...,n− 1
1 − τ 1,n−i+1
is used.
To compare the effect of the Gumbel dependence structure with the Gaussian, let
X i ∼Lognormal(0, 1) for all i, k i = k for all i and τ(X i ,X j )=0.5 for all i ≠ j.
This gives
( )
( )
n n
P[N = n] =1+(−1) 1 C 1 (F (k)) + ···+(−1) n C n (F (k),...,F(k)),
1
n
where C 1 (u) =u and C m (u 1 ,...,u m )isthem-margin of C n (u 1 ,...,u n )form ∈
{2,...,n− 1}.
In the Gumbel case
C m (F (k),...,F(k)) = exp{−[(− ln F (k)) θ + ···+(− ln F (k)) θ ] 1/θ } = F (k) m1/θ
and in the Gaussian case
C m (F (k),...,F(k)) = Φ m ρ l
(Φ −1 (F (k)),...,Φ −1 (F (k))),
where (to avoid complicated notation) ρ l = sin( π 2
τ) denotes the single parameter of
the equicorrelated Gaussian copula. Φ m ρ l
(Φ −1 (F (k)),...,Φ −1 (F (k))) can be calculated
by numerical integration using the fact that (see Johnson and Kotz p.48 [8])
∫ ∞
Φ m ρ l
(a,...,a)= φ(x) [ ( √ ) a − ρl x ]m
Φ √ dx,
1 − ρl
−∞
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