Modelling dependence in finance using copulas - Thierry Roncalli's ...

2.1 Pearson correlation and dependence
Pearson correlation ρ = linear dependence measure.
For two given asset prices processes S 1 (t) and S 2 (t) which are GBM,
the range of the correlation is
ρ − ≤ ρ (S 1 (t) , S 2 (t)) ≤ ρ +
with
ρ ± =
exp (±σ 1 σ 2 (t − t 0 )) − 1
√
exp ( σ1 2 (t − t ) √
0) − 1 · exp ( σ2 2 (t − t 0))
− 1
ρ (S 1 (t) , S 2 (t)) = ρ − (resp. ρ + ) ⇔ C 〈S 1 (t) , S 2 (t)〉 = C − (resp.
C + ) ⇔ S 2 (t) = f (S 1 (t)) with f a decreasing (resp. increasing)
function
Perfect dependence ≠ |ρ| = 1
Modelling dependence in finance using copulas
Copulas and multivariate financial models 2-2