202 E. Luciano and P. Semeraro
approximate the actual copula with the Gaussian one, at least over long horizons,
life becomes much easier. Closed or semi-closed formulas exist for pricing and risk
measurement in the presence of the Gaussian copula (see for instance [1]). Standard
linear correlation can be used for the model joint calibration.
In a nutshell, one can adopt anaccurate,non-Gaussian model and safely ignore
non-linear (and non-analytical) dependence, in favour of the linear dependence rep-
resented by the familiar Gaussian copula, provided the horizon is long enough. In
the stock market case analysed here, one year was quite sufficient for non-Gaussian
dependence to be ignored.
Appendix
Simulated measure of dependence
The simulated version of Spearman’s rho at timet,ρ ̃S(t), can be obtained from a
sample ofNrealisations of the processes at timet(xi 1 (t),x 2 i(t)),i= 1 ,...,N:
ρ ̃S(t)= 1 − 6∑N
i= 1 (Ri−Si)
2
N(N^2 − 1 ), (13)
whereRi=Rank(x 1 i(t))andSi=Rank(x 2 i(t)). Similarly for Kendall’s tau,τ ̃C(t):
τ ̃C(t)=c−d
(
N
2), (14)
wherecis the number of concordance pairs of the sample anddthe number of discor-
dant ones. A pair(x 1 i(t),x 2 i(t))is said to be discordant [concordant] ifxi 1 (t)x 2 i(t)≤ 0
[x 1 i(t)xi 2 (t)≥0]. TheNrealisations of the process are obtained as follows:
- SimulateNrealisations from the independent lawsL(Y 1 ),L(Y 2 ),L(Z); let them
be respectivelyy 1 n,yn 2 ,znforn= 1 ,...,N. - ObtainNrealisations(gn 1 ,g 2 n)ofGthrough the relationsG 1 =Y 1 +Zand
G 2 =Y 2 +Z. - GenerateNindependent random draws fromeach of the independent random
variablesM 1 andM 2 with lawsN( 0 ,G 1 )andN( 0 ,G 2 ). The draws forM 1 in
turn are obtained fromNnormal distributions with zero mean and varianceg 1 n,
namely
M 1 (n)=N( 0 ,g 1 n).
The draws forM 2 are from normal distributions with zero mean and varianceg 2 n,
namely
M 2 (n)=N( 0 ,g 2 n).