The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 189 Wednesday, February 4, 2004 3:00 PM

Concepts of Probability 189

correlations coefficients. In many instances, in particular in risk man-

agement, it is important to arrive at a quantitative measure of the

strength of dependencies.

The correlation coefficient provides such a measure. In many instances,

however, the correlation coefficient might be misleading. In particular, there

are cases of nonlinear dependencies that result in a zero correlation coeffi-

cient. From the point of view of risk management this situation is particu-

larly dangerous as it leads to substantially underestimated risk.

Different measures of dependence have been proposed, in particular

copula functions. We will give only a brief introduction to copula func-

tions.^11 Copula functions are based on the Theorem of Sklar. Sklar dem-

onstrated^12 that any joint probability distribution can be written as a

functional link, i.e., a copula function, between its marginal distribu-

tions. Let’s suppose that F(x 1 ,x 2 ,...,xn) is a joint multivariate distribu-

tion function with marginal distribution functions F 1 (x 1 ), F 2 (x 2 ), ...,

Fn(xn). Then there is a copula function C such that the following rela-

tionship holds:

Fx( 1 , , , x 2 ...xn ) = CF[ 1 ()x 1 ,F 2 ()x 2 , ..., Fn()xn]

The joint probability distribution contains all the information

related to the co-movement of the variables. The copula function allows

to capture this information in a synthetic way as a link between mar-

ginal distributions. We will see an application of the concept of copula

functions in Chapter 22 on credit risk modeling.

SEQUENCES OF RANDOM VARIABLES

Consider a probability space (Ω,ℑ,P). A sequence of random variables is an

infinite family of random variables Xi on (Ω,ℑ,P) indexed by integer num-

bers: i = 0,1,2,...,n... If the sequence extends to infinity in both directions, it

is indexed by positive and negative integers: i = ...,–n,..., 0,1,2,...,n....

A sequence of random variables can converge to a limit random

variable. Several different notions of the limit of a sequence of random

variables can be defined. The simplest definition of convergence is that

(^11) The interested reader might consult the following reference: P. Embrechts, F. Lind-
skog, and A. McNeil, “Modelling Dependence with Copulas and Applications to
Risk Management,” Chapter 8 in S.T. Rachev (ed.), Handbook of Heavy Tailed Dis-
tributions in Finance (Amsterdam: North Holland, 2003).
(^12) A. Sklar, “Random Variables, Joint Distribution Functions and Copulas,” Kyber-
netika 9 (1973), pp. 449–460.