The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 328 Wednesday, February 4, 2004 12:59 PM

328 The Mathematics of Financial Modeling and Investment Management

pt = pt– 1 ++μμμμ εεεεt

The key difference with respect to univariate random walks is that

one needs to consider cross correlations as the random disturbances εεεεt

will be characterized by a covariance matrix ΩΩΩΩwhose entries σi,jare the

covariances between asset iand asset j.Covariance and correlation are

one way of expressing the notion of functional dependence between ran-

dom variables. Consider two random variables X,Y.

The covariance between the two variables is defined as

σXY, =Cov(XY, ) =EX EX{[ – ()][YEY– ()]}=EXY( )–EX()EY()

The correlation coefficient is the covariance normalized with the prod-

uct of the respective standard deviations:

Cov(XY, )

ρXY, =Corr(XY, )=-------------------------------------------

Var ()XVar () Y

σXY,

=--------------

σXσY

The correlation coefficient expresses a measure of linear dependence.

Suppose that the variables X,Yhave finite mean and variance and that

are linearly dependent so that

Y= aX+ b+ ε

The above relationship is called a linear regression (see Chapter 6). It

can be demonstrated that the correlation coefficient between Xand Yis

related to the parameter ain the following way:

σX

a=ρXY, -------

σY

The correlation coefficient can assume values between –1 and +1

inclusive. It can be demonstrated that the variables X,Yare propor-

tional without any noise term if and only if the correlation coefficient is

+/–1. If the regression has a noise term, then the correlation coefficient

assumes a value intermediate between –1 and +1. If variables are inde-

pendent, then the correlation coefficient is zero. The converse is not

true. In fact, it is possible that two variables exhibit nonlinear depen-