The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 330 Wednesday, February 4, 2004 12:59 PM

330 The Mathematics of Financial Modeling and Investment Management

EXHIBIT 12.1 Fluctuations of the Variance-Covariance Matrix

It can then be demonstrated that the density of eigenvalues of the ran-

dom matrix tends to the following distribution:

ρλ

Q (λmax – λ)(λmin – λ)

() = ------------- -------------------------------------------------------

2 πσ^2 λ

MN, → , ∞ Q= M N ⁄ ≥ 1

1 1

λmax min , = σ^21 + -----± 2 -----

Q Q

where σ^2 is the average eigenvalue of the matrix. Exhibit 12.2 illustrates

the theoretical function and a sample computed on 500 simulated inde-

pendent random walks. The shape of the distribution of the eigenvalues

is the signature of randomness.