The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 374 Wednesday, February 4, 2004 1:00 PM

374 The Mathematics of Financial Modeling and Investment Management

It turns out that these three problems cannot be easily separated. In

fact, there is no reliable constructive theory for solving all these problems

automatically. In particular, the choice of the statistical model (i.e., the

distribution that best describes data) is a classical problem of formulating

and validating a scientific hypothesis in a probabilistic context. However,

there are many tools and tests to help the modeler in this endeavor.

The first fundamental tool is the graphical representation of data, in

particular the quantile plot or QQ-plot defined as the following set:

 ←nk– + 1  

Xkn,F ----------------------: k=^12 ,,...,n



,

 n+ 1  

The quantile transformation and the Glivenko-Cantelli theorem

allow concluding that this plot must be approximately linear. Should F

be a Pareto distribution, the linearity of the QQ-plot is another state-

ment of Zipf’s law. The quantile plot allows a quick verification of a sta-

tistical hypotheses by checking the approximate linearity of the plot. It

also allows the modeler to form a preliminary opinion on where the tail

begins and whether the model fails at the far end of the tail.

Though invaluable as an exploratory tool, graphics rely on human

judgment and intuition. Rigorous tests are needed. A starting point is

parameter estimation for the Generalized Extreme Value (GEV) Distri-

bution that we write as

  x– μ–^1 ⁄ ξ x– μ

Hξμ ψ; , ()x = exp– 1 + ξ------------ , 1 + ξ------------> 0

  ψ   ψ

with the convention that the case ξ= 0 corresponds to the Gumbel dis-

tribution:

 – x------------– μ

 ψ 

H 0 ;μψ, ()x = exp–e , x∈R

 

 

We saw above that these distributions are the limit distributions, if

they exist, of the normalized maxima of IID sequences. Suppose that the

data to be estimated are independent draws from some EGV. This is a

rather strong assumption that we will progressively relax. This assump-

tion might be justified in domains where long series of data are available