The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 378 Wednesday, February 4, 2004 1:00 PM

378 The Mathematics of Financial Modeling and Investment Management

behavior, or at least of exact Pareto tail, naturally leads to fitting a linear

model in a logarithmic scale. There is an ample literature on this topic

with a number of useful discussions, though empirical studies based on

Monte Carlo simulations are still limited.^15

The estimation methods reviewed above are based on the behavior

of maxima and upper order statistics; another methodology uses the

points of exceedances of high thresholds. Estimation methodologies

based on the points of exceedances require an appropriate model for the

point process of exceedances that was defined in general terms previ-

ously in this chapter.

ELIMINATING THE ASSUMPTION OF IID SEQUENCES

In the previous sections we reviewed a number of mathematical tools

that are used to describe fat-tailed processes under the key assumption

of IID sequences. In this section we discuss the implications of eliminat-

ing this assumption. However, in finance theory the assumption of sta-

tionary sequences of independent variables is only a first approximation;

it has been challenged in several instances. Consider individual price

time series. The autocorrelation function of returns decays exponen-

tially and goes to near zero at very short-time horizons while the auto-

correlation function of volatility decays only hyperbolically and remains

different from zero for long periods. In addition, if we consider portfo-

lios made of many securities, price processes exhibit patterns of cross

correlations at different time-lags and, possibly, cointegrating relation-

ships. These findings offer additional reasons to consider the assump-

tion of serial independence as only a first approximation.

If we now consider the question of stationarity, empirical findings

are more delicate. The non-stationarity that can be removed by differ-

encing is easy to handle and does not present a problem. The critical

issue is whether financial time series can be modeled with a single Data

Generation Process (DGP) that remains the same for the entire period

under consideration or if the model must be modified. Consider, for

instance, the question of structural breaks. At a basic level, structural

breaks entail nonstationarity as the model parameters change with time

and thus the finite-dimension distributions change with time. However,

at a higher level one might try to model structural changes, for instance

(^15) Francis X. Diebold, Til Schuermann, and John D. Stroughair, “Pitfalls and Oppor-
tunities in the Use of Extreme Value Theory in Risk Management,” The Journal of
Risk Finance (Winter 2000), pp. 30–36.