green) on a double logarithmic paper. For ± < 2 the tails form

straight lines with slope ’±.

STFstab05.xpl

1.3.1 Tail exponent estimation

The simplest and most straightforward method of estimating the tail index is

to plot the right tail of the (empirical) cumulative distribution function (i.e.

1 ’ F (x)) on a double logarithmic paper. The slope of the linear regression for

large values of x yields the estimate of the tail index ±, through the relation

± = ’slope.

This method is very sensitive to the sample size and the choice of the number of

observations used in the regression. Moreover, the slope around ’3.7 may in-

dicate a non-±-stable power-law decay in the tails or the contrary “ an ±-stable

20 1 Stable distributions in ¬nance

Tails of stable laws for 10^6 samples Tails of stable laws for 10^4 samples

-2

-4

-5

log(1-F(x))

log(1-F(x))

-6

-10

-8

-5 0 -4 -2 0 2

log(x) log(x)

Figure 1.6: A double logarithmic plot of the right tail of an empirical symmetric

1.9-stable distribution function for sample size N = 106 (left panel)

and N = 104 (right panel). Thick red lines represent the linear

regression ¬t. Even the far tail estimate ± = 1.9309 is above the

ˆ

true value of ±. For the smaller sample, the obtained tail index

estimate (ˆ = 3.7320) is close to the initial power-law like decay of

±

the larger sample (ˆ = 3.7881).

±

STFstab06.xpl

distribution with ± ≈ 1.9. To illustrate this run quantlet STFstab06. First sim-

ulate (using equation (1.3) and quantlet rndsstab) samples of size N = 104

and 106 of standard symmetric (β = µ = 0, σ = 1) ±-stable distributed vari-

ables with ± = 1.9. Next, plot the right tails of the empirical distribution

functions on a double logarithmic paper, see Figure 1.6.

The true tail behavior (1.6) is observed only for very large (also for very small,

i.e. the negative tail) observations, after a crossover from a temporary power-

like decay. Moreover, the obtained estimates still have a slight positive bias,

which suggests that perhaps even larger samples than 106 observations should

be used. In Figure 1.6 we used only the upper 0.15% of the records to estimate

1.3 Estimation of parameters 21

10^4 samples 10^6 samples 10^6 samples

2.1

2

2.5

2.5

1.9

alpha

alpha

alpha

1.8

2

2

1.7

0 500 1000 0 50000 100000 0 1000 2000

Order statistics Order statistics Order statistics

Figure 1.7: Plots of the Hill statistics ±n,k vs. the maximum order statistic k

ˆ

for 1.8-stable samples of size N = 104 (left panel) and N = 106

(middle and right panels). Red horizontal lines represent the true

value of ±. For better exposition, the right panel is a magni¬cation

of the middle panel for small k. A close estimate is obtained only

for k = 500, ..., 1300 (i.e. for k < 0.13% of sample size).

STFstab07.xpl

the true tail exponent. In general, the choice of the observations used in the

regression is subjective and can yield large estimation errors, a fact which is

often neglected in the literature.

A well known method for estimating the tail index that does not assume a

parametric form for the entire distribution function, but focuses only on the

tail behavior was proposed by Hill (1975). The Hill estimator is used to estimate

the tail index ±, when the upper (or lower) tail of the distribution is of the

form: 1 ’ F (x) = Cx’± . Like the log-log regression method, the Hill estimator

tends to overestimate the tail exponent of the stable distribution if ± is close

to two and the sample size is not very large, see Figure 1.7. For a review of the

extreme value theory and the Hill estimator see Chapter 13 in H¨rdle, Klinke,

a

and M¨ller (2000) or Embrechts, Kl¨ppelberg and Mikosch (1997).

u u

22 1 Stable distributions in ¬nance

These examples clearly illustrate that the true tail behavior of ±-stable laws is

visible only for extremely large data sets. In practice, this means that in order

to estimate ± we must use high-frequency asset returns and restrict ourselves

to the most ”outlying” observations. Otherwise, inference of the tail index may

be strongly misleading and rejection of the ±-stable regime unfounded.