where tk is an appropriate set of real numbers, m = log(2σ ± ), and k denotes

an error term. Koutrouvelis (1980) proposed to use tk = πk , k = 1, 2, ..., K;

25

with K ranging between 9 and 134 for di¬erent estimates of ± and sample sizes.

Once ± and σ have been obtained and ± and σ have been ¬xed at these values,

ˆ ˆ

estimates of β and µ can be obtained using (1.16). Next, the regressions are

ˆˆˆ

repeated with ±, σ , β and µ as the initial parameters. The iterations continue

ˆ

until a prespeci¬ed convergence criterion is satis¬ed.

Kogon and Williams (1998) eliminated this iteration procedure and simpli¬ed

the regression method. For initial estimation they applied McCulloch™s (1986)

method, worked with the continuous representation (1.2) of the characteristic

function instead of the classical one (1.1) and used a ¬xed set of only 10 equally

spaced frequency points tk . In terms of computational speed their method

compares favorably to the original method of Koutrouvelis (1980). It has

a signi¬cantly better performance near ± = 1 and β = 0 due to the elimina-

tion of discontinuity of the characteristic function. However, it returns slightly

worse results for very small ±.

The quantlet stabreg ¬ts a stable distribution to sample x and returns param-

eter estimates. The string method determines the method used: method="k"

denotes the Koutrouvelis (1980) method with McCulloch™s (1986) initial param-

eter estimates (default), method="km" denotes the Koutrouvelis (1980) method

with initial parameter estimates obtained from the method of moments, and

method="kw" denotes the Kogon and Williams (1998) method. The last two op-

tional parameters are responsible for computation accuracy: epsilon (default

epsilon=0.00001) speci¬es the convergence criterion, whereas maxit (default

maxit=5) denotes the maximum number of iterations for both variants of the

Koutrouvelis (1980) method.

26 1 Stable distributions in ¬nance

1.4 Financial applications of ±-stable laws

Distributional assumptions for ¬nancial processes have important theoretical

implications, given that ¬nancial decisions are commonly based on expected

returns and risk of alternative investment opportunities. Hence, solutions to

such problems like portfolio selection, option pricing, and risk management

depend crucially on distributional speci¬cations.

Many techniques in modern ¬nance rely heavily on the assumption that the

random variables under investigation follow a Gaussian distribution. However,

time series observed in ¬nance “ but also in other econometric applications “

often deviate from the Gaussian model, in that their marginal distributions are

heavy-tailed and, possibly, asymmetric. In such situations, the appropriateness

of the commonly adopted normal assumption is highly questionable.

It is often argued that ¬nancial asset returns are the cumulative outcome of

a vast number of pieces of information and individual decisions arriving almost

continuously in time. Hence, in the presence of heavy-tails it is natural to

assume that they are approximately governed by a stable non-Gaussian dis-

tribution. Other leptokurtic distributions, including Student™s t, Weibull and

hyperbolic, lack the attractive central limit property.

Stable distributions have been successfully ¬t to stock returns, excess bond

returns, foreign exchange rates, commodity price returns and real estate returns

(McCulloch, 1996; Rachev and Mittnik, 2000). In recent years, however, several

studies have found, what appears to be strong evidence against the stable model

(for a review see: McCulloch, 1997; Weron, 2001). These studies have estimated

the tail exponent directly from the tail observations and commonly have found

± that appears to be signi¬cantly greater than 2, well outside the stable domain.

Recall, however, that in Section 1.3.1 we have shown that estimating ± only

from the tail observations may be strongly misleading and for samples of typical

size the rejection of the ±-stable regime unfounded. Let us see ourselves how

well the stable law describes ¬nancial asset returns.

The quantlet STFstab08 ¬ts the stable and Gaussian laws to the USD/GBP

exchange rate returns from the period January 2, 1990 “ November 8, 2000

using the regression method of Koutrouvelis (1980). Next, it and compares

both ¬ts through Anderson-Darling (see Stephens, 1974) and Kolmogorov test

statistics. The former may be treated as a weighted Kolmogorov statistics

which puts more weight to the di¬erences in the tails of the distributions. The

obtained results, presented in Table 1.1, clearly show that the 1.71-stable law

1.4 Financial applications of ±-stable laws 27

Table 1.1: ±-stable and Gaussian ¬ts to the USD/GBP exchange rate returns

from the period January 2, 1990 “ November 8, 2000.

Parameters ± σ β µ

±-stable ¬t 1.7165 0.0033 -0.0986 0.0000

Gaussian ¬t 0.0059 0.0000

Test values Anderson-Darling Kolmogorov

±-stable ¬t 2.1209 0.9181

Gaussian ¬t 20.5433 3.1816

STFstab08.xpl

Table 1.2: ±-stable and Gaussian ¬ts to the Dow Jones Industrial Average

(DJIA) index from the period July 6, 1984 “ May 17, 1996.

Parameters ± σ β µ

±-stable ¬t 1.6723 0.0048 0.0999 0.0009

Gaussian ¬t 0.0102 0.0006

Test values Anderson-Darling Kolmogorov

±-stable ¬t 1.0624 0.8046

Gaussian ¬t +INF 4.9778

STFstab09.xpl

o¬ers a much better ¬t to the data than the Gaussian.

The quantlet STFstab09 ¬ts both distributions to the Dow Jones Industrial

Average (DJIA) index from the period July 6, 1984 “ May 17, 1996. Recall,

that this period includes the biggest crash in stock market™s history “ the Black

Monday of October 19, 1987. Clearly the 1.67-stable law o¬ers a much better

¬t to the DJIA returns, see Table 1.2. Its superiority, especially in the tails of

the distribution, is even better visible in Figure 1.8.

The quantlet STFstab10 ¬ts both laws to the Deutsche Aktienindex (DAX)

index from the period January 2, 1995 “ December 11, 2002. Like for two

previous data sets also here the ±-stable law o¬ers a much better ¬t, see Table

28 1 Stable distributions in ¬nance

Stable and Gaussian fit to DJIA returns Stable, Gaussian and empirical left tails

1

-5

log(CDF(x))

CDF(x)

0.5

-10

0

-0.04 -0.02 0 0.02 -7 -6 -5 -4 -3 -2

x log(x)

Figure 1.8: 1.67-stable (cyan) and Gaussian (dashed red) ¬ts to the DJIA re-

turns (black circles) empirical cumulative distribution function from

the period July 6, 1984 “ May 17, 1996. Right panel is a magni¬ca-

tion of the left tail ¬t on a double logarithmic scale clearly showing

the superiority of the 1.67-stable law.

STFstab09.xpl