Levy-stable distributions revisited: tail index > 2 does not exclude the Levy-stable regime

Power-law tail behavior and the summation scheme of Levy-stable distributions is the basis for their frequent use as models when fat tails above a Gaussian distribution are observed. However, recent studies suggest that financial asset returns exhibit tail exponents well above the Levy-stable regime (0 < alpha <= 2). In this paper we illustrate that widely used tail index estimates (log-log linear regression and Hill) can give exponents well above the asymptotic limit for alpha close to 2, resulting in overestimation of the tail exponent in finite samples. The reported value of the tail exponent alpha around 3 may very well indicate a Levy-stable distribution with alpha close to 1.8.