Approximation of skewed and leptokurtic return distributions

Abstract

There is considerable empirical evidence that financial returns exhibit leptokurtosis and nonzero skewness. As a result, alternative distributions for modelling a time series of the financial returns have been proposed. A family of distributions that has shown considerable promise for modelling financial returns is the tempered stable and tempered infinitely divisible distributions. Two representative distributions are the classical tempered stable and the Rapidly Decreasing Tempered Stable (RDTS). In this article, we explain the practical implementation of these two distributions by (1) presenting how the density functions can be computed efficiently by applying the Fast Fourier Transform (FFT) and (2) how standardization helps to drive efficiency and effectiveness of maximum likelihood inference.

Keywords:

• fast Fourier transform
• stable Paretian distribution
• tempered stable distribution
• tempered infinitely divisible distribution
• classical tempered stable distribution
• rapidly decreasing tempered stable distribution

JEL Classification::

• C22
• C32

Notes

¹For a review of these studies, see Rachev et al. (2005).

²See Rachev and Mittnik (2000).

³See DuMouchel (1975).

⁴See Sato (1999) for the definition and further discussion of infinitely divisible distributions.

⁵In general, for TS and TID distributions it may not be possible to find a closed-form expression for the corresponding characteristic function.

⁶See Fritsch and Carlson (1980).

⁷See Kim et al. (2008a) for the definition of the stdCTS.

⁸See Kolmogorov (1933).

⁹See Cramér (1928).

¹⁰See Anderson and Darling (1952).