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.
Notes
1For a review of these studies, see Rachev et al. (Citation2005).
2See Rachev and Mittnik (Citation2000).
3See DuMouchel (Citation1975).
4See Sato (Citation1999) 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 (Citation1980).
7See Kim et al. (Citation2008a) for the definition of the stdCTS.
8See Kolmogorov (Citation1933).
9See Cramér (Citation1928).
10See Anderson and Darling (Citation1952).