Stable and Tempered Stable Distributions

Abstract

In financial models for asset pricing and asset allocation, asset returns and prices are assumed to follow a normal or Gaussian distribution. However, the properties of the normal distribution are not consistent with the observed behavior found for real-world asset returns. More specifically, the symmetric and rapidly decreasing tail properties of asset return distributions cannot describe the skewed and fat-tailed properties of the empirical distribution of asset returns. The alpha-stable distribution or α-stable distribution has been proposed as an alternative to the normal distribution for modeling asset returns because it allows for skewness and fat tails. Recent research since the turn of the century has introduced alternative distributions such as the tempered stable distributions to better describe asset returns. Keywords: stable distribution; fat tails; heavy tails; volatility clustering; tempered stable distributions; classical tempered stable; distribution; generalized classical tempered stable; distribution; modified tempered stable; distribution; normal tempered stable; distribution; rapidly decreasing tempered stable; distribution; infinitely divisible; tempered stable distributions; log-Laplace transform