Encyclopedia of Financial Models III

Stable and Tempered Stable Distributions

SVETLOZAR T. RACHEV, PhD, Dr Sci

Frey Family Foundation Chair-Professor, Department of Applied Mathematics and Statistics, Stony Brook University, and Chief Scientist, FinAnalytica

YOUNG SHIN KIM, PhD

Research Assistant Professor, School of Economics and Business Engineering, University of Karlsruhe and KIT

MICHELE LEONARDO BIANCHI, PhD

Research Analyst, Specialized Intermediaries Supervision Department, Bank of Italy

FRANK J. FABOZZI, PhD, CFA, CPA

Professor of Finance, EDHEC Business School

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.

In finance, the normal or Gaussian distribution has been the underlying assumption in describing asset returns in major ...