Encyclopedia of Financial Models III

Continuous Probability Distributions Dealing with Extreme Events

MARKUS HÖCHSTÖTTER, PhD

Assistant Professor, University of Karlsruhe

SVETLOZAR T. RACHEV, PhD, Dr Sci

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

FRANK J. FABOZZI, PhD, CFA, CPA

Professor of Finance, EDHEC Business School

Abstract: Continuous probability distributions are commonly the preferred candidates when modeling financial asset returns. The most popular of them is unquestionably the normal distribution because of its appealing properties as well as the fact that it serves as the limit distribution for many sums of random variables such as, for example, aggregated returns. The normal distribution generally renders modeling easy because all moments exist. However, the normal distribution fails to reflect stylized facts commonly encountered in asset returns, namely, the possibility of very extreme movements and skewness. To remedy this shortcoming, probability distributions accounting for such extreme price changes have become increasingly popular. Some of these distributions concentrate exclusively on extreme values and others permit any real number, but in a manner that is capable of reflecting market behavior. Consequently, there is a selection of probability distributions that can realistically reproduce asset price changes. Their common shortcoming is generally that they are mathematically difficult ...