Encyclopedia of Financial Models III

Continuous Probability Distributions

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 needed when the random variable of interest can assume any value inside of one or more intervals of real numbers such as, for example, any number greater than zero. Asset returns, for example, whether measured monthly, weekly, daily, or at an even higher frequency are commonly modeled as continuous random variables. In contrast to discrete probability distributions that assign positive probability to certain discrete values, continuous probability distributions assign zero probability to any single real number. Instead, only entire intervals of real numbers can have positive probability such as, for example, the event that some asset return is not negative. For each continuous probability distribution, this necessitates the so-called probability density, a function that determines how the entire probability mass of one is distributed. The density often serves as the proxy for the respective probability distribution.

In this entry, we introduce the concept of con-tinuous probability distributions. We present the continuous distribution function with its corresponding density ...