Encyclopedia of Financial Models II

Important Functions and Their Features

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: Probability theory can be understood as a particular field in mathematics. Hence, it is only to be expected that it relies intensely on theory from analysis and algebra. For example, the fact that the cumulative probability over all values a random variable can assume has to be equal to one is not always feasible to check for without a profound knowledge of mathematics. Continuous probability distributions involve a good deal of analysis and the more sophisticated a distribution is, the more mathematics is necessary to handle it.

In this entry, we review the functions that are used in financial modeling: continuous functions, the indicator function, the derivative of a function, monotonic functions, and the integral. Moreover, as special functions, we get to know the factorial, the gamma, beta, and Bessel functions as well as the characteristic function of random variables. (For a more detailed discussion of these functions, see Khuri [2003], MacCluer [2009], and Richardson [2008].)

CONTINUOUS FUNCTION

In this section, we introduce general continuous functions.

Figure 1 Continuous Function f(x)

Note: For ...