Gamma Function Uses in Bayesian Finance, SDEs, and Credit Risk
Summary
The document describes several ways the gamma function can appear in quantitative finance. It extends factorials to noninteger arguments and, for integer inputs, satisfies the factorial identity. A beta integral involving powers of a probability and its complement can be evaluated using gamma functions; the resulting expression may occur in Bayesian calculations for binomial models.
It also notes that some stochastic differential equations have closed-form solutions expressed through special functions, including the gamma function, and that gamma distributions are used to model credit portfolio default rates. These are examples of where the function can arise, rather than a worked model or derivation. The discussion gives no data, implementation guidance, or comparison of modeling choices, and it does not establish how often these applications are useful in practice.
Key ideas
- The gamma function extends the factorial to noninteger arguments and reproduces factorial values at positive integers.
- A beta integral can be written as a ratio of gamma functions and used in Bayesian calculations involving binomial models.
- Some stochastic differential equations have closed-form solutions that involve the gamma function.
- Gamma distributions are used to model default rates in credit portfolios.
Tags
Full text
# Does the gamma function have any application in quantitative finance?
# Does the gamma function have any application in quantitative finance?
I was looking into the factorial function in an R package called gregmisc and came across the implementation of the gamma function, instead of a recursive or iterative process as I was expecting. The gamma function is defined as:
$$ \Gamma(z)=\int_{0}^{\infty}e^{-t}t^{z-1}dt $$
A brief history of the function points to Euler's solution to the factorial problem for non-integers (although the equation above is not his). It has some application in physics and I was curious if it is useful to any quant models, apart from being a fancy factorial calculator.
## Answer by bill_080 (score 4, accepted)
https://quant.stackexchange.com/a/704
It shows up in Bayes Analysis where a binomial distribution is involved (integer values apply):
$$ \Gamma(k + 1) = k! $$
That allows the following integral to be evaluated in closed form:
$$ \int_{0}^{1}p^{j-1}(1-p)^{k-1}dp = \frac{\Gamma(k)\Gamma(j)}{\Gamma(j+k)} $$
That integral can easily show up in the numerator and/or denominator of Bayes Equation.
## Answer by glyphard (score 2)
https://quant.stackexchange.com/a/701
In certain cases some stochastic differential equations(SDE's) have closed form(deterministic) solutions in the form of well known ordinary differential equations (ODE's), partial differential equations(PDE's), and special functions like the gamma function.
Here's an example from a paper where an SDE has a closed form solution in terms of the gamma function: http://www.siam.org/books/dc13/DC13samplechpt.pdf
Solving SDE's (preferably quickly), like with a closed form solution (when one is available), is a core activity in quantitative finance.
## Answer by Owe Jessen (score 0)
https://quant.stackexchange.com/a/702
Gamma distributions are being used to model the default rate of credit portfolios by CreditRisk.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.