< List of probability distributions

Probability theory is a field that has applications in various fields ranging from finance to physics. A probability distribution is a function that describes the likelihood of obtaining different values of a random variable. The **Lévy distribution** is one such probability distribution that is both continuous and stable. In this blog post, we will explore the properties of the Lévy distribution, its applications in finance, and how it relates to other probability distributions.

## Properties of the Lévy Distribution

The Lévy distribution is named after French mathematician Paul Lévy. It is a continuous probability distribution for non-negative random variables.

The probability density function (PDF) is

Where

- μ = the location parameter
- c = the scale parameter.

The cumulative distribution function (CDF) is:

where

- erfc = the complementary error function,
- Φ = the Laplace Function (CDF of the standard normal distribution),
- μ = a shift parameter. This parameter is sometimes denoted α in the literature.
- c = a scale parameter (c > 0).

The characteristic function is

*φ *(*t*; *a*, *c*) = *e ^{iat }*

^{– √(-2ict)}

Where *a *and *c *are scale and location parameters.

The Lévy distribution is *stable*, which means that when you sum two random variables x and y in a distribution, their sum (i.e., *x *+ *y*) is a random variable from that same distribution. The Lévy distribution is one of only three distributions with this property: the other two are the normal distribution and the Cauchy distribution.

## Lévy distribution applications

The Lévy distribution has a *heavier tail *than the normal distribution, which implies that it has a higher probability of extreme values.

As such, the distribution has found various applications in **finance**. In particular, it is useful for modeling asset returns, especially in cases where the returns are extreme. One example of such an application is the modeling of the distribution of log returns on financial assets.

The Lévy distribution has also been used for modeling the distribution of large losses in **insurance risk**. One of the advantages of using the Lévy distribution for modeling asset returns is that it can handle fat-tailed returns, which are a common feature of financial data.

In **physics**, the distribution can be used to model the length of a photon’s path traveling in a turbid medium,. The frequency of geomagnetic reversals (“pole shifts”) appears to follow this distribution as well [1]. And when a particle is in Brownian motion, the time it will take to hit a single point α can be modeled by the Levy distribution, where c, the scale parameter, is α^{2} .

## Relationship to Other Probability Distributions

The Lévy distribution is closely related to other probability distributions. For instance, the Cauchy distribution is a special case of the Lévy distribution with alpha = 1 and c = 1. The Lévy distribution is also related to the stable distribution and the infinitely divisible distribution. The stable distribution is a generalization of the Lévy distribution that includes other distributions such as the Gaussian distribution and the Cauchy distribution.

## Estimation of Parameters

In practice, the parameters alpha and c of the Lévy distribution are estimated from data. One way to do this is by using the method of maximum likelihood estimation. This involves finding the values of alpha and c that maximize the likelihood of the observed data. Another method is the method of moments, which involves equating the sample moments of the data to the theoretical moments of the Lévy distribution.

In conclusion, the Lévy distribution is a continuous and stable probability distribution that has found various applications in finance. Its heavier tail than the Gaussian distribution makes it useful for modeling extreme events such as large asset returns and losses in insurance risk. The Lévy distribution is closely related to other probability distributions such as the Cauchy distribution, the stable distribution, and the infinitely divisible distribution. Estimation of its parameters is done using methods such as maximum likelihood estimation and the method of moments. Understanding the Lévy distribution and its properties is useful for anyone interested in probability theory, finance, or data science.

## References

[1] Carbone, V.et. al (2006). “Clustering of Polarity Reversals of the Geomagnetic Field”. Physical Review Letters. 96 (12): 128501.