Hyperbolic distribution

< List of probability distributions

The hyperbolic distribution offers an alternative to the traditional normal distribution when modeling phenomena with numerically large values. In this blog post, we will take a closer look at the hyperbolic distribution, its characteristics and properties, and its applications in finance and meteorology.

What is the Hyperbolic distribution?

The hyperbolic distribution is a subclass of the generalized hyperbolic distribution that is commonly used for modeling financial returns and asset prices. The distribution is characterized by a hyperbola for a log probability function, in the same way that the normal distribution’s is a parabola [1]. The distribution decreases exponentially, although more slowly than the normal distribution, so is suited for data where you’ll likely see larger values than occur naturally such as those seen in financial asset returns or hurricane force wind speeds. When you increase the number of trials, you also increase the probability of seeing outliers.

The hyperbolic distribution probability density function (PDF) is:

hyperbolic distribution pdf


  • K_{\lambda } is a modified Bessel function of the second kind.

Hyperbolic distribution applications

One of the significant advantages of the hyperbolic distribution is its flexibility, making it suitable for modeling different types of financial data. For example, the distribution can accurately model returns with high kurtosis values or returns with skewed distributions. Additionally, the hyperbolic distribution can be used to construct models that incorporate time-varying volatility, making it suitable for modeling financial assets with variable risks.

Another application of the hyperbolic distribution is in meteorology, specifically in modeling turbulent wind speeds. The distribution has been shown to be a better fit for wind speed data, especially in regions with high turbulence, than the traditional normal distribution. Using the hyperbolic distribution, it is possible to model wind speeds with high kurtosis values and skewed distributions.


Although the hyperbolic distribution has gained popularity in recent years, it is worth noting that it is not a new concept. In fact, the distribution was first introduced in the 1930s by Bagnold [2] during his search for a way to optimally represent grain data. It wasn’t until the 1970s that the distribution gained traction, with various authors such as Barndorff-Nielsen [3] reintroducing the distribution to the literature. The distribution has been studied extensively in the field of actuarial science. However, with the increasing popularity of data science and machine learning, the distribution has found new applications and is gaining popularity across different fields.:

In conclusion, the hyperbolic distribution is a versatile probability distribution that is gaining popularity among data scientists and statisticians. The distribution’s flexibility makes it suitable for modeling financial data, such as returns and asset prices, and meteorological data, such as turbulent wind speeds. Understanding the hyperbolic distribution’s characteristics and properties is essential for college students pursuing a career in data analysis, finance, or meteorology. By incorporating the hyperbolic distribution into their models, data analysts and financial experts can make better-informed decisions based on accurate and reliable data.


[1] Christiansen, C. & Hartmann, D. (1991). The hyperbolic distribution.

[2] RA Bagnold. (1937). The Size-Grading of Sand by Wind. Royal Society Publishing.

[3] Barndorrf-Nielson, O. E. (1977). Exponentially decreasing distributions for the logarithm of particle size. Proceedings of the Royal Society (London), 353: 401-9

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