Probability and Statistics for Finance

Chapter 12. Continuous Probability Distributions Dealing with Extreme Events

In this chapter, we present a collection of continuous probability distributions that are used in finance in the context of modeling extreme events. While in Chapter 11 the distributions discussed were appealing in nature because of their mathematical simplicity, the ones introduced here are sometimes rather complicated, using parameters that are not necessarily intuitive. However, due to the observed behavior of many quantities in finance, there is a need for more flexible distributions compared to keeping models mathematically simple.

While the Student's t-distribution discussed in Chapter 11 is able to mimic some behavior inherent in financial data such as so-called heavy tails (which means that a lot of the probability mass is attributed to extreme values), it fails to capture other observed behavior such as skewness. Hence, we decided not to include it in this chapter.

In this chapter, we will present the generalized extreme value distribution, the generalized Pareto distribution, the normal inverse Gaussian distribution, and the α-stable distribution together with their parameters of location and spread. The presentation of each distribution is accompanied by some illustration to help render the theory more appealing.

GENERALIZED EXTREME VALUE DISTRIBUTION

Sometimes it is of interest to analyze the probability distribution of extreme values of some random variable rather than the entire distribution. ...