In the preceding chapter, we introduced the concept of continuous probability distributions. In this chapter, we discuss the more commonly used distributions with appealing statistical properties that are used in finance. The distributions discussed are the normal distribution, the chi-square distribution, the Student's t-distribution, the Fisher's F-distribution, the exponential distribution, the gamma distribution (including the special Erlang distribution), the beta distribution, and the log-normal distribution. Many of the distributions enjoy widespread attention in finance, or statistical applications in general, due to their well-known characteristics or mathematical simplicity. However, as we emphasize, the use of some of them might be ill-suited to replicate the real-world behavior of financial returns.
The first distribution we discuss is the normal distribution. It is the distribution most commonly used in finance despite its many limitations. This distribution, also referred to as the Gaussian distribution (named after the mathematician and physicist C. F. Gauss), is characterized by the two parameters: mean (μ) and standard deviation (σ). The distribution is denoted by N(μ,σ2). When μ = 0 and σ2 = 1, then we obtain the standard normal distribution.
For x ∈ R, the density function for the normal distribution is given by
The density in equation (11.1)