There are several probability distribution families that are commonly used in quantitative finance, depending on the specific problem and the underlying assumptions. Two examples of probability distribution families that are frequently used in quantitative finance are the normal (or Gaussian) distribution and the Student’s t-distribution.
The normal distribution is often used in finance to model the behavior of stock prices, interest rates, and other financial variables. The normal distribution is characterized by its bell-shaped curve and its two parameters: mean (denoted ) and standard deviation (denoted ). The mean represents the center of the distribution, while the standard deviation represents its width. The probability density function (PDF) of the normal distribution is given by:
$$f(x) = (1/\sigma\sqrt{2\pi}) * \exp(-(x-\mu)^2/(2\sigma^2)),$$
where x represents the random variable, μ is the mean, σ is the standard deviation, and π the mathematical constant.
An example where the normal distribution is used in quantitative finance is in the Black-Scholes-Merton model, which is a widely used mathematical model for pricing options. In this model, the underlying asset’s price is assumed to follow a log-normal distribution, which is a variant of the normal distribution.
The Student’s t-distribution is another probability distribution family that is commonly used in quantitative finance. The t-distribution is often used when the sample size is small or when the population standard deviation is unknown. The t-distribution has a similar shape to the normal distribution, but with heavier tails, which means that it is more likely to observe extreme values or outliers. The t-distribution has three parameters: mean (denoted ), scale (denoted ), and degrees of freedom (denoted ), which determines the shape of the distribution. The PDF of the t-distribution is given by:
$$f(x) = \frac{\Gamma((n+1)/2}{\Gamma(n/2)} \frac{1}{\sqrt{n\pi}} \left(1+(x-\mu)^2/n\right)^{(-(n+1)/2)},$$
where x represents the random variable, μ is the mean, n is the degrees of freedom, τ is the gamma function, and π is the mathematical constant.
An example where the t-distribution is used in quantitative finance is in the estimation of VaR (Value at Risk), which is a statistical measure of the maximum potential loss that an investment portfolio may suffer within a given time frame, with a certain level of confidence. The t-distribution is used to generate the critical values of VaR under the assumption that the returns of the assets in the portfolio are normally distributed.