Log-t distribution

In probability theory, a log-t distribution or log-Student t distribution is a probability distribution of a random variable whose logarithm is distributed in accordance with a Student's t-distribution. If X is a random variable with a Student's t-distribution, then Y = exp(X) has a log-t distribution; likewise, if Y has a log-1distribution, then X = log(Y) has a Student's t-distribution.

The log-t distribution has the probability density function:

$p(x\mid \nu ,{\hat {\mu }},{\hat {\sigma }})={\frac {\Gamma ({\frac {\nu +1}{2}})}{x\Gamma ({\frac {\nu }{2}}){\sqrt {\pi \nu }}{\hat {\sigma }}\,}}\left(1+{\frac {1}{\nu }}\left({\frac {\ln x-{\hat {\mu }}}{\hat {\sigma }}}\right)^{2}\right)^{-{\frac {\nu +1}{2}}}$,

where ${\hat {\mu }}$ is the location parameter of the underlying (non-standardized) Student's t-distribution, ${\hat {\sigma }}$ is the scale parameter of the underlying (non-standardized) Student's t-distribution, and $\nu$ is the number of degrees of freedom of the underlying Student's t-distribution. If ${\hat {\mu }}=0$ and ${\hat {\sigma }}=1$ then the underlying distribution is the standardized Student's t-distribution.

If $\nu =1$ then the distribution is a log-Cauchy distribution. As $\nu$ approaches infinity, the distribution approaches a log-normal distribution. Nonetheless, for any finite degrees of freedom, the mean and variance and all higher moments are infinite or do not exist.

The log-t distribution is a special case of the generalized beta distribution of the second kind.

The log-t distribution has applications in finance. For example, the distribution of stock market returns often shows fatter tails than a normal distribution, and thus tends to fit a Student's t-distribution better than a normal distribution. While the Black-Scholes model based on the log-normal distribution is often used to price stock options, option pricing formulas based on the log-t distribution can be a preferable alternative if the returns have fat tails.