# Simulating t-distributed returns by calibrating degrees of freedom $\nu$ from variance or kurtosis

A slight twist (I hope) on the familiar problem of simulating log returns from a t-distribution. My two questions concern calibration to sample data.

1. First, one can infer the degrees of freedom, $$\nu$$, of the t-distribution by equating the kurtosis of a sample of log returns with the kurtosis of the t-distribution, which is $$k_\nu = \frac{3(\nu-2)}{(\nu-4)}.$$ Alternatively, one could do the same thing with the variance, which is given by $$\sigma^2_\nu = \frac{\nu}{(\nu-2)}.$$ In general, the two procedures will not yield the same value for $$\nu$$. Which is better? Or should one take a GMM approach?
2. My second concern involves scaling. It is often suggested that when simulating stock prices using a t-distribution, one should scale the sample volatility by $$\sqrt{\frac{(\nu-2)}{\nu}}$$. I have found that this scaling can produce a density which (although fatter tailed) is more peaked than the normally distributed returns for the same sample. This seems wrong.
• for the t-distribution, $\sigma^2_\nu = \frac{\nu}{(\nu-2)}$ for $\nu> 2$ only. $\sigma^2_\nu = \infty$ for $\nu \in (1,2]$, and is undefined otherwise. There are similar rules for the t-distribution's excess kurtosis, but not sure how to express them in terms of (non-excess) kurtosis. – develarist Nov 18 '20 at 0:03