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. First, one can infer the degrees of freedom in the t distribution, v, by equating the kurtosis of a sample of log returns with the kurtosis of the t distribution, which is 3(v-2)/(v-4). Alternatively, one could do the same thing with the variance, which is given by v/(v-2). In general, the two procedures will not yield the same value for v. Which is better? Or should one take a GMM approach?
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 the square root of (v-2)/v. I have found that this scaling can produce a density which (athough fatter tailed) is more peaked than the normally distributed returns for the same sample. This seems wrong.