Calibrating Student-t Return Simulations from Sample Data

Summary

The document raises two issues in fitting a Student-t distribution to observed log returns. It compares estimating the degrees of freedom by matching sample kurtosis with estimating them by matching variance, noting that the two calibrations generally disagree. It asks whether one estimate is preferable or whether a generalized method of moments approach could combine information from both statistics.

It also questions the common volatility adjustment used when simulating Student-t returns. The author reports that the adjustment can yield a distribution with heavier tails but a higher central peak than a normal distribution at the same sample volatility, and asks whether that outcome is appropriate. The document presents these as open questions rather than resolving them: it gives no sample data, comparison study, or recommended estimator. The calibration and scaling choices therefore need further analysis, including attention to the Student-t distribution's moment constraints and to what is held fixed when comparing simulated distributions.

Key ideas

• Degrees of freedom can be estimated by matching sample kurtosis or sample variance to their Student-t counterparts.
• The two moment-matching procedures need not produce the same degrees-of-freedom estimate.
• A generalized method of moments approach is raised as a way to use multiple sample statistics.
• Volatility scaling may affect both tail thickness and the concentration of probability near the center.

Tags

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


# 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.

- 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?

- 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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.