Student-t measure of return volatility and time scaling

Question

I have a series of price returns of an asset (4 days worth of data). They are relatively high-frequency.

My ultimate goal is to calculate realized volatility, but using a student's t-distribution.

I have fit a two-scale realized volatility (TSRV) model to the returns, then scaled that by sqrt(252) to annualized volatility. The results look reasonable and are close to industry reported numbers. However, I want a student's t-distribution instead.

And the returns don't look normally distributed. So, I'd like to fit a student's t-distribution. Following the advice I have found online, the degrees-of of freedom can be calculated from the excess kurtosis:

k <- np.mean(rets**4) / np.mean(rets**2)**2
excessK <- k-3
df <- 6/excessK + 4
variance <- nu / (nu-2)
sd <- sqrt(nu-2/2)

My questions:

1. How do I scale it to an annualized basis?
2. How do I determine the optimal sampling frequency? (Obvious 1 second has too much noise, but 1 day is missing data.)
3. With an assumed Gaussian distribution, the TSRV methods work well. Is there an equivalent process for a t-distribution?

Thank you!

Answer 1

I can only offer some thoughts on your first question:

First of all, the dof parameter of Student's t-distribution is commonly found by maximum-likelihood methods, implemented via gradient-descent, EM algorithm or the like.

Second, annualizing (daily) returns means convoluting the daily return distributions, thereby evening out any non-normalities, converging to the Normal distribution. Do note, however, that intratemporal dependencies (autocorrelation) are driving the non-normality in annualized returns.