Student-t Return Volatility, Estimation, and Annualization
Summary
The question asks how to model high-frequency asset returns with a Student-t distribution after estimating two-scale realized volatility, and how to annualize the result and choose a sampling frequency. The response focuses on degrees-of-freedom estimation and annualization rather than resolving the sampling-frequency or realized-volatility estimator questions.
It recommends estimating the Student-t degrees of freedom with maximum likelihood, using a numerical method such as gradient descent or EM. For annualization, it describes aggregating daily return distributions through convolution; repeated aggregation can make the resulting distribution more nearly normal. It also cautions that autocorrelation and other dependence across periods affect the shape of annualized returns. The exchange gives conceptual guidance rather than a tested procedure, and supplies no answer for adapting the two-scale estimator to Student-t returns or selecting an optimal observation interval.
Key ideas
- Maximum likelihood is suggested for estimating the Student-t degrees of freedom.
- Gradient descent and EM are examples of methods that can perform the estimation.
- Annualizing daily returns involves aggregating their distributions across periods.
- Aggregation can make annual returns more nearly normal, while dependence can shape their distribution.
- The response leaves sampling frequency and a Student-t version of the realized-volatility estimator unresolved.
Tags
Full text
# Student-t measure of return volatility and time scaling # Student-t measure of return volatility and time scaling 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: - How do I scale it to an annualized basis? - How do I determine the optimal sampling frequency? (Obvious 1 second has too much noise, but 1 day is missing data.) - With an assumed Gaussian distribution, the TSRV methods work well. Is there an equivalent process for a t-distribution? Thank you! ## Answer by Kermittfrog (score 1) https://quant.stackexchange.com/a/73412 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.
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