# Calculating volatility of inhomogeneous time series

I am reading an article by Zumbach and Müller whose name is Operators on Inhomogeneous Time Series. This is interesting in general, but my main goal is to learn a good and efficient method to calculate volatility of inhomogeneous time series (of prices).

So far I was calculating volatility in the following way: I have created an artificial regular time series, spaced by 5 seconds (say), and calculated the simple standard deviation of the returns of these intervals, and annualized. In the paper I'm reading, the authors explain that this method is noisy and that it loses data (and this is clear). In addition, they claim that while this method uses $L^2$ norm, it may be a better option to use $L^1$ norm.

Question 1. why? If I'm going to treat volatility as some sense of standard deviation, why would I use a definition which is less similar?

Later, the authors provide their own volatility operator, which uses three parameters, $\tau$, $\tau'$ and $p$. $\tau$ is the time being checked, but $\tau'$, which is more important, is the time interval in question (for example, in my method, $\tau'$ would be 5 seconds). $p$ is the norm dimension - that is, if we "work" in $L^2$, $p$ would be 2.

Their definition involves two previously defined operators - the norm operator which is the square root of the moving average of the square values, and the differential operator, which is a very weird operator involving empirical constants which the authors claim that are much better than the common return operator $r[\tau](t) = x(t) - x(t-\tau)$. It looks like this: $$\Delta[\tau] = \gamma(EMA[\alpha\tau,1] + EMA[\alpha\tau,2] - 2EMA[\alpha\beta\tau,4])$$ where $\gamma=1.22208$, $\beta=0.65$ and $\alpha^{-1}=\gamma(8\beta-3)$.

Question 2. why? Should I really use an empirical formula, instead of simply use a (perhaps a little bit noisy) interpolation scheme and simply calculate returns and their standard deviations?

Do you see any real profit in the above mentioned idea?

Question 3. Don't you think it worth working with log-returns instead of returns in general?

• Interesting approach, I was looking for that paper on inhomogeneous operators but this time I think I will to through there derivations myself! And I'll be keeping an eye on this question :) – experquisite May 29 '14 at 18:18