Estimating Volatility from Irregularly Spaced Price Data
Summary
The document asks how to estimate volatility when prices arrive at irregular times. Its author contrasts resampling prices onto fixed five-second intervals and taking the standard deviation of returns with a method described in a paper on inhomogeneous time series. The paper’s approach uses a volatility operator built from a norm operator and a specially constructed differential operator, with parameters for the observation time, interval length, and norm. The author asks why an L1 norm might be preferable to the usual L2 measure and whether the empirical differential formula offers a real advantage over interpolating prices and calculating returns.
The discussion is framed as open questions rather than a tested recommendation: no comparison, data, or performance results are supplied. It also raises whether log returns would be more appropriate than simple returns. The document therefore identifies issues in sampling and volatility measurement, but does not resolve them or establish that the proposed operator is more accurate or efficient in practice.
Key ideas
- Resampling irregular price observations onto fixed intervals can introduce noise and discard information.
- The paper’s proposed volatility operator combines a norm with a specialized differential operator.
- The author questions whether an L1 norm is useful when volatility is intended to represent dispersion similar to standard deviation.
- No empirical comparison is provided to establish advantages over interpolated returns.
- The document also raises log returns as an alternative, without reaching a conclusion.
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Full text
# Calculating volatility of inhomogeneous time series
# 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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.