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I am looking at the market impact paper here (http://www.cims.nyu.edu/~almgren/papers/costestim.pdf) and I had a question about volatility on page 11.

On page 11 it is stated: "For volatility, we use an intraday estimator that makes use of every transaction in the day. We find that it is important to track changes in these variables not only between different stocks but also across time for the same stock."

I am not sure what the intraday estimator is. Is it only the current day's volatility or it is an average of previous days volatility?

It states "...makes use of every transaction in the day" but if you are trading 10 minutes after the open and want to estimate market impact it seems like having only 10 minutes of tick data to calculate a volatility would be concerning. I was thinking he meant an average of a number of days of volatility using all the ticks in EACH day.

Any idea?

Thank you.

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This is actually a deceptively good question because, as we all know, estimates of variance are extremely sensitive to sampling frequency, sampling intervals, and lags. This is because not all stock prices perfectly adhere to Brownian Motion (i.e., the variance doesn't adhere strongly to the root time rule). It is also not entirely clear from the paper how exactly intraday volatilty is being measured. Therefore, real-world interpretations and implementations could vary significantly.

Taking your excerpt and comparing to the following expert, from page 12, suggests that Almgren intends intraday volatilty to mean one which is sampled from intraday data across 15-minute intervals:

Ten-day average intraday volume profile (upper) and volatilty profile (lower), on 15-minute intervals.

I infer this to mean that "intraday volatility" for the puprose of this paper means something which resembles the following:

${P_t} = $ average logarithmic price sampled over each $t$ minute interval

$\mu_{(P,t)} = $ average logarithmic price of each $t$ minute interval sampled from $t$ to $T$.

$$\mathbb{E}[\sigma^2_{intraday}] \approx \frac{ \sum_{t=0}^{15*24*60} \big( ({P_{15t}}-{P_{15t-15}}) -({\mu_{P,15t }-\mu_{P,15t -15}}) \big)^2}{T}$$

This verbose implementation is supported by Almgren's footnotes on high frequency volatility estimation: http://cims.nyu.edu/~almgren/timeseries/notes7.pdf.

Almgren also references simplified implementations using the Garman-Klass (GK) and Yhang-Zhang (YZ) estimators.

In practice, I implement the Almgren impact model using the YZ OHLC estimator. For one, it has been shown to be more efficient than GK. In addition, it's a great example of the Pareto 80/20 rule: YZ gets about 80% of the benefit of intraday data but it requires roughly 20% of the effort (i.e., 5x less work) to implement -- plus OHLC data is low-low-cost.

A precis on the YZ estimator is found here: Understanding Yang-Zhang Volatility Estimator.

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In the posted script it is stated that:

$$ V = \text{Average daily volume in shares, and} $$ $$ σ = \text{Daily volatility} $$ $V$ is a ten-day moving average. For volatility, we use an intraday estimator that makes use of every transaction in the day.

Hence, I guess from this statement that volatility is the change of traded volume per day, which is plotted in the graphs on page 12.

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    $\begingroup$ Where are you getting "volatility is the change of traded volume". The author mentions and estimator that makes use of every transaction in the day. Typically that means "price" transacations used to calculate a volatility. $\endgroup$ – joesyc Oct 22 '15 at 18:58

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