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Why, in general, is the variance of volume changes higher than variance of price changes?

I understand that these two quantities are functions of some very different factors, but I don't understand fully why the variance of volume is changing so much.

I've heard that even if one accounts for seasonal changes in volatility (by standardizing daily stock volume by daily volume of basket of stocks/volume of index), the variance of the volume changes is much higher.

Edition : I meant quantities measured in percentage. High jumps in volume are very common, whereas high price changes aren't.

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aren't you "pushing"/"forcing" the assumed relation between variance of volumes (variance?) and variance of prices? i mean you (yourself) say different factors are in action behind. – edouard Apr 25 '12 at 14:14

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Not to over simplify, but there is the different scaling to consider here as well.

  • Volumes and volume changes are observed in 1000s and 100s, while prices and price changes are observed in 100s and 1s. For most medium and smaller stocks prices and price changes are observed in 1s and 0.01s.

  • Clarify this through the identity Var(aX) = a²Var(X).

  • Since the numerical values you are comparing are an order of magnitude in difference, naturally their variance, and thus changes in local variance, will also be quite different.

Additionally, price changes are directional and volume changes are not.

  • Quite possibly you are observing that many consecutive minutes are 'confused', being reported with high volume and comparatively little change in price.

  • Clarify this through the inequality |E[X]| <= E[|X|], and its relation to variance as Var(X) = E[X²] - E[X]².

  • This means that even if the volume was somehow scaled down to the same magnitude of price, that the expectation of volume (being non-negative) would always be higher than the expectation of price.

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Volume changes LEAD to price changes. That is to say that prices go up or down because larger (or smaller) volumes of people want to buy or sell.

So volumes change FIRST, and in larger amounts. It is as a result of those volume changes that prices change.

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Could you write more about it, are there any research about shape of function f(x) that : (delta_Price)^2=f(delta_Volume) ? local depth of the market have to be accounted somehow – Qbik Apr 26 '12 at 11:56
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Volume is a flow, whereas price is a stock. As wikipedia notes,

Stocks and flows have different units and are thus not commensurable – they cannot be meaningfully compared, equated, added, or subtracted.

It makes no sense to compare the variance of price to the variance of volume. They are simply two fundamentally different things.

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Accepting that this is true would the implication be that indicators that combine price and volume, such as OBV, money flow index etc. are fundamentally flawed? – babelproofreader Apr 25 '12 at 22:27
@babelproofreader that is good question – Qbik Apr 26 '12 at 11:56
@babelproofreader while we cannot meaningfully compare the level or volatility of price and volume, the product still has a meaning, specifically it is the "dollar volume", and that is the basis of the money flow index. OBV refers only to the sign of changes in price, which is not so bad. In short, comparing price to volume directly BAD, taking the product of price and volume OK. – Tal Fishman Apr 26 '12 at 14:46
You can compare them, e.g. in % change. – LazyCat Apr 26 '12 at 14:55
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@LazyCat just because you can doesn't mean you should. – Tal Fishman Apr 26 '12 at 14:56
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