Can Price–Volume Correlations Always Be Zero? A Preprint Puts Averaging Under the Spotlight

Two ways to average a price

Olkhov starts with a basic accounting relationship: a trade’s monetary value equals its price multiplied by its volume. Here, volume means the quantity traded, while value means the money exchanged.

The paper allows observations to represent either individual transactions or groups of transactions collected into successive time intervals. For grouped observations, price is total value divided by total volume within each interval.

Across a longer period, prices can be averaged in different ways. An arithmetic average gives every observed price equal weight. A volume-weighted average price, or VWAP, instead gives greater influence to observations involving larger quantities. It equals the total money exchanged divided by the total quantity traded.

These averages answer different questions: one describes the average observed price; the other describes the average price per unit traded. They coincide when every observation has the same volume. Olkhov argues that VWAP is the economically appropriate average when volumes vary.

Why the paper obtains zero

The central calculation subtracts VWAP multiplied by average volume from the average of price multiplied by volume.

The accounting relationship makes the first term average monetary value. The definition of VWAP makes the second term exactly the same quantity: total value divided by total volume, multiplied by volume per observation, gives value per observation. Subtracting the two therefore produces zero.

That cancellation is an algebraic identity under the paper’s definitions. It does not require observing a particular market pattern, and it is not evidence that prices and trading quantities move independently.

Olkhov explicitly acknowledges the latter point. Even a zero covariance need not mean that two variables have no statistical dependence.

The crucial statistical qualification

Although the paper calls its quantity “correlation,” its displayed definition is a covariance: an average product minus the product of averages. The familiar Pearson correlation additionally divides covariance by the standard deviations of both variables.

More importantly, the zero calculation combines different weighting rules. Price is averaged using volume weights, whereas volume and the price–volume product are averaged by giving observations equal weight. The usual covariance identity uses a consistent averaging rule throughout.

The paper itself displays the consequence. Its calculation that averages products of deviations gives a generally different result from the expression that produces zero. Olkhov interprets that discrepancy as a failure of the conventional empirical calculation. But the discrepancy also identifies the limitation of his conclusion: replacing the equally weighted price average with VWAP does not establish that an equally weighted covariance is invalid.

There is another distinction in the constant-volume case. Covariance with a constant is zero, but Pearson correlation is undefined because the volume standard deviation is zero.

What a nonzero calculation represents

One useful identity in the preprint connects the conventional, equally weighted covariance to the two price averages. Using the paper’s normalization, that covariance equals the difference between VWAP and the arithmetic average price, multiplied by average volume.

If VWAP exceeds the arithmetic average, the covariance is positive; if it falls below it, the covariance is negative. The relationship links a statistical calculation to an understandable feature of the transactions: how weighting by traded quantity shifts the average price.

Olkhov argues that this difference is what earlier researchers have actually calculated, rather than a genuine price–volume correlation. That is the author’s interpretation, not an empirical finding demonstrated by reanalysing those studies.

Proposed alternatives and remaining limits

To investigate dependence, Olkhov proposes quantities involving price and squared volume, or squared price and squared volume. Squaring volume gives larger observations greater numerical influence. The appendix derives expressions for these quantities within his framework, including a relationship between them that uses a price-variance formula drawn from his earlier work.

The supplied paper contains no new market dataset, numerical validation or comparison showing that these alternatives outperform conventional measures. It also does not demonstrate that they can reveal every form of dependence.

The defensible takeaway is narrower than the title: averaging choices matter, and VWAP has a precise accounting interpretation. The preprint’s universal zero is a consequence of its definitions—not a demonstrated universal absence of conventional price–volume correlation.