To what degree is volume correlated with price impact?

Question

There is question from 2017 here but the answer given doesn't really go into any detail.

How correlated is volume with price impact? Especially with regard to wash trading; For example, wash trading is extremely prevalent in crypto markets and yet, despite this, one can still a clear correlation between volume and price impact.

Is there a nice formula that can break down the relationship?

Answer 1

Market impact corresponds to price moves due to the intensity of trading in one direction. As you mention: when a market participant trades in size in one direction, the price usually follows.

They are two reasons:

1. the first one is informational: the trader is right to buy (or sell), he or she understood the price will go up (or down). Hence in this case, a relation with volume is not needed: the price will move.
2. the second one is mechanical: it is the one you mention. The pressure of the buying flow moves the price. In such a case the relation between the traded quantity $Q$ and the price move $\Delta P$ is square root (Bacry, Emmanuel, Adrian Iuga, Matthieu Lasnier, and C-A L. "Market impacts and the life cycle of investors orders" Market Microstructure and Liquidity 1, no. 02 (2015)): $$\Delta P\propto \sigma\sqrt{Q\over V} + a\,\psi,$$ where $\sigma$ is the volatility of the specific instrument, $\psi$ its bid-ask spread, and $V$ its traded volume.

JP Bouchaud explains the two effects quite well in his Introduction to Part II. Price Impact: Information Revelation or Self- Fulfilling Prophecies? to Capponi, Agostino, and C-A L, eds. Machine Learning and Data Sciences for Financial Markets: A Guide to Contemporary Practices Cambridge University Press, 2023.