Interpreting Square-Root Market Impact Limits Across Execution Days

Summary

The document discusses how to interpret a participation threshold in a square-root market-impact model. It presents impact as proportional to volatility and the square root of order size relative to daily volume, and notes that the cited framework assumes the quantity executed remains small compared with volume. It also raises a separate caveat: if execution takes too long, impact may instead become linear in total quantity.

The central question is whether the threshold should be assessed against one day’s volume or cumulative volume across a multi-day schedule. The example considers executing a fixed quantity in equal daily clips and asks whether the square-root regime would end once accumulated execution crosses a fraction of total volume over the full period. The document provides no resolution or evidence for either interpretation. It is therefore a useful statement of the modeling ambiguity, but not a definitive execution rule; applying the threshold requires consulting the underlying model and its assumptions about horizon and volume.

Key ideas

• The square-root impact relation scales with volatility and the square root of quantity relative to volume.
• The cited applicability condition requires executed quantity to remain small relative to volume.
• Long execution horizons may call for a linear impact relationship instead.
• The document asks whether a participation threshold uses daily or cumulative multi-day volume.
• It does not resolve the interpretation or provide validation for a particular rule.

Tags

# Market impact measured through several days


# Market impact measured through several days












According to different papers in the literature, the standard method of market impact is the square-root equation that satisfied that, after a quantity $q<Q$ is executed, the average adverse price move is given by:

$$ I(q) = \epsilon ⋅\sigma ⋅ \sqrt{q/V} $$

where $\sigma$ is the daily volatility, $V$ is daily share transaction volume, and $Y$ a numerical constant of order unity. According to A proposal for impact-adjusted valuation: Critical leverage and execution risk. For the above impact formula to be valid, the execution time $T$ needs to be large enough that $Q$ remains much smaller than $V$ (20% is a typical upper limit). The execution time should not be too long either, otherwise impact is necessarily linear in Q.

The question is, how should we interpret the mentioned threshold of 20% on $V$ to apply either the previous equation or a linear equation?

As example, suppose we want to execute 1,000 shares, divided in daily orders of 100 shares each one (thus, with a total of 10 days) and total daily share transaction volume $V$ is 200. Should we interpret this 20% as 20% out of the total $V$ executed during all 10 days (i.e., 20% out of $200·10 days =2,000$ shares? In that case, starting from the $4^{th}$ day ($0.2·2,000 = 400$, which is equivalent to 4 consecutive days executing 100 daily shares), the impact would be linear. Is this interpretation right or wrong?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.