How Square Root Market Impact Relates to Optimization Cost Exponents

Summary

The document asks why a mean-variance optimization formulation uses a cost exponent of five-thirds when incorporating a square root market impact model. The author suspects the exponent may result from converting impact costs between units, but does not derive it. An excerpt from the cited paper states that market impact is modeled with an exponent p between one and two, reflecting the view that impact grows more slowly than a quadratic cost.

This offers context for connecting nonlinear market impact assumptions to portfolio optimization penalties, but the excerpt alone does not explain how the square root impact relationship produces the five-thirds objective exponent. No derivation, implementation details, or performance evidence are included. The text is therefore a focused question about the mathematical mapping between an impact model and an optimization objective, rather than a complete method or recommendation.

Key ideas

• The document asks how a square root market impact model leads to a five-thirds objective exponent.
• The cited excerpt considers objective cost exponents between one and two.
• The stated motivation is to represent market impact costs growing more slowly than quadratically.
• The document provides no derivation connecting the square root model to the specific exponent.

Tags

# Market impact in optimization objective function


# Market impact in optimization objective function












Can anyone help to explain why when the square root market impact model is used in the standard mean-variance optimization, the exponent becomes $\frac{5}{3}$ in the objective function? I suspect this has to do with expressing market impact in different units but I could not find a clear explanation.

EDIT: Near the top of page 8, the authors state:

> ... show that the square root function is more appropriate for modelling market price impact, thus suggesting market impact costs grow at a rate slower than quadratic. Therefore in this section we consider the case with p ∈ (1, 2) in objective function (1).

https://pdfs.semanticscholar.org/b96d/c0273a25d27ab1aef9f8e300701f1689d738.pdf

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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