Why Square-Root Market Impact Produces a 3/2 Trading-Cost Term
Summary
The document examines the relationship between a square-root market-impact model and the three-halves power often used for trading-cost penalties in portfolio optimization. The initial question is whether the higher exponent comes from integrating an instantaneous square-root impact over execution. The answer instead treats the quoted square-root expression as the price impact associated with the full order, rather than an instantaneous impact rate.
To obtain total dollar cost, the price difference from the initial price is multiplied by the order quantity. This produces a component proportional to order size and another proportional to its three-halves power. The distinction matters in mean-variance optimization: a three-halves penalty is convex in traded quantity, while a square-root penalty is concave and does not preserve the same convexity guarantee. The post gives an algebraic explanation, but does not examine alternative impact models, empirical calibration, or uncertainty in transaction costs.
Key ideas
- A square-root impact expression for a full order is distinct from an instantaneous impact rate to be integrated over time.
- Multiplying the order’s price impact by its quantity yields a linear cost component and a three-halves-power component.
- A three-halves-power trading-cost penalty is convex in traded quantity.
- A square-root penalty is concave, so adding it to a convex mean-variance objective can remove convexity guarantees.
- The explanation depends on the stated impact model and does not establish its empirical fit.
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# Market Impact: Going from 1/2 power to 3/2
# Market Impact: Going from 1/2 power to 3/2
So the standard market impact model (please correct me if I am wrong) says that for an order of size $V_O$ executed from time $0$ to time $T$: $$ S_{T} = S_0\left(\alpha + \beta \sigma\sqrt{\frac{V_O}{V_T}} \right) $$where $S_0$ is the price at the start of the execution, $V_T$ is the total volume, $\sigma$ is some measure of daily volatility (in percentage terms), and $\alpha$, $\beta$ are coefficients, which are unknown but estimated from data.
When reading literature on optimization, I see that there is sometimes a penalty due to trading costs, of the form: $$ \propto |w - w_0| ^ {3/2} $$ where $w$ is the new proposed set of weights and $w_0$ is the initial set of weights. The exponent $3/2$ seems to come from integrating $x^{1/2}$ (the square root above), suggesting that the square-root model is some "instantaneous" expression, and by integrating it, we get the full impact of the order. But that doesn't seem to be the case in what I've read and what I've written above, as it seems to describe the impact of a full order.
I ask about this distinction, because if we are doing classical mean-variance optimization, and want to add an expected cost term (let's drop the variance of the cost for now, as that's a little bit more challenging to calculate, although feasible w/ delta-method style approximations), then if we add a $3/2$ penalty to the problem, the objective remains convex as a function of $w$ (as power laws with exponent > $1$ are convex). However, if we use $1/2$ as the exponent, this is a concave function, and then adding it to the convex mean-variance problem no longer guarantees convexity nor concavity.
## Answer by rubikscube09 (score 1)
https://quant.stackexchange.com/a/74061
So I just realized that the $3/2$ doesn't come from an integral, but from a "totaling".
If we have some quantity $V_O$ that we want to execute at price $S_0$, and we receive a price $S_T$ rather than $S_0$, our notional difference (in some currency) is: $$ (S_T - S_0) V_O = S_0\left(\alpha + \frac{\beta \sigma}{\sqrt{V_T}}\sqrt{V_O}\right)V_O - S_0 V_O = S_0\left(c_1 V_O+ c_2 V_O^{\frac{3}{2}}\right) $$ Thus the dollar difference is a linear function of $V_O$, plus a $3/2$ power.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.