# Is there a standard model for market impact?

Is there a standard model for market impact? I am interested in the case of high-volume equities sold in the US, during market hours, but would be interested in any pointers.

There is a family of models that is so commonly used among practitioners that it can be almost regarded as standard. For a survey, check out Rob Almgren's entry in the Encyclopedia of Quantitative Finance. Check out also Barra, Axioma and Northfield's handbooks. In general, the impact term per unit traded currency is of the form

$$MI \propto \sigma_n \cdot \text{(participation rate)}^\beta$$

where the exponent is somewhere between 1/2 and 1, depending on the model being used, and the participation rate is the percentage of total volume of the trade, during the trading interval itself. When including the total MI in optimization, the models commonly used are the "3/2" model and the "5/3" model, in which the costs are proportional to (dollar value being traded for asset i)^{3/2, 5/3}. Since the term is not quadratic (and not solvable by a quadratic optimizer) some people approximate it by a linear term plus a quadratic one, or by a piece-wise linear convex function.

I don't believe that there is a "standard" model (per se); in fact, there are many considerations around market impact models, so you would need to be more specific. At the most basic level, you might define market as $P_{first fill} - P_{last fill}$ once your order in actually in the order book (e.g. not including other costs like "opportunity cost"). This doesn't take into account any other trades that may be taking place at the same time or other events that might be impacting the price beyond your order. It doesn't help you to forecast market impact on an impending order (which would require some knowledge of time of day, volume, volatility, etc.).

That being said, I would certainly recommend reading "Optimal Trading Strategies" (Kissell, Glantz 2003) which gives a good overview (in addition to covering other transaction cost subjects).

In practice all impact models are sub-linear. Despite this is fact (seen in many academic publications, commercial and proprietary models), there is an interesting argument for using a linear impact models (other than being careful and pessimistic). Would anyone try to build a model, this approach would be also more parsimonious with less parameters to fit.

Imagine 10 VWAP orders of 10 different traders making up the buy trades of a day for a security, each of them trading the same amount. If impact would be sub-linear (concave) then only one of them trading $10 \times$ more, would result in a different impact. Assuming they are using the same VWAP algorithm (or same broker even), this would lead to contradiction as the impact should be the same.

You might also find this discussion useful: Quantopian Slippage Model.

The square root law is a quite simple and popular model for price impact estimation:

$$\Delta p = Y\sigma\sqrt{\frac{Q}{V}}$$

where: $\Delta p$ is the price impact, $Y$ is a constant (needs to be calibrated). $\sigma$ annualized daily volatility of the returns $Q$ daily trading volume.

There are a lot of papers around this model (e.g Gomes and Walbroeck 2015, Zarinelli et al. 2015)

• I had thought the "square root law" of market impact was the standard, from these answers I guess not. – pyCthon Jul 26 '18 at 18:07

The easiest way ,i suppose, would be to analyze the market depth. If there is a 20 cent gap between each 100 shares on the bid then to sell 1000 shares instantly would have an impact of \$2. Your average price is the midpoint. There are more complicated formulations, but this seems to be how it works on simple examples such as bitcoin exchanges.

• What you describe (more or less) is the instantaneous market impact of a single, aggressive fill (execution) of a given size. If you can fill your entire order like this, then you now have a benchmark "worst" cost. On average, you should be able to do better by e.g. posting some passive liquidity, taking liquidity for smaller portions of your order and then allowing the order book to refresh, et cetera. Very often (most?) of the time, institutional orders are too large to fill instantaneously simply by walking the order book. – afekz Dec 24 '15 at 8:13
• Your USD2 'market impact' reflects how much you've instantaneously moved the bid price. If you're measuring your impact against the mid-price, you have only moved the mid USD1. Wider-than-normal spreads are likely to result in more passive orders refilling the book. Roughly speaking, under most circumstances, there are likely to be more refills on the bid side that you just cleaned out than on the offered side. – afekz Dec 24 '15 at 8:28