# 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.

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## 2 Answers

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.

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I don't believe that there is a "standard" model (per say); 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 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).

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