# Typical coefficients uses in square-root model for market impact

The square-root model is widely used to model equity market impact. It assumes that volatility, traded volume, total volume, and a spread cost are the drivers of slippage.

Jim Gatheral has an excellent summary of the process here.

There is a coefficient that is estimated via regression on a realized trade schedule. This is defined as a constant (alpha) on the page I excerpt below:

What are typical values of alpha as estimated via regressions on realized trade schedules? Jim uses 3/4 later in the paper but this seems a bit high, and I have seen another paper by Rob Almgren that assumes 1/2.

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The short answer for models like BECS (formerly StockFacts Pro) is: it depends. They fit separate models for each exchange, and even within the US they fit NYSE vs. NASDAQ and S&P 500 vs. non-S&P 500 separately. They also use a model that contains both temporary (as above) and permanent impact terms, which would affect the regression coefficients, but in my experience the temporary coefficient ranged between ~1-1.5 for North America. Asia xJP was slightly higher (~2-3) and Japan was north of 4. I no longer use this model at my current firm, so I'm not sure how much this may have changed. – michaelv2 Jan 22 '13 at 16:33
Similar question to: quant.stackexchange.com/questions/43/… – Mark Horvath Jul 18 '15 at 23:05

Find out yourself? -> why not solving for alpha as a function of the difference between the model delta and your true market impact of past trades and subsequent market impact. You obviously need the accompanying data to run such computation but it should be rather straight forward.

From my own research I have not found "stiff" formulae to fit the bill at most times. Market impact between highly liquid stocks and other stocks that trade at a daily huge percentage of public float follow completely different dynamics in my opinion. To be honest I chuckle each time I see similar to above formula because, come on, even you and I, sitting drunk at the bar counter and being asked to come up with a formula will at least account for a) spread cost, b) PLUS something, c) a constant "to save our ass" * , d) some volatility adjusted ratio of our to be traded quantity to some sort of daily measure of trading volume. There are way smarter ways to approach this, but again, I think you will not end with one formula that fits it all.

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There are many ways to calibrate the coefficient for the square root impact. One of the easiest ways is to actually just run a regression between the Q, number of shares to be traded, and V (you can use 20 day average for this) to show how much the market moved. First, you should however do a log log plot to show first that it is indeed a square root market impact. Another thing you can do is actually use the roll model as a way of predicting market impact as well. Jim mentions this in some of his other lectures I believe. Also, the coefficient for the market impact really can depend largely on when you ran the regression as well.

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Depends on the execution algorithm and market.

I have heard of many funds spending few PhD years of research finding out the answer wrt their in house algorithms. Ususally they failed as they do not have enough trades for this (even big funds). Impact is on the order of magnitude of bps, while daily volatility is few orders of magnitues higher. Attributing asset movements to your trades is not an easy task.

One rule of thumb I apply for permanent one-way impact for a VWAP order, is as follows:

$$0.3 \times spread + 2 \times IVP \times bps$$

where IVP is Interval Volume Percentage (Q/V). The formula is clearly flawed, but represents an average of models across markets.

On a side note (although the square root term is popular):

• If we had a non-linear impact, how would the coefficients change if participants would have traded differnt relative sizes?

• Would the impact would be still sub linear if we traded huge sizes?

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

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The constant factor should be a proportional to the current price. A stock with a high price per share would have a higher absolute spread cost. The volatility and volume ratio parameters are invariant

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