# Market impact, why square root?

The standard method of market impact is the square-root formula $$\begin{equation} \Delta P = c \cdot\sigma \cdot \sqrt{\frac{n}{\nu}} \end{equation}$$ where $$\Delta P$$ is the price change from executing a trade for $$n$$ shares, with market volatility $$\sigma$$, average market turnover $$\nu$$ and some constant $$c$$. This is empirically justified across a wide range of markets (even Bitcoin).

Is there any good theoretical justification as to why the square-root formula is so consistent across different markets?

• I don't think there is yet. There is some work in trying to generalize this formula and understand where such market impact formulas come from by Kyle and Obizhaeva papers.ssrn.com/sol3/papers.cfm?abstract_id=3124502 (Mainly they come from dimensional analysis and some basic economic assumptions.) Sep 27 '18 at 8:45
• Some more intuition can be found in The amazing power of dimensional analysis: Quantifying market impact, by Pohl, Ristig, Schachermayer, Tangpi. In chapter 4 you might find something.
– Ric
Sep 27 '18 at 9:08
• A derivation of the rule is given here papers.ssrn.com/sol3/papers.cfm?abstract_id=2412761 . Visually, the idea is to convert stock chart to a triangle, which is the simplest possible shape and then derive a relationship between price and volume that obeys an equilibrium condition. The formula can be generalized for different shapes. a convex curve has a greater impact . A concave one will have less impact initially but then greater a consequence of a rising derivative as one moves down the curve. The steep convexity of market bubbles can explain why they end suddenly. Jan 9 '20 at 23:33

I found this power point and this paper to be an excellent source on this topic.

Here is a quote from the paper:

A square-root singularity for small traded volumes is highly non-trivial, and certainly not accounted for in Kyle’s classical model of impact , which predicts a linear impact ∆ ∝ Q. A concave impact function is often thought of as a saturation of impact for large volumes. We believe that the emphasis should rather be placed on the anomalous high impact of small trades. Numerically, Eq. (1) means that trading one hundredth of the daily volume moves the price by a tenth of its daily volatility, which is indeed a huge amplification. Mathematically, Eq. (1) implies that marginal impact diverges for small volumes as $$Q^{-1/2}$$ , meaning that the susceptibility of the market to trades of vanishing size is formally infinite. In most systems, the response to a small perturbation is linear, i.e. small disturbances lead to small effects. The breakdown of linear response often implies that the system is at, or close to, a critical point, where very special properties emerge, such as long-range memory or scale invariant avalanches, that accompany this diverging susceptibility.

It goes on to say that besides being empirically robust (it appears to hold in a suprisingly wide number of settings), the square root law arises according to the authors from the very peculiar nature of the order book (the collection of all buy and sell orders) near the boundary between buying and selling. Closer to the "current price", the order book rapidly thins in density.

Indeed if this thinning in price-space of the order book is approximately linear, then the window in price space required to fill an order of dollar size Q will grow with the square root of Q (my own illustration): Their model to explain this thinning supposes that orders undergo a diffusion process in price space (a diffusion associated with volatility), and therefore, the order book thins in density near the critical point where buy and sell orders meet each other and annihilate (execute).

My understanding (devoid of any mathematical grounding) is as follows.

v = Turnover PER UNIT TIME
n = Shares you need to execute


therefore

n/v = Number of units of time required to execute your size at the normal turnover rate


Realized vol follows a SQRT(T) heuristic.

Given that we can now rewrite the transaction cost formula purely in terms of vol and time units.

This translates to the observation that the cost of execution over a period is directly proportional to the realized vol over the period

• While I do enjoy this observation, this only serves to rephrase the question to "why is the cost of execution proportional to the volatility?". Sep 28 '18 at 11:36
• I believe at the most fundamental level, volatility is completely determined by the ratio of market taking demand vs market making supply. 2 stocks can have the same turnover but different volatility reflecting a different 'base' level of market making liquidity. It is this 'base' liquidity that is the determining factor for both volatility and transaction cost. It therefore makes a lot of sense for volatility and transaction cost to move in the same direction. But as to why it is linear and not exponential or higher order, I have no idea.
– hjw
Sep 29 '18 at 16:32