# How rapidly should estimated volatility and volume change for estimating market impact in small markets?

The cost of market impact is usually modeled as:

$$\Delta{P} = \delta \sigma (\frac{Q}{V})^{1/2}$$

Where:

• $$\Delta{P}$$ is the change in price of the asset caused by the transaction size $$Q$$
• $$\sigma$$ is a measure of price volatility (units of price)
• $$V$$ is a measure of trading volume
• $$Q$$ and $$V$$ have the same units (both are dollars, or number of shares)
• $$\delta$$ is a dimensionless coefficient of order 1

How one produces estimates of these parameters affects how quickly market impact costs fluctuate.

If, for instance, price volatility $$\sigma$$ is estimated from just the last $$n$$ 10-minute-frequency open, high, low, and close ("ohlc") prices (using Yang-Zhang volatility or similar), estimated volatility will vary rapidly if $$n$$ is small, or will vary slowly if $$n$$ is large. The same applies to estimating the volume $$V$$.

The goal should be to model the real market impact of a trade well. In my case, I want to accurately model it in small markets (daily dollar volume ~\$100K).

So how rapidly should I expect market impact costs (estimated from $$\sigma$$ and $$V$$) to fluctuate? In other words, what should $$n$$ be, or how should it be chosen?

This is a difficult problem, especially since estimating the volatility faces a number of issues:

• the classic "pollution" of realized variance by bid-ask bounce when using intraday data (cf Aït-Sahalia, Mykland, and Zhang);
• including overnight gap effects if using daily or less frequent data;
• volatility changing (hence the utility of GARCH and related models); and,
• the possibility of structural breaks (cf Timmerman) or jumps in the mean return and volatility (cf Todorov and Tauchen).

You also need to determine what is a reasonable way to estimate $$V$$ since that is usually an average. (Hence why the $$V$$ is usually written as $$\bar{V}$$.)

That said, estimating price impact is imprecise, so to some extent these concerns are less important than the uncertainty of estimation.

In my experience, price impact modeling quickly gets into "secret sauce" which means getting a specific answer about estimation is difficult to impossible. What I will say (i.e. what I will allow myself to say) is to try a recent estimate of $$\sigma$$ that it stable and to use a similar estimation period for your average volume $$\bar{V}$$.

Finally, I would be remiss if I did not suggest you consider other price impact models. I mention a few better models here.