# 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?