# Comparison between Effective Bid-Ask spreads

I understood that given two listed assets, the one with the lower effective spread is more liquid, and if one has effective spread lower than the quoted one, it means there has been a price improvement. Which other conclusions could be figured by comparing one asset’s effective spread with another?

## 1 Answer

in general, the bid-ask spread is proportional to the volatility and inverse proportional to the number of trades per day: $$\psi\propto \frac{\sigma}{\sqrt{N}}.$$

This is true for small tick securities, for large tick securities you need to introduce a "bid-ask bound term" (the Rosenbaum's $$\eta$$). For details see Section 2.2 p143 of Market Microstructure in Practice (L and Laruelle, 2nd edition).

Qualitatively

• the larger the volatility $$\sigma$$, the more risk to bare for liquidity providers, hence the larger the BA-spread to get more money in front of this risk
• the more trade per day, the more occasions to unwind your inventory per day, hence the more market makers can afford to offer an attractive spread.

Hence you can check the consistency of this formula for your different assets, what is interesting is to

1. perform a linear regression over all your securities to obtain $$\psi=a + b\frac{\sigma}{\sqrt{N}} + \epsilon.$$
2. for each security $$i$$, computes the residuals $$\epsilon_i := \psi_i-\left(a + b\frac{\sigma_i}{\sqrt{N_i}}\right).$$
3. now you can compare the $$\epsilon_i$$ that are the effective bid-ask spread corrected from the volatility and liquidity of each security.

Otherwise you can think that one security is more attractive than another, but indeed it has a smaller volatility of more trades per day.