Using Roll's Model to Estimate Bid-Ask Spreads from Return Bounce
Summary
The document explains the Roll model, which estimates the bid-ask spread from the negative autocovariance of consecutive high-frequency price changes. The negative relationship arises when trades alternate between the bid and ask: transaction prices move even if the underlying quote midpoint remains unchanged. In the model, this bid-ask bounce is the source of the signal used to infer the spread.
The discussion corrects the idea that negative return autocovariance means dealers face little adverse-selection risk and therefore charge a narrow spread. In this setting, negative autocovariance is expected and is precisely what allows the Roll estimate. If ultra-high-frequency returns instead show no negative autocovariance, the model provides no useful spread estimate; the answer suggests the minimum tick as a fallback. That guidance is limited to the stated model and does not establish a general spread estimator for other data frequencies or market conditions.
Key ideas
- Bid-ask bounce can create negative autocovariance in consecutive high-frequency returns.
- The Roll model uses this negative autocovariance to estimate the bid-ask spread.
- Alternating trades at bid and ask can generate price changes without a change in the quote midpoint.
- A lack of negative autocovariance leaves the Roll model without a useful spread estimate.
- The suggested minimum-tick fallback is a limited rule of thumb, not a general estimator.
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# Bid-Ask spread in Roll's model: Negative autocovariance of returns and informational content
# Bid-Ask spread in Roll's model: Negative autocovariance of returns and informational content
Currently studying on techniques to estimate the bid-ask spread. Perhaps the most widely known model is the Roll model (1984). Let $P_t$ indicate log prices
$\begin{cases} Bid_t=P_t-c, \\ Ask_t=P_t+c, \end{cases}$
Where $c=\sqrt{-Cov(\Delta P_t, \Delta P_{t-1})}$.
My question comes down to the interpretation of autocovariance. My intuition is that positive autocorrelation in returns, imply presence of informed traders (Probability of having a trade in the same direction after a buy\sell is higher), and dealer sets a higher spread due to adverse selection. On the other, with negative autocorrelation, a trade is likely to be followed by a trade in the opposite direction (Sell after a Buy and vice versa) and dealer does not have adverse selection risk, so he\she should set a low spread. In the literature, some researchers take the absolute autocovariance to deal with undefined spread. Does this imply overestimation bias ("Expensive" spread when there is negative autocorrelation)?
## Answer by kurtosis (score 4, accepted)
https://quant.stackexchange.com/a/57276
This does not imply overestimation bias. We expect a negative autocorrelation in high- and ultra-high-frequency (every trade) data due to bid-ask bounce. Bounce occurs when buy and sell orders trading at the offer and bid are interspersed; that yields what seems to be returns even when the bid, ask, and midpoint do not change.
The Roll (1984) model examines this bounce in a theoretical market and determines that the negative autocovariance can be used to estimate the bid-ask spread.
What if ultra-high-frequency returns do not exhibit a negative autocovariance? That is rare and suggestive of very strong trending behavior. In that case, the best estimate of the bid-ask spread would be the minimum tick size -- since the Roll model offers you no useful information.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.