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I am reading a paper "A Simple Way to Estimate Bid-Ask Spreads from Daily High and Low Prices" cf.A Simple Way to Estimate Bid-Ask Spreads from Daily High and Low Prices

The authors proposed the method of estimation the bid-ask spread from high and low prices of consecutive two days.

From what I can understand, there is an important assumption there that the prices follow geometric Brownian motion and, therefore, the true variance over a 2-day period is twice as large as the expectation of the variance over a single day. This property is used for the spread estimation.

Next, assume that I have more data, than just high and low prices, say, 10 min bars.

Will it improve the spread estimator if I use high and low prices of consecutive two 10 min bars instead on days? Does it contradict to the derivation for daily case?

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If you have access to intraday data, they are better ways to estimate the bid-ask spread. If you have Open, High, Low and Close price on each 5min bin $b$ (or any other interval): the Close of the previous bin and the Open of this one are consecutive. Hence $dP(b)=C(b-1)-O(b)$ allows to define an estimate $\psi(b)$ of the bid-ask spread $$\psi(b):=\min_{b:\, |dP(b)|>0} |dP(b)|.$$

It is not defined on every bin of each day (sometimes $dP(b)=0$), but often you have several of them. You can average them to obtain an estimate for the bid-ask of the day.

Of course you can add an estimate deduced from High and Low of the bin, but it is clearly worst than this $\phi$.

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