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They are a lot of ways to compute an "estimated bid-ask spread". The most straightforward one is to sample the bid-ask on a regular time grid (for instance every second), but that for you need all the quote changes (a quote is the best bid and ask, price and quantity).

Usually it is easier (and cheaper) to get the quotes sampled just before the transactions of the day. Hence it is natural the compute the average bid-ask spread on such a database. Unfortunately this sample time is a stopping time and not a deterministic good that is known "a priori". Worst than that it is probable that the occurence of a trade is not independent of the bid-ask spread: most probably traders or algorithms are making a trade-off between the cost of crossing the spread and some predictors or a waiting cost. Because of that the bid-ask spread compute just before a trade is in general smaller than the "average bid-ask spread" (on a regular time grid).

In this context, the original question is: "is the bid-ask spread sampled just before a trade always lower than the bid-ask spread sampled on a regular time grid?"

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  • $\begingroup$ After a trade is before the next trade. It don't get at all what you are saying. $\endgroup$
    – Kurt G.
    Commented Mar 4 at 13:27
  • $\begingroup$ I edited the question , I hope is clearer $\endgroup$
    – XY0
    Commented Mar 4 at 13:47
  • $\begingroup$ Not at all. If I understand correctly this means the bid-ask spread should decrease forever as long as new trades are closed. An absurd claim. $\endgroup$
    – Kurt G.
    Commented Mar 4 at 13:57
  • $\begingroup$ I was not clear ,I am sorry . Is the spread smaller when a trade (X_5) occurs than at the preceding instants? by precendent instants I mean the instants after trading X_4 . this does not mean that the spread observed at the occurrence of trading X_6 (and later) are always smaller and smaller than that sampled spread at trading X_5. $\endgroup$
    – XY0
    Commented Mar 4 at 14:08
  • $\begingroup$ If the spread is smaller at $T_5$ than at $T_4$ because of $T_4<T_5$ then, for the life of me, I don't understand why the spread should not be even smaller at $T_6$ that you said is later than $T_5\,.$ Oh my goodness! $\endgroup$
    – Kurt G.
    Commented Mar 4 at 14:22

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Consider that sometimes the spreads at the exact moment a trade is executed can be larger than the spread before that trade, due to the effect of removing hit orders from the order book, specifically the best bid or best ask orders (assuming they have been completely filled). If you have bids of 10-11-12 and offers of 13-14-15, the spread is 1. However, when, let's say, the 3rd offer crosses the spread and hits the 1st bid, it removes both orders, and the bids are now 10-11 while the offers are 13-14. Now the spread is 2. If the trade size sweeps through many orders, the spread will be even larger at the time of the trade.

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