Stambhika joins the bid at Rs 250.00 behind 40,000 shares with a 10,000 share buy order. Fills arrive at 2,000 shares a minute, and 60% of the selling that reaches the bid comes just before a 5 paise drop. How long until you fill, what is the chance the price moves first, and what is the adverse selection per fill worth?

You reach the front after 20 minutes and finish after 25 if nothing moves, but the 60% figure implies the quote moves first 75% of the time: 37.5% it steps up and leaves you behind, 37.5% it sweeps you and drops. A fill earns the 2.5 paise half spread but carries 0.6 x 5 = 3 paise of expected markout, a realised spread of -0.5 paise. Against crossing now, the queue is still worth about 1.25 paise a share, Rs 125 on the order, not the Rs 500 the spread suggests.

Start with the arithmetic the queue itself gives you. 40,000 shares stand ahead of you and sellers take 2,000 a minute, so you reach the front after 20 minutes and your own 10,000 take another 5, 25 minutes to be done. That number is the ceiling on your wait, not the expected wait, because it assumes the quote sits still for 25 minutes, and the study says it does not. It also ignores cancellations ahead of you, which shorten the queue, and new orders behind you, which do not matter. The real question is what else happens in those 20 minutes.

Think of a bus queue where the bus sometimes leaves early with everyone squeezed on and sometimes gets cancelled. If most people who board do so in the last-second crush, then crushes must be common relative to orderly boarding. If 60% of fills at the bid are sweep fills, then over your wait the chance of a sweep must be 1.5 times the chance of a quiet drain. Let the quote step up or down with equal hazard. If q is the chance nothing moves in your 20 minutes, the chance of a sweep first is (1 - q) / 2, and the sweep share of fills is (1 - q) / (1 + q). Setting that to 0.6 gives q = 0.25: a 25% chance of a quiet fill, 37.5% that the offer is lifted and the bid steps to 250.05 with you still waiting, and 37.5% that the bid is swept, which fills you and then drops the quote to 249.95. The quote moves on average every 14 minutes, and your expected time in the queue is about 11 minutes, not 20.

Score the fill the way a market maker scores it, against the mid a few minutes later. You bought at the bid, 2.5 paise below the mid of 250.025, so you earned the half spread. But 60% of the time the bid drops 5 paise right after you are filled, an expected markout of 0.6 x 5 = 3 paise, so the realised spreadThe half spread earned on a passive fill minus the price move against you after the fill. It is what a liquidity provider actually keeps. is 2.5 - 3 = -0.5 paise a share. The queue is where the adverse selection lives: the people who sweep the bid are the ones who know the price is going lower, and the deeper you sit, the larger the share of your fills that come from them. The curve shows the realised spread turning negative once more than about 31,699 shares stand ahead of you. A market maker in this stock should not be joining a 40,000 share queue to earn the spread.

A buyer who must own the stock scores differently from a market maker, against the price of crossing now. A quiet fill or a sweep fill both buy at 250.00 and save 5 paise against crossing at 250.05; being left behind costs 5 paise, because you chase at 250.10; so the queue is worth 0.625 x 5 - 0.375 x 5 = 1.25 paise a share, Rs 125 on 10,000 shares. That is a quarter of the Rs 500 a candidate quotes by saying the passive order saves the spread. The same answer comes from comparing with someone who crosses after 20 minutes at whatever the offer then is: only the quiet drain beats them. Joining still wins, narrowly, and the margin shrinks to nothing if the step up is likelier than the sweep, or if the chase is more than one tick. The limitation to state: this treats one tick as the whole move, and the fills that sweep you tend to be followed by further falls, which hurts a market maker and does not change the buyer's arithmetic.