Why a Spread Exists at All

Someone quoting a two-sided market posts a bid below their estimate of value and an offer above it. The obvious story is that the gap is their margin, and that competition should erode it.

But the puzzle survives every attempt to remove the obvious costs. Suppose exchange fees are zero, the quoting firm has no capital costs, no technology costs, and competes so hard that it earns nothing in expectation. There is still a spread.

The classical result is due to Lawrence Glosten and Paul Milgrom, in "Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders", Journal of Financial Economics, volume 14, 1985, pages 71 to 100. Their conclusion is exactly the counterintuitive one: a positive spread arises with zero transaction costs and zero expected profit for the quoting participant.

Something other than cost is being paid for. Working out what it is explains most of what market makers do.

Here is the mechanism, and it is entirely about information rather than money.

Some participants know something the quoting firm does not: a pending order, a model that has updated, an announcement read faster. Call them informed. Everyone else trades for reasons unrelated to value: a fund rebalancing, a saver investing, a hedger offsetting exposure. Call them uninformed.

The quoting firm cannot tell them apart. What it can observe is the direction of the trade, and direction is evidence. An informed buyer only buys when the asset is worth more than the quote. So the act of someone lifting your offer is itself news that your offer was too low.

This is adverse selection in its exact sense: the counterparties you attract are systematically the ones you least want. The trades you win are disproportionately the ones you should have declined.

The resolution is to stop thinking of the bid and ask as one estimate plus and minus a margin, and to treat each as a separate forecast.

Set the ask equal to the expected value of the asset given that someone buys at it. Set the bid equal to the expected value given that someone sells at it. Each quote is conditioned on the trade it would attract.

Because a buy is evidence of higher value and a sell is evidence of lower value, those two conditional expectations differ. The ask lands above the unconditional estimate and the bid below it. The gap between them is the spread, and it appeared without anyone adding a margin.

The firm now breaks even by construction: every trade is executed at the fair value conditional on that trade occurring. It loses to the informed and recovers exactly that much from the uninformed, and the spread is the size of the transfer.

The argument is small enough to run numerically. Value is 100 or 200 with equal probability. A fraction of traders are informed and always trade the right way; the rest buy or sell at random.

def quotes(p_high=0.5, informed=0.30, v_low=100.0, v_high=200.0):
    u = (1 - informed) / 2          # chance an uninformed trader buys (or sells)
    # P(buy | value) : informed buy only when value is high
    p_buy_hi, p_buy_lo = informed + u, u
    p_sell_hi, p_sell_lo = u, informed + u

    def posterior(p_hi_given, p_lo_given):
        num = p_hi_given * p_high
        return num / (num + p_lo_given * (1 - p_high))

    ask = v_high * posterior(p_buy_hi, p_buy_lo) + v_low * (1 - posterior(p_buy_hi, p_buy_lo))
    bid = v_high * posterior(p_sell_hi, p_sell_lo) + v_low * (1 - posterior(p_sell_hi, p_sell_lo))
    return bid, ask

quotes(informed=0.00)   # (150.0, 150.0)   no informed traders -> no spread
quotes(informed=0.30)   # (135.0, 165.0)   spread 30
quotes(informed=0.60)   # (120.0, 180.0)   spread 60

With nobody informed the spread is exactly zero. It widens in direct proportion to the informed fraction. No cost term appears anywhere in the calculation.

A spread has three conventional components, and they behave completely differently.

Order processing is the mundane part: exchange fees, connectivity, staff. It has fallen for decades and keeps falling.

Inventory compensates the quoting firm for holding a position it did not want. Buy from a seller and you are now long, exposed until you can offset. Better hedging instruments shrink this.

Adverse selection is the irreducible one. It is not a cost the firm incurs; it is a transfer to better-informed counterparties. No technology removes it, because it exists precisely to the extent that someone else knows more.

That is why spreads have compressed enormously and not to zero. Two components were costs and shrank. The third is an information asymmetry, and asymmetry does not get cheaper.

The quoted spread is a poor measure of cost, for two reasons that pull in opposite directions. Orders sometimes execute inside the quotes, and orders in size execute outside them.

The standard replacement is the effective spread: twice the distance from the mid at the time of the order to the price actually obtained. Buy at 100.02 against a mid of 100.00 and the effective spread is 4 cents, whatever the screen said.

Doubling looks arbitrary and is not. It expresses the cost as a full round trip, so it is comparable to a quoted spread, which also spans both sides.

The convention to be careful with is which mid. Using the mid after the trade builds your own impact into the benchmark and flatters the result. The mid immediately before the order is sent is the honest choice, and using the decision-time price rather than the arrival-time price is stricter still.

The measurement that makes the theory operational separates what the quoting firm keeps from what it gives back.

After a trade, wait a defined interval and look at the mid again. If a buyer lifted the offer and the mid then rose, the buyer knew something: that move is price impact, and it is the quoting firm's loss. Whatever is left of the effective spread is the realised spread, the part the firm actually keeps.

effective spread  =  realised spread  +  price impact
     what you paid       market maker's       information
                            revenue           you revealed

The decomposition is a diagnostic for both sides. A trader whose orders show large price impact is leaking intent, and slowing down or splitting the order will help. A trader whose costs are almost entirely realised spread is paying for immediacy and would benefit from resting orders instead.

Same total cost, two different problems, two different fixes.

A model earns its place by predicting things you can check, and this one does.

• Spreads widen before scheduled announcements. The share of informed flow rises, so quotes must widen even though nothing about costs changed.
• Spreads widen in less-followed instruments. Thin coverage means any given counterparty is more likely to know something specific.
• Retail flow is valuable. It is disproportionately uninformed, so quoting against it is safer than quoting against anonymous flow, which is the economic basis for arrangements that route it separately.
• Being early in the queue is not purely good. Priority means you trade first, including first against whoever just learned something.

The last point is the one that separates people who have thought about this from people who have not. A resting order is a free option granted to everyone else, and it gets exercised when it is worst for you.

The spread is now accounted for. A trade is evidence, quotes price that evidence, and the gap is the transfer from uninformed to informed participants.

One thing is still missing. The argument above treats each trade in isolation and asks what it reveals. It does not say how much the price should move per unit of quantity, or why a large order moves the price more than a small one beyond simply consuming more of the book.

That is a question about the rate at which information enters the price, and it is what makes trading in size fundamentally different from trading small. It also determines how much capital a strategy can absorb before its own trading destroys its edge, a limit the lesson Costs, Capacity, and a Protocol You Can Trust approaches from the backtesting side.

The next lesson takes it from the market's side.