Why the spread has to be as wide as it is
A buy order is a small piece of evidence that the stock is worth more. A market maker who cannot tell informed traders from the rest must price every order as if it might be informed, and that is where the spread comes from.
- Update the odds that a stock is worth more, given that someone wants to buy it
- Derive the break-even offer, bid and spread from the informed share and the size of the news
- Read the informed share of the flow off a quoted spread
- Explain why spreads widen before earnings and why retail flow is worth paying for
The intuition
Tomorrow a stock will be worth $40.50 or $39.50, equally likely. One trader in five already knows which. A buy order arrives. Before it, the odds were even. But the informed always buy when the news is good and never when it is bad, so a buy is more likely to come on a good day: the odds are now 60–40 in favour of $40.50, and the stock is worth $40.10 to whoever just sold it. A market maker who offered it at $40.00 gave away 10 cents on average.
So the break-even offer is the expected value given that someone wants to buy, 10 cents above the mid, and the break-even bid is 10 cents below: a 20-cent spread, on a stock whose news is worth 50 cents either way. The uninformed traders pay that spread, the informed collect it, and the market maker is the conduit. More informed flow, or bigger news, and the spread must widen; that is what happens before earnings. Flow that is almost never informed, retail, can be filled inside the spread and still make money, which is why wholesalers pay brokers for it.
P(high | buy) = (1 + π) ÷ 2. Break-even offer = mid + π × δ, bid = mid − π × δ, spread = 2 × π × δ. Quoting a half-spread h earns h against the uninformed and h − δ against the informed, so the expected edge is h − π × δ. Informed share implied by a spread = spread ÷ (2 × δ).
Why it works
- The conventions here: a one-trade model. The stock will be worth mid + δ or mid − δ with equal odds. Each trader who arrives is informed with probability π and trades in the direction of the news; everyone else buys or sells with equal chance. The competitive market maker sets the offer at the expected value given a buy and the bid at the expected value given a sell, so it breaks even on average. One share a trade unless a size is stated; quotes in cents from the mid.
- Bayes in one line. A buy happens with probability ½ whatever the news, by symmetry. A buy and good news together happen with probability ½ × (π + (1 − π) × ½). Divide and the odds of good news after a buy are (1 + π) ÷ 2.
- From odds to a price. The expected value after a buy is the mid plus δ × (P(high) − P(low)) = mid + δ × π. That is the lowest offer that does not lose money on average, and the bid is the mirror image.
- The spread is a transfer. Uninformed traders lose the half-spread on every trade; informed traders gain δ less the half-spread. Weight each by how often it trades and the two cancel exactly at the break-even quote.
- Quoting away from break-even. A half-spread above π × δ earns the difference on every share and invites a competitor to undercut; a half-spread below it loses the difference, and more volume only loses faster.
- Run it backwards. If the spread only covers adverse selection, the informed share is spread ÷ (2 × δ). Real spreads also pay for inventory risk, fees and capital, so that figure is an upper bound.
| P(high | buy): (1 + 0.20) ÷ 2 | 60% |
| Break-even offer: 0.20 × 50 cents above the mid | 10 cents |
| Break-even spread: 2 × 10 | 20 cents |
| Quoting 5 cents a side: expected edge 5 − 10 | −5 cents a share |
| Informed share rises to 40%: spread 2 × 0.40 × 50 | 40 cents |
| Nobody informed: P(high | buy) | 50%, and the spread can be zero |
The formulas
A buy order raises the odds of good news by half the informed share.
The expected value given the order; the market maker breaks even on average.
Proportional to the informed share and to the size of the news.
What you quote less what informed flow costs you.
The backwards reading: an upper bound, since real spreads pay for more than adverse selection.
Worked example
Update the odds after a buy, turn them into an expected value, and double the offset for the spread. The follow-up asks who pays that spread and who receives it.
See it move
Same stock. Change the share of the flow that is informed, the size of the news, and the half-spread you quote.
- Raise the informed share. The odds of good news after a buy rise, the break-even half-spread and spread rise, and the edge on your quote falls.
- Make the news bigger. The break-even spread rises in proportion and your edge falls; the odds after a buy do not change, because they depend only on who is trading, not on how much is at stake.
- Widen your own half-spread. Only the edge moves: the break-even figures belong to the flow, not to you.
Run it backwards
Competitive market makers quote a stated spread, and the news is worth a stated move. If the spread only covers adverse selection, what share of the flow do they think is informed?
Break-even spread = 2 × π × δ, so π = spread ÷ (2 × δ). Half the spread divided by the size of the move.
It is an upper bound. Real spreads also pay for inventory risk, exchange fees and the market maker's capital, so the adverse-selection part is smaller than the whole spread and the true informed share is at most this figure.
Traps
Say it in the interview
“Why does a market maker quote a spread at all, and why does it widen before earnings?”
Check yourself
5 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- A buy order moves the odds of good news to (1 + π) ÷ 2.
- Break-even half-spread = π × δ; spread = 2πδ, proportional to the informed share and the news.
- Expected edge = your half-spread − πδ. Above it a rival undercuts you; below it volume loses faster.
- Spread ÷ (2δ) is an upper bound on the informed share.