Adverse selection

Idea: Some orders know more

When a customer may know the value, the trade itself is news. The spread must pay for it.

• The orders that reach you are tilted against you
• That is adverse selection
• The uninformed pay, you pass it along

Example: Count buys in each world

Worth $90$ or $110$, equally likely. $20\%$ of traders know and buy only when high, the rest pick a side at random. Break-even ask?

1. Of $1000$ arrivals: high world $100+\frac{400}{2}=300$ buys, low world $200$
2. A buy means high $\frac{300}{500}=\frac{3}{5}$ of the time
3. Ask: $\frac{3}{5}\times 110+\frac{2}{5}\times 90=102$

Formula: The break-even ask

$\text{ask}=F+a\times \frac{H-L}{2}$

Equally likely values L and H, informed share a: start at fair value F and add a times half the range.

Common mistake: Shifting by the full range

$100+0.2\times 20=104$ uses the whole range, but the value can only rise $10$ above the start.

Instead Shift by $a$ times half the range: $100+0.2\times \frac{20}{2}=102$. At $104$ a rival undercuts you.

Idea: Who pays

The uninformed overpay, the informed collect, the maker breaks even.

• Uninformed: $400\times 2=800$ overpaid
• Informed: $100\times 8=800$ collected
• No fees or risk in the model, so a real quote adds a margin

Formula: Each buy multiplies the odds

$\text{ratio}=\frac{1+a}{1-a}$

A buy is this much likelier in the high world. Use odds when the values are not equally likely.

Quick check: After one buy

Same market, one buy already seen. The next ask allows for the next buy. What is $P(H)$ then?

Idea: Say it out loud

The ask is what the contract is worth given that someone bought.

• Count buys in each world, take the share, price it
• Name who pays: the uninformed