Market Making: Fair Value, Spread & Adverse Selection

What a market maker does

A market maker continuously shows a bid (the price at which they will buy) and an ask (the price at which they will sell). The mid is their estimate of value; the spread $=$ ask $-$ bid is their compensation for providing liquidity.

In interviews you will often be asked to "make a market" on an unknown quantity, such as the sum of some dice or the number of cars in a city. The interviewer then trades with you and reveals information. You are judged on three things:

1. A fair value that is a sensible conditional expectation.
2. A width that reflects your uncertainty and the risk of trading with someone better informed.
3. How you update after trades and new information.

Fair value is a conditional expectation

If $V$ is the quantity you are quoting and $\mathcal{I}$ is everything you know, your mid should be $\mathbb{E}[V \mid \mathcal{I}]$. For three dice with one revealed showing 5:

$$\mathbb{E}[V \mid \text{first die} = 5] = 5 + 2 \times 3.5 = 12 .$$

Sampling without replacement changes this: if the first of five cards dealt from a deck is a King, the remaining four cards come from a deck whose average is $\frac{364 - 13}{51} \approx 6.88$, not 7.

Why spreads exist: adverse selection

Suppose a fraction $\alpha$ of counterparties know the true value $V$ and the rest trade for unrelated reasons. Informed traders only buy when $V >$ ask and only sell when $V <$ bid, so every informed trade loses money for you. The half-spread $s$ you earn from uninformed traders has to pay for it:

$$(1-\alpha)\,s \;\gtrsim\; \alpha \cdot \mathbb{E}\big[\,|V - \text{price}|\;\big|\;\text{informed trade}\,\big].$$

More informed flow, or more uncertainty about $V$, means a wider spread. As information is revealed and uncertainty falls, you can tighten.

Inventory and skew

Every fill changes your position. A large position exposes you to the final value: if you are long 10 lots and the truth comes in 3 below your fair value, you lose 30. Market makers manage this by skewing: shifting both bid and ask down when long (so you sell more readily and buy less) and up when short. The skew is usually small relative to the spread but persistent.