Theoretical value, width and skew

A market is three decisions, made in order: what you think it is worth, how sure you are, and what you are already holding. Candidates who quote well answer them separately and out loud; candidates who quote badly collapse all three into a guess.

Definition 2.1

What you are being asked for

A two-sided market, $b \ @ \ a$ — A bid you would genuinely buy at and an ask you would genuinely sell at. The interviewer then trades against whichever side suits them, so both have to be prices you are happy to be filled on.

Input	What it decides	Failure mode
Theoretical value	Where the market sits	Guessing instead of estimating, or refusing to commit to a number
Uncertainty	How wide it is	Quoting tight on something you cannot price, then being picked off
Inventory	Which way it leans	Holding the same market after three fills on the same side

Table 2.2 · The three inputs.

Proposition 2.3

Start with a theoretical value

Say the number and the reasoning that produced it. On a known distribution this is an expectation you can compute; on an unknown quantity it is a Fermi estimate you decompose out loud. Either way, name your assumptions before you quote — an interviewer who disagrees with an assumption will tell you, and that is far cheaper than being wrong about the answer.

Holds when

On a die, a deck or a coin, the theoretical value is exact and there is no excuse for a wide market.
On an estimate, the theoretical value is a midpoint of a range you should also be able to state.

Proposition 2.4

Width is a statement about your uncertainty

The width of your market should track how wrong your theoretical value could be, not how much profit you would like. A tight market on something you know is confidence; a tight market on something you do not know is how you lose money to whoever does.

Holds when

Tighter markets win more of the flow, which is the whole business — an absurd market never trades and earns nothing.
Wider markets survive being wrong. The trade-off is fill rate against adverse selection.

Why both extremes lose. Quote zero wide and every trade you get is one where the counterparty knew better — you capture no spread and absorb every mistake. Quote absurdly wide and you never trade, so you earn nothing at all. The width that maximises expected profit sits between, and it moves with how informed you think the flow is.

Equation 2.5

Turning uncertainty into a width

Treat your own estimate as a distribution. A market roughly one standard deviation either side of your theo is a sensible default; when asked explicitly for a confidence interval, use the z-score for the level requested.

\text{half-width} \approx z \cdot \sigma_{\text{your estimate}}

$\sigma$: How far out your estimate could plausibly be.
$z$: $0.67$ for 50%, $1.28$ for 80%, $1.64$ for 90%, $1.96$ for 95%.

Figure 2.6 · Inventory shifts both sides without widening the quote. An illustrative market around theo 100 with width 1 and a maximum quarter-point skew. As inventory moves from short to long, bid and ask both move down by the same amount. Their vertical gap stays exactly 1.

Theoretical value, width and skew

After this lesson you should be able to

What you are being asked for

Start with a theoretical value

Width is a statement about your uncertainty

Turning uncertainty into a width

The rest of this lesson is in Premium