You are making a market on the number of sixes in twelve dice. After you quote, six of the dice are revealed and exactly one of them is a six. Reprice, and explain what happens to your width even though the fair value does not move.

Fair value stays at 2, but the standard deviation falls from 1.29 to 0.91, so the market should tighten from 1.5 / 2.5 to about 1.65 / 2.35. Before the reveal the count was binomial on twelve dice; after it, one six plus a binomial on six. The reveal landed exactly on its expectation, so the mean did not move, but half the dice are now certain and the variance halved. A market's width prices risk, not just the mean, so it narrows.

Twelve dice, each a six with probability one sixth: the count is binomial with mean 12/6 = 2 and variance 12 x 1/6 x 5/6 = 1.67, a standard deviation of 1.29. A 1.5 / 2.5 market is centred on fair value with half a point of edge each side, about 0.39 of a standard deviation, which is the usual shape for a first quote in a game where nobody yet knows more than you. The distribution is wide: there is a 11% chance of no sixes at all and a 13% chance of four or more, so a lot sold at 2.5 could lose Rs 350 if five sixes turn up.

Think of a cricket chase where the first half of the innings goes exactly to the required rate: the projected total is unchanged, but the range of likely outcomes is much narrower than it was before a ball was bowled. Six dice shown with one six is exactly what was expected of them, so the fair value stays at 1 + 6/6 = 2; but the count is now 1 plus a binomial on six dice, with variance 6 x 5/36 = 0.83 and standard deviation 0.91, half the variance of before. The chance of an extreme outcome collapses: zero sixes is now impossible, and four or more has a 6% chance against 13%. Had the reveal shown no sixes, fair value would have dropped to 1; two sixes would have lifted it to 3. This reveal happened to move the mean by nothing and the risk by a lot.

A market maker's width pays for two things: the risk of holding the position until it settles, and the chance that the other side knows more. In this game nobody knows more, so the width is about risk alone, and risk has halved: the expected size of the miss from fair value falls from 0.99 points to 0.67. A width that scales with the standard deviation goes from 1.0 to about 0.71, so quote something like 1.65 bid, 2.35 offered, or 1.75 / 2.25 in the quarter-point steps of the game. Keeping the old 1.5 / 2.5 would be leaving half a point of unnecessary edge on the table, and in a game with other players someone will quote inside you and take the flow. The reverse holds too: if the interviewer had revealed nothing but added six more dice to the cup, fair value would rise and the market should widen with the extra variance.

Say the limitation. Scaling the width with the standard deviation is a rule of thumb; a real desk also prices inventory, the size the counterparty wants and how long the position lasts. The principle survives: price the mean from what you know and what you expect, and price the width from what you do not know.