Make a market on the product of two dice. The fair value is 12.25 with a standard deviation near 8.9 and a long right tail. How wide are you for one lot and for ten, and which way do you lean?

Fair value is 12.25; quote about 11.5 at 13.5 for one lot and about 10.8 at 14.8 for ten. The two dice are independent, so the mean is 3.5 squared. Ten lots are wider because a size request is more likely to come from someone who has seen a die, which alone pushes the break-even quotes to 10.94 and 13.56. Both markets lean up, because a short position owns the tail to 36.

Independence does the first step: the expected product is the product of the expectations, 3.5 x 3.5 = 12.25. The variance needs one more line: the mean of the square of one die is 91/6, so the mean square of the product is (91/6) squared, about 230.0, and the variance is 230.0 - 150.06, about 80, a standard deviation of 8.94. The shape matters as much as the mean: products bunch at small values and stretch thinly up to 36, so the median is 10 and 23 of 36 outcomes sit below fair value. Think of household incomes in a town with a few very rich families: the average is pulled above what most households earn.

With no reason to think the counterparty knows anything, any market centred on 12.25 earns its half-width on average, so the width only has to pay for the risk of one roll. A one-lot carries a standard deviation of 8.9 points, Rs 894, which is small. A two-point market is reasonable, and a careful candidate leans it slightly up: 11.5 at 13.5. The lean comes from asking what each side can lose. Short at the offer, the worst one in ten rolls costs you on average 17.3 points; long at the bid, the worst one in ten costs 9.6. Setting the market so that each side earns the same edge per point of that tail loss moves its centre up by 0.28.

Two reasons, and interviewers want both. First, ten lots on one roll is ten times the money on the same coin flip; the risk does not diversify. Second, and larger here, size is a signal. If a quarter of ten-lot requests come from someone who has seen one die, so knows the contract is worth 3.5 times that die, a market tighter than 10.94 at 13.56 loses money on average before any risk charge. The informed trader lifts your offer only when the die is 4 or more and hits your bid only when it is 3 or less, so every trade with them is a loser. Break-even is where the edge from the uninformed three-quarters covers that.

Add a risk margin on top and apply the same tail-matching lean with a half-width of 2: 10.8 at 14.8 for ten lots, centred 0.57 above fair value. At that offer, a short ten-lot loses Rs 21,200 if double six comes up; at that bid, a long ten-lot loses at most Rs 9,800, on double one. The two worst cases are not mirror images, and the quote should not be either.

Say the limits. The quarter chance of an informed counterparty is an assumption for the round, not a fact, and the right width moves with it; the lean depends on judging tail loss the same way on both sides, and a desk with a different risk measure would lean by a different amount. What does not change is the direction: on a right-skewed contract, being short is the side with the long tail, so the offer goes further from fair value than the bid.