Derivatives Foundation puzzles, solved step by step

About 1,096 in the middle, so quote 1,000 bid, 1,200 offered, and say the share of golds is where you are least sure. Summer: about 29 Games attended, times about 190 events, times about 18% won by the United States, gives about 992. Winter: 24 Games, about 62 events, about 7%, adds about 104. The honest range is 756 to 1,491; the quote sits around the middle and moves as soon as someone trades on it.

What is the interviewer actually marking?

If a friend asks how much your monthly grocery bill is, you do not recall a number; you think of trips a week, spend a trip, four weeks, and say a figure with a range. A market on an unknown quantity is marked on the structure behind the middle, the honesty of the range, and what you do when someone trades with you, not on knowing the answer. The interviewer knows the real count. Treat every input below as an assumption to be checked against the published medal table, and say so; inventing precision is worse than a wide range.

How do you build the middle?

Split it into Summer and Winter, because they differ by an order of magnitude. Modern Summer Games run every four years from 1896, less three wartime cancellations, about 30 Games, and take one off for the boycotted 1980 Games, 29. Events grew from a few dozen to over three hundred, so call the average about 190, with 170 to 210 as the bracket. The share of golds is the input you own: the United States wins more than any other country, so something like a sixth to a fifth of all golds, 18% in the middle, is defensible but soft. Winter adds 24 Games at about 62 events and a smaller share, about 7%.

How wide should the quote be, and what happens when someone trades?

Running all lows together and all highs together gives 756 to 1,491, but every input is unlikely to sit at its extreme at once, so the quote can be much tighter than that range. Quote about 10% either side of the middle, 1,000 bid and 1,200 offered, and treat the first trade as information from someone who may know the answer. If the interviewer pays 1,200, move both sides up, perhaps to 1,150 at 1,350, and ask yourself which link you underestimated; the share is the first suspect. The limitation to admit is that a quote built from memory of rough facts can be confidently off-centre, which is exactly why the market moves when it is traded.

Centre on 50 and quote something like 48 at 52; after 8 heads in 10, move to 51 at 55, around 53. The mid is the expected count, 100 x 1/2. The standard deviation is sqrt(100 x 1/4) = 5, which tells you how much one lot can swing. Nobody can know more than you about a fair coin, so the quote can be tight. Once 8 heads are banked, the 90 flips left average 45, so the mid is 53 and the sd is sqrt(22.5), about 4.74.

Where does the middle of the quote come from?

If a friend asks how many of the next 100 cars past a junction will be white, and one car in two is white, you say 50 and you are not embarrassed by it. The middle of a market is your expected value, and for 100 fair flips that is 100 x 1/2 = 50 heads; everything else in the answer is about the width. The spread of the outcome comes from the binomial: variance n x p x (1 - p) = 25, so the standard deviation is 5. The count lands between 45 and 55 about 72.9% of the time, and outside 40 to 60 only about 3% of the time. Say both numbers out loud; the interviewer is checking that you know the centre and the scale before you name a price.

How wide should the quote be, and why not plus or minus one standard deviation?

Width is what you charge for two risks: that the person trading with you knows something you do not, and that you have to carry a position whose outcome swings. On a fair coin nobody can know more than you, so the first risk is zero and the quote can be tight, a point or two either side of 50; the standard deviation of 5 sets how many lots you are willing to show, not how far apart your bid and offer are. A quote of 45 at 55 is not wrong, but it says you think the other side may be informed, and an interviewer will ask why. Open at 48 at 52, say you would tighten to 49 at 51 against someone who clearly has no edge, and say what would make you widen: hidden information, a size much bigger than you want to hold, or a coin you have not inspected.

What exactly changes when I tell you 8 of the first 10 were heads?

Two things, and candidates usually get only one. The centre moves to 8 plus the expected heads on the 90 flips left, 45, which is 53. The coin is still fair, so 80% is not the new rate; the 8 are simply banked. The uncertainty also falls, because 10 of the flips are no longer random: the standard deviation goes from 5 to sqrt(90 x 1/4) = 4.74, about 5% lower. Move the whole quote up by 3 to 51 at 55, and notice what the news did to your old offer: a counterparty who had seen those flips would have lifted your 52 and made 1 a lot on average. That is the lesson of the follow-up, and it is why market makers widen or pull quotes when they suspect the other side has seen something they have not. The limitation of the tidy answer is that it trusts the coin; after a run of 8 heads a careful trader at least asks whether the coin is fair.

First quote around 12.25: say 11.75 bid, 12.75 offer. After the 4 is shown, requote around 14: say 13.50 at 14.50. Two independent dice have an expected product of 3.5 x 3.5 = 12.25. Once one die is known to be a 4, the product is 4 times the other die, with expectation 4 x 3.5 = 14. The second market can be the same width or tighter, because only one die of uncertainty is left: the standard deviation of the outcome falls from about 8.9 to about 6.8.

Why does fair value come from multiplying the averages?

A canteen's daily takings are the number of customers times the average spend, and if the crowd size has nothing to do with how hungry people are, the average takings are the average crowd times the average spend. For two independent quantities the expectation of the product is the product of the expectations, so two dice thrown separately have an expected product of 3.5 x 3.5 = 12.25. You can check it on the grid: row i of the multiplication table averages 3.5 i, and the six row averages, 3.5 through 21, average 12.25. The product is skewed, with most of the 36 cells below the mean and a few large ones pulling it up.

How does the reveal change the quote, and why can the market tighten?

Replace the revealed die with its value and keep the other at its expectation. The product is now 4 times one unknown die, so fair value is 4 x 3.5 = 14 and the only uncertainty left is a single die, with a standard deviation of 4 times 1.71, about 6.8, against about 8.9 before. Less uncertainty means less risk per trade, so a market maker can quote the same width with more confidence or tighten it. What you must not do is anchor on the old 12.25: a quote that still straddles 12 after a 4 has been shown is a free trade for the other side, who lifts your offer and collects the difference.

What is the interviewer watching for in the requote?

Speed and direction first, then the width. A trader who says 14 inside a second, moves the market up without hesitation and gives a reason for the width is passing; one who recomputes from the grid, or leaves the old market up, is failing. The next step is usually a trade: if the interviewer lifts your 14.50 offer, you are short at above fair and should hold or edge the market up slightly rather than chase; if they hit your bid at 13.50, you are long below fair. The limitation is that this is a one-shot game with a known distribution; in a real market the reveal would itself be a signal about what else the counterparty knows.

Open around 35, say 33 bid at 37 offer. After the king and the 3, requote around 36.9, say 35.5 at 38.5. Each card averages 7, so five average 35. Once a 13 and a 3 are known, 16 points are fixed and three cards remain from a 50-card deck whose average is now (364 - 16)/50 = 6.96, so the total is 16 + 3 x 6.96 = 36.88. Three unknown cards carry less spread than five, so the market can tighten.

Why is the opening fair value simply five times seven?

Five friends each pick a sweet from a jar without looking; the expected total weight is five times the average sweet, even though each pick changes what is left for the next. Linearity of expectation holds whether or not the draws are independent, so the expected total of five cards is 5 x 7 = 35 and the dealing-without-replacement detail changes only the spread, not the centre. The standard deviation of the total is about 8.0, slightly below the independent-draw figure because the finite deck pulls the cards apart, and a market four wide around 35 is a reasonable opening quote.

What does the reveal change, and what do people get wrong?

Two things move. The two revealed cards swap their expectation of 7 each for their actual values of 13 and 3, which lifts the total by 2, and the remaining deck has lost a high card and a low card, so its average drops from 7 to 6.96, which trims 0.12 off the three unknown cards. The careful centre is 36.88; the quick answer of 16 + 21 = 37 is off by only 0.12, and in the room 37 with a note that the deck is slightly poorer is a fine answer. What is not fine is leaving the market at 35, or widening it when the uncertainty has fallen from five cards to three.

How should the width change, and what trade do you expect next?

Width tracks the remaining uncertainty. The standard deviation of the total falls from about 8.0 to about 6.2 once only three cards are unknown, so a market that was 4 wide can go to 3 wide without taking more risk per trade. The next step is usually a trade: if the interviewer lifts your 38.5 offer, you are short at above fair and hold; if they hit 35.5 you are long below fair. The limitation is that the puzzle assumes a fair deck and honest reveals; in the real version of this game the counterparty may have seen a card you have not, and a run of trades in one direction is the tell.

About 10 million a day, so quote 8 million bid, 12 million offered. Take a population of roughly 67 million, births around 1.1% a year, so about 737,000 babies, children in nappies for about two and a half years, giving 1.84 million children, at five or six nappies a day each: 10.1 million. Running every input at its low gives 5.2 million and at its high 17.6 million, so the quote sits inside that range with room to be wrong.

How do you turn a guess into a market?

If a friend asks you to bet on how many samosas a canteen sells a day, you would not name one number. You would count the seats, guess the sittings, guess how many order a samosa, and then say a range you would bet either side of. A market is that range with a price on each end. Decompose the quantity into inputs you can bound, carry a low and a high through each step, and let the spread of the outputs set the width of your quote. Population times birth rate gives babies a year; times years in nappies gives children in nappies; times nappies per child per day gives the answer. The population and birth rate are the kind of round figures you carry in your head; the interviewer accepts any sensible anchor and marks the structure.

How wide should the market be?

Not as wide as the full range. Every input at its low or every input at its high is unlikely, because the errors are mostly independent and partly cancel. A quote about one third as wide as the all-low to all-high range, centred on the point estimate, is tight enough to be a real market and wide enough that you expect to be inside it. Here the range runs from 5.2 to 17.6 million, so 8 at 12 is a working quote. Say the limitation: the tree ignores adult incontinence products and children over three, both of which push the true figure up, so if anything you would skew the market higher rather than lower.

Then expect the interviewer to trade. If they lift your offer at 12 million, ask yourself what they know: perhaps the adult market, perhaps a higher nappies per day. Move your quote up and tighten, do not freeze it. If they hit your bid, do the reverse. The exercise is not the answer; it is whether you update your market when someone trades against you, which is what market making is.