Derivatives Foundation puzzles, solved step by step

Raise the market and widen it; do not sell a third time near the old level. With only your 10 known, the fair total is 24 and the two unknowns give a standard deviation of about 5.3. A player who sees his own card and lifts your offer is telling you his card is high: if it is 8 or more the fair total is 27.5; after a second lift, 11 or more, it is 29. You are short 2 at an average near 27 against a value near 29. Quote something like 28 at 32.

What does a lift tell you that the cards do not?

If you are selling a second-hand bike and the first viewer pays your asking price without haggling, you have probably priced it low; if he immediately asks to buy a second one, you certainly have. A counterparty who can see something you cannot and keeps buying is telling you the value is higher than your offer, and every trade you do with him before repricing is a trade you will regret. This is adverse selection, and a market-making game exists to see whether you notice it in time.

How much do you move, and how much do you widen?

Move at least as far as the information says and widen because your uncertainty about his behaviour has grown. After one lift a quote of 26 at 29 sits around the new estimate of 27.5; after two lifts 28 at 32 sits around 29 and is twice as wide, because the next lift would mean his card is 12 or 13 and the total near 30 or more. The width is not a penalty on him; it is the price of your own blindness. A standard deviation of 5.3 on the total from the two unknown cards is the natural scale for the width before any lifts.

What about the position you already have?

You are short 2 at an average near 27 and the value is near 29, so you are losing about 4 on paper. Do not try to earn it back by selling more at a worse price, and do not flip to buying from the other player at any cost; skew your quote up so the next trade is more likely to reduce the short than add to it. Say your position out loud when the interviewer asks; the game checks whether you can hold the fair value, the quote and the inventory in your head at once. The limitation to state: the thresholds 8 and 11 are a rough model of his behaviour, and a player who bluffs changes the inference.

You finish short 4, you lose Rs 2 on the settlement, and your expected profit when you stopped trading was Rs 6. Sold 5, bought 3, sold 2 is a net short of 4. Cash is +37.5 - 19.5 + 16 = +34. Settling at 9 costs 4 x 9 = 36, so the result is 34 - 36 = -2. Against the fair value of 7 the short would have cost 28, leaving +6: the edge of 2.5 + 1.5 + 2 captured on the three trades.

Why keep three numbers in your head and not one?

A shopkeeper who sells umbrellas at a markup has a profit on each sale, a stock count, and a worry about whether it rains. Three separate things. A market maker tracks the same three: edge per trade against fair value, net position, and the exposure to the final number, and the game checks that you never let one of them slip. Say the fair value, 7, first. Then say each trade's edge as you do it: +2.5 on selling 5 at 7.5, +1.5 on buying 3 at 6.5, +2 on selling 2 at 8. Then say the position: short 4.

Was the loss a mistake?

No. Every trade was done at a better price than fair value, so the trading was right; the dice came in high. Expected profit of +6 is what you controlled, and the realised minus 2 is what the dice did; an interviewer wants to hear you separate the two without being asked. The sum of two dice has a standard deviation of about 2.4, so a short of 4 carries a one-standard-deviation swing of nearly 10, far larger than the 6 of edge. The real question is whether a short of 4 was more risk than you wanted to carry against 6 of edge.

What would you have done differently in the game?

Skewed the quote as the short grew. After selling 5, you are short 5 and should lower both your bid and your offer so that the next trade is more likely to be a buy that cuts the position, which is exactly what buying 3 at 6.5 did. Selling 2 more at 8 added to the short again; at that point a wider or higher quote would have protected you. The limitation of the puzzle: with two interviewers trading against you, their trades carry information about nothing, because the dice are not rolled yet, so here the only reason to skew is inventory, not adverse selection.

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 1.75 and quote something like 1.6 bid, 1.9 offered. The die averages 3.5 coins and each coin gives half a head, so the mean is 1.75. The spread comes from two sources: the coin flips, E[N]/4 = 0.875, and the uncertain number of coins, Var(N)/4 = 0.729, a variance of 1.604 and a standard deviation of 1.27. The fair value is exact, so the quote can be tight; the spread tells you how hard to size it.

How do you get the centre?

Suppose a shop's daily customers vary and each spends Rs 200 on average. Average takings are average customers times Rs 200, whatever the day-to-day mix. When a random number of random things are added up, the mean is the expected count times the expected size of each, so here 3.5 coins times half a head = 1.75. That is the law of total expectation in one line. Note that 1.75 is not a possible outcome, and the single most likely outcome is one head, with a chance of 0.312; a market is centred on the mean because that is where neither side has an edge.

Why is the spread bigger than the coins alone suggest?

Because you are uncertain about two things at once: how many coins, and how they land. The variance of a random sum is the average of the inner variance plus the variance of the inner mean: E[N] x 1/4 from the coins, plus Var(N) x 1/4 from the die, 0.875 + 0.729 = 1.604. If you knew the die would show 3.5 coins and ignored its own wobble, you would quote a standard deviation of 0.94 instead of 1.27 and size too large. The die's contribution is almost half the total, which is the step most candidates leave out.

How tight should the market be, and what if the other side saw the die?

Width pays you for two risks: not knowing the fair value, and trading with someone who knows more. Here the fair value is exact and nobody has seen anything, so a tight quote around 1.75, such as 1.6 at 1.9, is right, and the standard deviation of 1.27 governs how many contracts you take, not where you centre. Change one fact and the answer changes: if the counterparty has seen the die, a buyer is telling you the die was high. A die of 6 implies 3 heads on average, and a die of 1 only 0.5, so widen sharply or ask to see the die before quoting. The limitation is that real games price in that information risk from the first quote.

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.

Skew the whole quote up, to about 50 at 52, keeping the width of 2. The short is a risk you want to shed, so make your bid attractive to sellers and your offer less attractive to buyers. A simple rule moves the mid 0.05 per lot of inventory, so 20 lots short moves it up 1. Widening to 48 at 52 also stops the buying, but it pushes sellers away too, so the short stays on your book and you earn less while you wait.

What is the position telling you to do?

A fruit seller who has run short of mangoes by mid-morning does not shut the stall; she raises the price she pays suppliers and nudges up the price to customers, so more mangoes arrive and fewer leave. Inventory changes what you want to trade next, not what the contract is worth: with fair value still 50, a short of 20 lots means your next trade should be a buy, so the quote should lean towards buying. You have used 80% of the risk limit, and five more lifts would put you at it. The question is testing whether you separate the two numbers a market maker carries: the fair value, which has not moved, and the price at which you want to trade, which has.

How far should you skew, and what does it cost?

A common rule moves the mid in proportion to the position: here 0.05 per lot, so 20 lots short lifts the mid by 1 and the quote becomes 50 at 52. The bid now sits at fair value, so you buy back with no edge, and the offer is 2 above fair, so a buyer who still lifts it pays you well for adding to the risk. Skewing trades some edge for risk reduction, and the closer the position is to the limit, the more edge you should be willing to give up. In a toy model where a quote d away from fair trades with probability 0.6 x e to the minus d each period, the unchanged quote earns 0.44 a period and never reduces the short; the full skew earns 0.16 but buys back a net 0.52 lots a period, about 39 periods to flatten; the half skew sits between at 0.38 and 0.23. The numbers are illustrative; the direction is not.

Why not simply widen?

Widening to 48 at 52 makes the offer as unattractive as the skew does, but it also moves the bid two points below fair, so sellers go elsewhere. In the toy model both sides fill 0.08 of the time and the expected change in the position is zero: a wider quote stops new risk arriving but does nothing about the risk you already hold, and the 20-lot short keeps moving with the market while you wait. Widening has its place, when you think fair value is uncertain or that the people lifting you know something, because then both sides are dangerous. Say that distinction, and then say the last resort: if skewing does not bring sellers fast enough, hedge the short in a related market, or cross the spread and buy, rather than drift up to the limit.

Fair value is about 3.99, so quote something like 3.9 bid, 4.1 offer. A given face is absent from all six rolls with probability (5/6) to the sixth, about 0.335, so it appears at least once with probability 0.665. The number of distinct faces is the sum of six such indicators, and linearity of expectation gives 6 x 0.665 = 3.99, without listing a single case.

Why does linearity let you skip the cases?

Six friends each toss a letter into one of six boxes at random, and you want the expected number of boxes that end up non-empty. Counting the ways the boxes can fill is a mess; asking each box whether it got anything is easy. The number of distinct faces is one plus one plus one over the six faces, each one counting if that face appeared, and the expectation of a sum is the sum of the expectations even though the six events overlap. Each face is missed on every roll with chance (5/6) to the sixth, so the expected count is 6 times one minus that, about 3.99.

How wide should the market be, and how do you defend it?

Width comes from how uncertain the outcome is and how much the other side may know. The exact distribution puts about 50% of the mass on four faces, 23% on three and 23% on five, with a standard deviation of 0.78, so a market 0.2 wide around 3.99 is tight relative to the noise but still symmetric around fair. If the interviewer lifts your offer at 4.1, you are short at a price above fair and should keep the quote where it is; if they lift it twice, ask yourself whether they know something, such as that the die is loaded, and move the market up rather than argue with the flow.

What is the check, and what changes with more rolls?

Check the ends. With one roll there is exactly one distinct face and the formula gives 6(1 - 5/6) = 1. With many rolls the missed chance collapses and the expectation approaches 6. At six rolls you are at 3.99, meaning two faces are typically missing, which surprises people who expect six rolls to nearly cover six faces. The limitation of the quote is that it prices only the mean; a counterparty who wants to bet on exactly six distinct faces needs a different market, and that chance is only 6!/6^6, about 1.5%.

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.