---
title: "The Final Round and Trading Games"
book: "The Interview Book"
subject: quant
language: en
chapter: 6
exercises: 0
source: https://one-course.com/books/quant/18/en/chapter/6-the-final-round-and-trading-games
---

# Chapter 6 — The Final Round and Trading Games

Six candidates sit around a table with chips and a deck of cards; the interviewer asks for a market on the sum of the next three cards. One quotes 17 at 23 and never moves it; one tightens after every trade and is picked off twice by the same player; one says “I am lifting the offer because two high cards are gone.” The assessor behind them writes down three different things, none of which is the chip count. A [final round](#def-iv-the-final-round-and-trading-games-final) is the most expensive stage of the funnel, and each of its parts records something specific.

## 6.1 The final round

**Definition 6.1 (Final round, superday).**

The *final round* is the last stage of an [interview loop](https://one-course.com/books/quant/18/en/chapter/1-what-each-interview-tests-by-role#def-iv-what-each-interview-tests-by-role-loop) before the decision: several interviews on the same day or over a few days, each measuring different abilities, often with a game, a [group exercise](#def-iv-the-final-round-and-trading-games-group) or a presentation. A *superday* is the name some banks give to a final round held on a single day with several interviewers ([Box 2.1](https://one-course.com/books/quant/18/en/chapter/2-the-process-by-firm-type#dat-iv-the-process-by-firm-type-published)).

The [final round](#def-iv-the-final-round-and-trading-games-final) is designed so that each interview measures a different part of the role’s analysis ([Chapter 1](https://one-course.com/books/quant/18/en/chapter/1-what-each-interview-tests-by-role#ch-iv-what-each-interview-tests-by-role)), and its decision is taken by a committee or a lead from the written scores. Two consequences follow for the candidate. The day is long, and the fourth interview counts as much as the first: energy is a resource to plan. And no single interview has to be perfect; a weak interview among strong ones is usually recoverable, whereas a pattern (no checks, no numbers, no questions) is not.

**Method 6.2 (Planning a final-round day).**

1. Ask in advance for the schedule: how many interviews, of what kinds, with whom, and whether there is a game or a presentation.
2. Prepare one-minute answers for the three questions every interviewer may ask: your background, why this role, and one piece of work you know deeply.
3. Treat each interview as a fresh start; do not carry a bad one into the next room.
4. Eat and drink between interviews, even when told there is no time.
5. Write down, after each interview, the questions asked: the debrief with the recruiter goes better with them, and so does the next process.

## 6.2 Mock trading and betting games

**Definition 6.3 (Trading game).**

A *trading game* is an interview exercise in which candidates make markets on, bet on or trade an uncertain quantity whose distribution can be reasoned about (dice, cards, an estimation question), against the interviewer or each other, while information is revealed.

The mechanics of making a market, its width and centre, and updating on trades, are in One Quant Book 2, chapter 30, and the betting and sizing arithmetic in its chapter 29; this book’s [Chapter 13](https://one-course.com/books/quant/18/en/chapter/13-betting-and-market-making-games#ch-iv-betting-and-market-making-games) is the question bank for them. What matters in a [final round](#def-iv-the-final-round-and-trading-games-final) is what the assessor records, which is behaviour: whether the candidate’s price is near fair value and moves when information arrives, whether the width reflects the uncertainty, whether the candidate notices being traded against by someone who knows more, and whether positions and risk are tracked.

**Example 6.4 (The three-card market).**

Cards count 1 (ace) to 13 (king), four of each, 52 in all. The sum of three cards drawn without replacement has mean $3 \times 7 = 21$ and variance $3 \times 14 \times \tfrac{49}{51} \approx 40.4$ (each card has variance $(13^2 - 1)/12 = 14$, reduced by the finite-population factor), so a standard deviation of about 6.35; an exact enumeration puts 90% of the outcomes between 10 and 32. A market of 19 at 23 is centred and about a third of a standard deviation wide on each side. When a card is shown, the fair value becomes that card plus twice the mean of the 51 remaining cards: a king moves it to $13 + 2 \times 351/51 \approx 26.76$. [Figure 6.1](#fig-iv-the-final-round-and-trading-games-game) follows one game’s fair value as its cards appear.

![One three-card game: the fair value of the sum (dots) and the range holding 90% of the outcomes (bars) after 0, 1, 2 and 3 cards are shown. The cards were 6, 3 and 7 (a seeded draw). Exact enumeration; data: fig_iv_game.py.](https://one-course.com/images/onecourse/chapters/quant-18/iv-the-final-round-and-trading-games/fig-bc993e232fec.svg)

***Figure 6.1.** One three-card game: the fair value of the sum (dots) and the range holding 90% of the outcomes (bars) after 0, 1, 2 and 3 cards are shown. The cards were 6, 3 and 7 (a seeded draw). Exact enumeration; data: `fig_iv_game.py`.*

**Remark 6.5 (What the three candidates of the hook showed).**

A fixed market of 17 at 23 is centred at 20, below the fair 21, and never moves: it shows no updating. A market tightened after every trade shows that the candidate treats trades as noise, whereas a player who keeps trading on the same side is telling the market something (the adverse selection of One Quant Book 1, chapter 1). Lifting an offer because two high cards are gone is conditional reasoning spoken aloud, which is what the assessor came to see.

## 6.3 Group exercises and presentations

**Definition 6.6 (Group exercise).**

A *group exercise* is a final-round task given to several candidates at once (an estimation, a case, a trading simulation with teams) and observed by assessors who score each candidate’s contribution, not the group’s result.

[Group exercises](#def-iv-the-final-round-and-trading-games-group) come from the assessment-centre tradition of personnel selection, in which several exercises are scored on a few dimensions by trained assessors. A meta-analysis of 34 studies reduced the dimensions to six (consideration of others, communication, drive, influencing others, organising and planning, problem solving) and found criterion-related validities of 0.25 to 0.39 for them (Arthur, Day, McNelly and Edens, 2003). The candidate’s lesson is that the group’s answer is not scored; the candidate’s part in reaching it is. Talking most is not the same as contributing: structuring the problem, bringing a number, asking the quiet member for theirs, and summarising the decision all score.

A presentation, often of the candidate’s own past work or of a case prepared in the day, is scored on structure, clarity and the response to challenge. The challenge is the point: an assessor who says “your sample is too small” wants to see whether the candidate can quantify the concern, concede what is true and defend what is not.

## 6.4 What assessors write down

The notes an assessor writes are short and concrete: the answer given, the time taken, whether a check was made, how the candidate responded to new information, and one or two quotations. They are written against the rubric’s dimensions, and the debrief afterwards compares them across assessors before the decision. A candidate can ask for feedback after the decision; many firms give a little, and what they give is usually a paraphrase of those notes.

## 6.5 Worked answers

**Example 6.7 (A second game: how many suits).**

*“Four cards will be dealt from a full deck. Make me a market on the number of different suits among them.”* The value is 1, 2, 3 or 4. By indicators ([Chapter 11](https://one-course.com/books/quant/18/en/chapter/11-probability-ii#ch-iv-probability-ii)), each suit is absent with probability $\binom{39}{4}/\binom{52}{4} \approx 0.304$, so the expected number of suits is $4 \times (1 - 0.304) \approx 2.78$; all four suits appear with probability $13^4/\binom{52}{4} \approx 0.11$. A market of 2.5 at 3.0 is centred. The first card is shown and is a heart: by symmetry nothing changes, since some suit had to come first, and the expectation stays at 2.78 (computed from the three remaining cards: $1 + 3(1 - \binom{38}{3}/\binom{51}{3})$). The second card is also a heart: now $1 + 3(1 - \binom{37}{2}/\binom{50}{2}) \approx 2.37$, and the market moves down to about 2.1 at 2.6. The assessor scores the candidate who says aloud why the first card changed nothing and the second changed a lot.

**Example 6.8 (Answering “your sample is too small”).**

A candidate presents a strategy with an annualised Sharpe ratio of 1.5 over eighteen months of daily returns, and the assessor says the sample is too small. The scoring response quantifies the objection before answering it: “Measured on $T$ years, a Sharpe ratio is uncertain by roughly $\sqrt{(1 + \mathrm{SR}^2/2)/T}$, here $\sqrt{2.125/1.5} \approx 1.19$. So 1.5 is only about 1.3 standard errors from zero: you are right that eighteen months cannot establish it. What I can defend is the mechanism, which I tested on the other markets in the appendix, and what I’d want is a live test sized to find out.” Conceding the true part with a number and defending the rest is what the assessor writes down ([Chapter 14](https://one-course.com/books/quant/18/en/chapter/14-statistics#ch-iv-statistics)).

## 6.6 Question bank

**Interview question 6.1 ★ trader • market maker.**

Make me a market on the sum of three cards drawn from a full deck, counting ace as 1 and king as 13. Give your centre and your width, and say why.

**Solution of Interview question 6.1.**

Centre 21: each card averages 7. Width: the sum’s standard deviation is $\sqrt{3 \times 14 \times 49/51}
\approx 6.35$; a market of 19 at 23 (two points each side, a third of a standard deviation) is tight enough to trade and wide enough to absorb an informed trade or two. Say that you will move it as cards appear or as trades come in.

*What the interviewer is looking for: a centred price from the mean, a width tied to the spread, and a stated plan to update.*

**Interview question 6.2 ★ trader, researcher, developer • bank.**

You have a [superday](#def-iv-the-final-round-and-trading-games-final) of five interviews tomorrow, from 9:00 to 15:00. What do you do tonight, and how do you manage the day itself?

**Solution of Interview question 6.2.**

Tonight: confirm the schedule and the interview types, prepare the three one-minute answers (background, why this role, one piece of work in depth), revise the core families for the role, not new topics, and sleep. Tomorrow: arrive early, eat and drink between interviews, treat each interview as independent, note the questions after each one, and keep two good questions for each interviewer. If one interview goes badly, say nothing about it in the next room.

*What the interviewer is looking for: preparation that matches the day’s structure and management of energy across it.*

**Interview question 6.3 ★ trader • market maker.**

In the three-card market, the first card is turned over and it is a king. Where is fair value now?

**Solution of Interview question 6.3.**

The king is one of the three; the other two are drawn from 51 cards whose values sum to $364 - 13 = 351$, mean $351/51 \approx 6.88$. Fair value $13 + 2 \times 6.88 \approx 26.76$.

*What the interviewer is looking for: conditioning on the revealed card and removing it from the deck.*

**Interview question 6.4 ★★ trader • market maker.**

You quote 19 at 23 on the three-card sum. The interviewer has been shown one of the three cards and buys at 23. If the interviewer buys whenever that card is above 7, what is the fair value given the trade, and how do you move your market?

**Solution of Interview question 6.4.**

Given the rule, the interviewer’s card is uniform on 8 to 13 (mean 10.5), and the other two cards average $(364 - c)/51$ each. Averaging $c + 2(364 - c)/51$ over $c = 8, \dots, 13$ gives about 24.36. The buy was informative: raise the market to around 23 at 27 or tighter around 24.4, and be wary of selling more at 23 to the same player.

*What the interviewer is looking for: a Bayesian update from the counterparty’s trading rule, not from the trade price alone.*

**Interview question 6.5 ★★ trader, risk • proprietary firm.**

You start a betting game with 100 chips. Ten times, you may bet any part of your chips at even money on a coin that lands heads with probability 0.6. What fraction do you bet each time if you want to maximise the expected logarithm of your final chips? What is the expected growth per bet, and what does it imply for the typical final stack?

**Solution of Interview question 6.5.**

The Kelly fraction at even money is $2p - 1 = 0.2$: bet 20% of current chips each time (One Quant Book 2, chapter 29). The expected log growth per bet is $0.6\ln 1.2 + 0.4\ln 0.8 \approx 0.0201$, so after ten bets the median outcome is about $100e^{0.201} \approx 122$ chips; betting 10% or 30% grows more slowly. The mean final stack is higher than the median and larger bets raise it, at the price of ruinous paths.

*What the interviewer is looking for: the Kelly fraction, the log-growth rate and the difference between mean and median wealth.*

**Interview question 6.6 ★★ trader, researcher • market maker.**

In a [group exercise](#def-iv-the-final-round-and-trading-games-group), five candidates must estimate the number of trading days on which a large equity index moved more than 2% over the last twenty years, and agree one answer in fifteen minutes. What role do you take, and what do you actually say in the first two minutes?

**Solution of Interview question 6.6.**

Take the structuring role without taking over: “Twenty years is about 5 000 trading days. Daily volatility is about 1% in calm years and 2 to 3% in crises; if returns were normal with 1.2% daily volatility, a 2% move is a 1.7-sigma event in either direction, about 10% of days, but fat tails and volatility clustering change that. Can we each give a number and a reason in thirty seconds, then combine?” Then ask the quietest member for theirs, write the numbers down, and summarise the decision at the end. The assessors score the structure, the number and the inclusion, not whether the group’s final answer is right.

*What the interviewer is looking for: structure, a first quantitative anchor, inclusion of others and a clear close.*

**Interview question 6.7 ★★ trader • market maker.**

You are long three contracts on the three-card sum at an average price of 24. Two cards are shown: 5 and 6. What is your expected P&L? Someone bids 18. Do you sell?

**Solution of Interview question 6.7.**

With 5 and 6 shown, the third card is one of the 50 remaining, whose values sum to $364 - 11 = 353$, mean $7.06$: fair value $11 + 7.06 = 18.06$. The position is worth $3 \times (18.06 - 24) \approx -17.82$ in expectation. A bid of 18 is 0.06 below fair: selling costs about 0.06 a contract in expectation and removes the remaining variance (one card, standard deviation about 3.8). Sell if the position is large against your limits or the game rewards risk management; otherwise holding is marginally better. Say both.

*What the interviewer is looking for: the conditional fair value, mark-to-market, and a risk-aware decision stated with its trade-off.*

**Interview question 6.8 ★★★ trader, risk • market maker.**

You make a market centred at 21 on the three-card sum. A player who has seen one of the three cards trades whenever your bid or offer is wrong in their favour. What is your expected loss per trade opportunity with a half-width of 0? Of 2? What half-width stops this player from ever trading?

**Solution of Interview question 6.8.**

The informed player’s fair value is $F(c) = c + 2(364 - c)/51$, between $15.24$ (an ace) and $26.76$ (a king). With a market of half-width $h$ around 21 the player buys when $F(c) > 21 + h$ and sells when $F(c) < 21 -
h$, and your expected loss is the average of the amounts by which $F$ crosses your quotes: about 3.10 per opportunity at $h = 0$ and 1.43 at $h = 2$ (exact enumeration over the 13 card values). The player never trades once $h \ge 98/17 \approx 5.76$. The cost of informed flow falls quickly with width; the game is about choosing a width at which uninformed trades pay for the informed ones.

*What the interviewer is looking for: the informed player’s conditional values, the loss as a function of width, and the break-even width.*

**Interview question 6.9 ★★★ trader, researcher • proprietary firm.**

A deck of 30 red and 22 black cards is shuffled and turned over one card at a time. At any moment before the last card you may say “next”, and you win if the next card is red; if you never say it, the last card decides. What stopping rule maximises your chance of winning, and what is the chance?

**Solution of Interview question 6.9.**

Every rule wins with probability $30/52 = 15/26$. The proportion of red cards among those left is a martingale as the deck is turned over, and a stopping rule chooses a stopping time; the chance of winning is the proportion at the moment you stop, whose expectation is the initial proportion by the optional stopping theorem (the deck is finite, so the conditions hold). A backward induction over all counts of red and black remaining confirms it, and so does a simulation of a rule that waits for black cards to run ahead.

*What the interviewer is looking for: recognising a martingale, and not being seduced by rules that “wait for a good moment”.*

**Interview question 6.10 ★★★ trader, researcher • market maker.**

You are the assessor for the hook’s three candidates. Write the one-line note you would record for each, and say which of them you would pass and why.

**Solution of Interview question 6.10.**

First candidate: “Quoted 17 at 23 throughout; centre 1 below fair; never updated on cards or trades.” Second: “Tightened after each trade; sold three times to the same informed player; did not infer from flow.” Third: “Lifted the offer after two high cards were shown, citing the removal; tracked position; price near fair.” Pass the third. The first two may be strong in other interviews, but the game measures updating and awareness of informed flow, and on those dimensions the notes are specific and negative; a note that quotes numbers and behaviour is what a committee can use.

*What the interviewer is looking for: concrete, behavioural notes tied to the rubric, and a decision that follows from them.*

Sources and further reading

- W. Arthur, E. A. Day, T. L. McNelly and P. S. Edens, “A meta-analysis of the criterion-related validity of assessment center dimensions”, *Personnel Psychology* 56(1), 2003, 125–153.
- International Task Force on Assessment Center Guidelines, “Guidelines and ethical considerations for assessment center operations”, *International Journal of Selection and Assessment* 17(3), 2009, 243–253.
- J. L. Kelly, “A new interpretation of information rate”, *Bell System Technical Journal* 35(4), 1956, 917–926.
- One Quant Book 2, chapters 29–30 (expected value, Kelly, making markets); One Quant Book 4, chapter 1 (martingales and stopping).