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Practice · Quant prep · Expected value & market making

Expected value & market making interview questions

34 questions with worked solutions. Fair value, betting and Kelly, options intuition and quoting against informed flow.

18 easy · 16 medium · 0 hard · Question type reported at: SIGOptiverIMC

Practise timed: 30 questions · 75s each All Quant prep topics

1 Easy Type asked atSIGOptiverIMC
You are asked to make a market on the sum of two dice. What is its fair value?
Show answer
Answer: 77
E[X+Y]=3.5+3.5\mathbb E[X+Y]=3.5+3.5. Quote around it, for example 6.5 at 7.5, and explain the width.
2 Easy Type asked atSIGOptiverIMC
Make a market on the product of two dice. What is the fair value?
Show answer
Answer: 12.2512.25
Independent dice: 3.5×3.53.5\times3.5.
3 Medium Type asked atSIGOptiverIMC
Make a market on the maximum of two dice. What is the fair value?
Show answer
Answer: ≈4.47\approx4.47
∑kk⋅2k−136=16136\sum_k k\cdot\tfrac{2k-1}{36}=\tfrac{161}{36}.
4 Easy Type asked atSIGOptiverIMC
Make a market on the number of heads in 10 fair tosses. What is the fair value?
Show answer
Answer: 55
Binomial mean np=10⋅12np=10\cdot\tfrac12.
5 Easy Type asked atSIGOptiverIMC
You pay £1 to roll a die and receive its face value in pounds. What is your expected profit?
Show answer
Answer: £2.502.50
3.5−13.5-1.
6 Easy Type asked atSIGOptiverIMC
A coin is tossed until the first tail; you earn £1 for every head. What is the expected payout?
Show answer
Answer: £11
The number of heads is geometric starting at 0 with p=12p=\tfrac12, mean 1−pp=1\tfrac{1-p}{p}=1.
7 Medium Type asked atSIGOptiverIMC
Cards are dealt from a shuffled deck until the first ace appears. What is the expected position of that ace?
Show answer
Answer: 10.610.6
The 4 aces split the 48 other cards into 5 gaps of expected size 485=9.6\tfrac{48}{5}=9.6, so the first ace is at 9.6+19.6+1.
8 Medium Type asked atSIGOptiverIMC
Under the Kelly criterion, what fraction of your bankroll should you stake on an even-money bet you win with probability 0.60.6?
Show answer
Answer: 20%20\%
For even odds, f∗=2p−1f^*=2p-1.
9 Medium Type asked atSIGOptiverIMC
A bet pays 2 to 1 and you win with probability 0.550.55. What is the Kelly fraction?
Show answer
Answer: 32.5%32.5\%
f∗=bp−qb=2(0.55)−0.452f^*=\dfrac{bp-q}{b}=\dfrac{2(0.55)-0.45}{2}.
10 Easy Type asked atSIGOptiverIMC
An event has probability 0.20.2. What are the fair odds against it, xx to 1?
Show answer
Answer: 44 to 11
Odds against are 1−pp=0.80.2\tfrac{1-p}{p}=\tfrac{0.8}{0.2}.
11 Easy Type asked atSIGOptiverIMC
A stock will be 90, 100, 110 or 120 with equal probability. What is the value of a call struck at 100 (zero rates)?
Show answer
Answer: 7.57.5
Payoffs 0,0,10,200,0,10,20; average 7.57.5.
12 Medium Type asked atSIGOptiverIMC
With zero interest rates, S=102S=102, K=100K=100 and the call is worth 5. What is the put worth?
Show answer
Answer: 33
Put–call parity C−P=S−KC-P=S-K gives P=5−2P=5-2.
13 Medium Type asked atSIGOptiverIMC
You quote a market and keep getting lifted on your offer. What is the most sensible first reaction?
  1. Keep the quote; the fair value hasn't changed
  2. Move your quotes up and ask what the buyer might know
  3. Widen only the bid
  4. Stop quoting entirely
Show answer
Answer: Move up; consider adverse selection
Persistent one-way flow is information. Repricing protects you from a better-informed counterparty.
14 Easy Type asked atSIGOptiverIMC
A straddle struck at 100 costs 8 in total. What is its upper break-even price at expiry?
Show answer
Answer: 108108
The payoff ∣S−100∣|S-100| must cover the premium of 8.
15 Easy Type asked atSIGOptiverIMC
Roughly what is the delta of an at-the-money call option?
  1. 00
  2. about 0.50.5
  3. 11
  4. −0.5-0.5
Show answer
Answer: About 0.5
Roughly the probability of finishing in the money, a little above 12\tfrac12 because of drift under the pricing measure.
16 Medium Type asked atSIGOptiverIMC
Two fair coins are tossed and you receive the square of the number of heads. What is the expected payout?
Show answer
Answer: 1.51.5
E[H2]=Var⁡H+(EH)2=12+1\mathbb E[H^2]=\operatorname{Var}H+(\mathbb EH)^2=\tfrac12+1.
17 Medium Type asked atSIGOptiverIMC
Three dice are rolled. What is the expected value of the highest?
Show answer
Answer: ≈4.96\approx4.96
E[max⁡]=∑k=16P(max⁡≥k)=6−∑k=15(k6)3=6−225216\mathbb E[\max]=\sum_{k=1}^6P(\max\ge k)=6-\sum_{k=1}^{5}\left(\tfrac k6\right)^3=6-\tfrac{225}{216}.
18 Easy Type asked atSIGOptiverIMC
You buy 100 shares at 50.10 and sell them at 50.20. What is your P&L in pounds (ignoring costs)?
Show answer
Answer: £1010
100×0.10100\times0.10.
19 Easy Type asked atSIGOptiverIMC
A digital option pays £1 if a fair die shows 5 or more. What is it worth?
Show answer
Answer: £13\tfrac13
Two favourable faces out of six.
20 Medium Type asked atSIGOptiverIMC
The "double after every loss" roulette strategy, with a finite bankroll, has expected profit:
  1. positive
  2. zero on a fair game (negative with a house edge)
  3. infinite
  4. guaranteed small and positive
Show answer
Answer: Zero (negative with an edge)
Optional stopping: no bounded strategy turns a fair game into a favourable one. Frequent small wins are paid for by rare ruin.
21 Easy Type asked atSIGOptiverIMC
A policy covers a loss of £5000 that happens with probability 1%. What is the actuarially fair premium?
Show answer
Answer: £5050
0.01×50000.01\times5000.
22 Easy Type asked atSIGOptiverIMC
A bet pays 3 to 1. What probability of winning makes it break even?
Show answer
Answer: 0.250.25
3p=1−p3p=1-p.
23 Easy Type asked atSIGOptiverIMC
You stake £100 at even money on a coin that lands heads 60% of the time. What is your expected profit?
Show answer
Answer: £2020
0.6⋅100−0.4⋅1000.6\cdot100-0.4\cdot100.
24 Medium Type asked atSIGOptiverIMC
Daily volatility is 1%. Roughly what is annual volatility over 252 trading days, in %?
Show answer
Answer: ≈15.9%\approx15.9\%
Variance adds over independent days, so volatility scales with 252\sqrt{252}.
25 Medium Type asked atSIGOptiverIMC
A game pays £10 on a six and nothing otherwise, and a market maker quotes 1.40 at 1.80 on it. At which price do you trade, and do you buy or sell?
  1. Buy at 1.80
  2. Sell at 1.40
  3. Buy at 1.40
  4. Don't trade
Show answer
Answer: Don't trade
Fair value is 106≈1.67\tfrac{10}{6}\approx1.67, inside the market. Buying at the 1.80 offer overpays and selling at the 1.40 bid undersells, so neither trade has positive expectation. (You cannot buy at the bid.)
26 Easy Type asked atSIGOptiverIMC
A die is rolled once and you are paid £10 if it shows a six. What is the fair value of the bet, in pounds?
Show answer
Answer: £1.67
16×10\tfrac16\times10.
27 Easy Type asked atSIGOptiverIMC
A market is 99.98 bid, 100.02 offered. What is the round-trip cost of buying and immediately selling 500 units, in pounds?
Show answer
Answer: £20
You cross the spread of 0.04 once: 0.04×500=200.04\times500=20.
28 Medium Type asked atSIGOptiverIMC
Roughly what is the delta of an at-the-money call with a short time to expiry?
  1. 0
  2. 0.25
  3. 0.5
  4. 0.75
  5. 1
Show answer
Answer: C. 0.5
The option is about as likely to finish in the money as out, so the delta sits near a half. It moves towards 1 as the option goes deeper in the money.
29 Medium Type asked atSIGOptiverIMC
With interest rates at zero, a one-year call struck at 100 on a stock trading at 100 is worth 8. What is the put worth?
Show answer
Answer: 88
Put–call parity: C−P=S−Ke−rT=0C-P=S-Ke^{-rT}=0 when S=KS=K and r=0r=0.
30 Medium Type asked atSIGOptiverIMC
Under the Kelly criterion, what fraction of your bankroll should you stake on an even-money bet you win with probability 0.55?
Show answer
Answer: 0.10.1
For even money, f∗=2p−1=0.10f^*=2p-1=0.10.
31 Easy Type asked atSIGOptiverIMC
You quote 99 bid, 101 offered in a contract and are lifted twice for one lot each. What is your position?
  1. Long 2
  2. Long 1
  3. Flat
  4. Short 1
  5. Short 2
Show answer
Answer: E. Short 2
Being lifted means someone bought at your offer, so you sold: two lots short, at 101.
32 Medium Type asked atSIGOptiverIMC
A bookmaker offers decimal odds of 1.9 on both sides of a fair coin toss. What is the overround, as a percentage?
Show answer
Answer: ≈5.26%\approx5.26\%
Implied probabilities are 11.9\tfrac1{1.9} each, totalling 21.9=1.0526\tfrac2{1.9}=1.0526: 5.26% above certainty. That margin is the bookmaker's edge.
33 Easy Type asked atSIGOptiverIMC
A bet wins or loses £1 with equal probability. What is the variance of the outcome?
Show answer
Answer: 11
The mean is 0 and the square of the outcome is always 1, so the variance is 1.
34 Medium Type asked atSIGOptiverIMC
A stock is 50. You are long 200 shares and short 4 call options, each on 100 shares, with delta 0.4. What is your net delta, in shares?
Show answer
Answer: 4040
Short calls give −4×100×0.4=−160-4\times100\times0.4=-160 deltas, so the net is 200−160=40200-160=40 shares.

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