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Stochastic Processes Quant Interview Prep & Practice

Random walks, Markov chains and stopping rules are standard in quant research and harder trading interviews. These questions test the tools for each.

Sample Stochastic Processes Questions

  1. Question 1 · SIG Problem-Solving Assessment

    A fair 3-sided die is rolled repeatedly. On average, how many rolls are needed before every face has appeared at least once?

    See the solution ↓

  2. Question 2 · Five Rings Quant Trading Screen

    A price moves up or down $0.25 on each of 723 independent ticks, 50/50. What is the expected absolute distance from where it started, in dollars?

    See the solution ↓

  3. Question 3 · Old Mission Quant Screen

    A machine is On or Off each day. From On it goes to Off with probability 17%, and from Off it goes to On with probability 18%. It is On today. What is the chance it is On two days from now?

    • 51.4%
    • 68.9%
    • 71.9%
    • 83%
    See the solution ↓

  4. Question 4 · Belvedere Technical Challenge

    A price moves up or down one tick on each of 8 independent ticks, 50/50. What is the chance it finishes where it started, as a percentage?

    See the solution ↓

  5. Question 5 · IMC Assessment

    A fair 6-sided die is rolled. You may keep that number or reroll once and take whatever comes up. Played properly, what is this worth?

    See the solution ↓

Step-by-Step Solutions

  1. Solution 1

    A fair 3-sided die is rolled repeatedly. On average, how many rolls are needed before every face has appeared at least once?

    Answer: 5.5

    Once i faces are seen, the wait for a new one is geometric with mean 3/(3-i). Summing: 3(1 + 1/2 + ... + 1/3) = 5.5.

  2. Solution 2

    A price moves up or down $0.25 on each of 723 independent ticks, 50/50. What is the expected absolute distance from where it started, in dollars?

    Answer: 5.36

    After 723 ticks the position is roughly normal with standard deviation 0.25 × √723 = 6.72. E|X| = σ√(2/π) ≈ 0.8σ ≈ 5.36.

  3. Solution 3

    A machine is On or Off each day. From On it goes to Off with probability 17%, and from Off it goes to On with probability 18%. It is On today. What is the chance it is On two days from now?

    Answer: 71.9%

    Two routes back to On: stay, stay at (1 − 0.17)² = 0.689, or leave and return at 0.17 × 0.18 = 0.031. Together 71.9%.

  4. Solution 4

    A price moves up or down one tick on each of 8 independent ticks, 50/50. What is the chance it finishes where it started, as a percentage?

    Answer: 27.3

    Finishing flat needs exactly 4 up-ticks out of 8: C(8,4) = 70 of the 256 paths, so 27.3%. The chance falls like 1/√8 as the walk lengthens, which is why a long walk almost never ends exactly at the start.

  5. Solution 5

    A fair 6-sided die is rolled. You may keep that number or reroll once and take whatever comes up. Played properly, what is this worth?

    Answer: 4.25

    A reroll is worth the plain mean, 3.5, so keep anything above 3 and reroll otherwise. That is (15 + 3 × 3.5) ÷ 6 = 4.25.

How to Pass the Stochastic Processes Screen

For a chain, write the transition matrix and the stationary equations before reaching for anything cleverer.

For hitting and ruin problems, condition on the first step. One equation per state usually solves it.

For a stopping rule, ask whether optional stopping applies. When it does, a martingale turns a hard expectation into a one-line answer.

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