PRACTICE GUIDE CITADEL SECURITIES
Citadel Securities Probability & Stochastic Processes Practice Test
The Citadel Securities probability & stochastic processes screen is 14 questions in 22 minutes - about 1.6 minutes each - answered by typing the number, with no calculator. There is no negative marking, so leaving an item blank gains you nothing over guessing.
Where it sits: Research-track screen. The practice sitting on this page runs the same item count, the same clock and the same marking rule, with questions generated by Outcry rather than taken from Citadel Securities.
Outcry is not affiliated with Citadel Securities and has no access to their assessment content. This guide describes an assessment format that candidates report publicly; the questions here are generated by Outcry and are not Citadel Securities’s own.
What it screens
A market maker quoting across equities, options, fixed income and ETFs globally.
- ✓Martingale arguments and optional stopping applied to games
- ✓Gambler's ruin: absorption probabilities and expected duration
- ✓Recognising when a process is a martingale (and when the argument breaks)
- ✓Random-walk intuition: recurrence, drift, and scaling
Where it sits at Citadel Securities
This section does not arrive on its own. It opens the paper, and the sitting around it runs 40 minutes and 26 items in total, with Inference & research integrity on the same continuous clock. Candidates who prepare only for this round tend to be caught by the pacing of the rest.
There is no going back. You answer or skip, and the next item loads, which means the decision to leave an item is final at the moment you make it.
The format
These are the numbers the Citadel Securities sitting on this site runs on, matching the format candidates report.
| Questions | 14 |
|---|---|
| Time | 22 minutes |
| Per question | 1.6 minutes |
| Negative marking | No |
| Answer style | Typed numeric answer, no calculator |
| Where it sits | Research-track screen |
What it tests, with a worked example
Every example below is generated by Outcry, drawn from the same question generators the timed drills run. None of them is Citadel Securities’s.
Markov chains
Transition matrices, absorbing states, and expected time to absorption.
Example
You draw 5 values independently and uniformly from [0, 1]. What is the expected value of the maximum?
- 1/5
- 4/5
- 1/2
- 5/6
Answer 5/6
For the maximum of n iid Uniform(0,1) draws, E[max] = n/(n+1) = 5/6.
Random walks and martingales
Stopping times, the optional stopping theorem, and gambler's ruin.
Example
5 strategies each have return variance 9, and every pair has correlation 0.2. What is the variance of the sum of all 5 returns?
- 45
- 81
- 225
- 52.2
Answer 81
Var(ΣX) = Σ Var(X) + Σ over ordered pairs Cov(X,Y). The variances give 5 × 9 = 45. There are 5 × 4 = 20 ordered pairs, each contributing 0.2 × 9 = 1.8, so the covariance term is 36. Total = 45 + 36 = 81. Stopping at 45 is the answer only when the strategies are independent.
Stating assumptions
Research screens mark the assumption you named as much as the number you produced.
Conditional expectation
Tower property and the expectation of a stopped process, which is most of what these rounds ask.
Example
Regime Prob Mean Variance Calm 0.6 2 1 Stressed 0.4 4 25
Daily P&L runs in one of two regimes, drawn fresh each day from the table above. What is the variance of the daily P&L?
- 14.6
- 10.6
- 11.56
- 30
Answer 11.56
Law of total variance. The within-regime part is E[Var(P | regime)] = 0.6×1 + 0.4×25 = 10.6. The regime means differ by 2, so the between-regime part is Var(E[P | regime]) = p(1−p)(Δmean)² = 0.6×0.4×4 = 0.96. Total = 10.6 + 0.96 = 11.56. Averaging the two variances and stopping there drops the 0.96 that comes from the regimes sitting at different levels.
Overfitting and multiple testing
Why a result at the five per cent level means very little after the twentieth test.
Example
A regression of returns on a signal gives t = 6.2 but R² = 0.004. What is the correct reading?
- The relationship is reliably non-zero but explains almost none of the variation
- R² of 0.004 means the t-statistic must be wrong
- The signal explains 40% of returns
- The model is misspecified and t is invalid
Answer The relationship is reliably non-zero but explains almost none of the variation
Statistical significance and explanatory power are different questions. With enough observations a tiny effect is measurable - and in trading a tiny reliable edge can still be valuable.
Variance decomposition
Signal against noise, and how much of a measured edge should be believed.
Example
An A/B test compares two equally sized groups, at the 5% two-sided level, with 80% power. The effect worth detecting is 0.25 standard deviations. How many observations are needed PER group?
Answer 252
n per group = 2(z_{α/2} + z_β)²/d² = 2(1.96 + 0.84)²/0.25² = 2 × 7.85/0.06 ≈ 252. The argument: the difference in means has standard error σ√(2/n), and the test detects the effect when the true difference sits far enough above the critical value that only 20% of draws fall short. That needs the effect to span z_{α/2} + z_β = 2.8 standard errors, not just 1.96 - the second term is what buys the power, and forgetting it is the most common way a study gets sized at half what it needs. The 1/d² is the part worth internalising: halving the effect you want to catch quadruples the sample. A 0.25 effect is 252 per arm; a 0.13 effect would be 1,008.
What a good score looks like
On a paper of 14 questions with no penalty for a wrong answer, the only thing an unanswered question can do is cost you. Research screens are marked by a person more often than a machine, so a numeric cutoff is rarely visible. Candidates consistently report that a wrong answer with a stated assumption scores better than a right answer with none.
How to train for it
- 01Rebuild the standard results rather than memorising them - a two-state chain, gambler's ruin, the regression slope from correlation and standard deviations.
- 02Practise writing the assumption down first. It is the part these screens grade and the part people skip.
- 03Get comfortable with conditional expectation specifically. More of these rounds reduce to the tower property than to anything else.
TRAIN IT HERE
The drills that match each section
Martingale Mutiny
Optional stopping played as a game. Try to find a betting strategy that beats a fair coin.
Ruin Walker
Gambler's ruin with live absorption probabilities.
Distribution Lab
Recognise a distribution from how its samples behave.
SIT THE FULL BATTERY
All the sections back to back on one clock, marked the way the real screen marks them, with a by-skill breakdown at the end. Included with any pass.
Also reported at Citadel Securities
Common questions
- Is the Citadel Securities probability & stochastic processes test multiple choice?
- Typed numeric answer, no calculator. You type the number, so there is nothing to eliminate your way to.
- How long is the Citadel Securities probability & stochastic processes test?
- 14 questions in 22 minutes, which is about 1.6 minutes each.
- Is there negative marking on the Citadel Securities probability & stochastic processes test?
- No. A wrong answer costs nothing beyond the mark you would have earned, so leaving an item blank is never better than guessing at it.
- How do I practise for it free?
- Every drill linked on this page is free to play, with no account, inside a daily run cap. Questions are generated fresh each run, so there is nothing to memorise between attempts. The full Quant Research Screen sitting puts the sections back to back on one clock.