Citadel Securities Interview Questions - Complete List

Quant Interview Questions

59 questions

You regress X1 against Y and obtain an R-squared value of 0.1. Then, you regress X2 against Y and obtain an R-squared value of 0.2. What is the maximum possible R-squared when regressing both X1 and X2 against Y together?

Quantitative Researcher Interview

Given a string, return all possible permutations of its characters as a list. You may use recursive backtracking or any relevant Python packages.

Quantitative Researcher Interview

Starting at the origin on a grid, you can move either up/down or left/right at each step. After N steps, how far can you be from the origin? Additionally, if there are two boundaries, what is the probability that you reach one boundary before the other?

Quantitative Researcher Interview

If you toss a fair coin N times, what is the probability that you will get an even number of heads? Follow-up: Now, what if you have N - 1 fair coins, and the Nth coin is biased and lands on heads with probability p?

Quantitative Researcher Interview

What happens if two features are highly collinear in Lasso regression?

Quantitative Researcher Interview

Explain the basics of logistic regression.

Quantitative Researcher Interview

What happens to the R-squared (R^2) value if you double the data in a regression analysis? What about the t-value?

Quantitative Researcher Interview

We have n switches, each with probability p_i of being on and 1-p_i of being off. What is the probability that C or more switches are on? What if C is a uniform random variable between 0 and n?

Quantitative Researcher Interview

Given an array representing the price of a stock on each day, and an integer k, determine the maximum profit you can achieve by completing at most k buy-sell transactions (you must sell before buying again).

Quantitative Researcher Interview

Given n balloons, each balloon is painted with a number on it represented by an array nums. You are to burst all the balloons. If you burst balloon i, you get nums[left] * nums[i] * nums[right] coins. Here, left and right are adjacent indices of i. After the burst, the left and right balloons become adjacent. Find the maximum coins you can collect by bursting the balloons wisely.

Quantitative Researcher Interview

Randomly generate points (x, y) on the circumference of a circle in two dimensions. What is Var(x)? What if you randomly generate points on the surface of a sphere instead?

Quantitative Researcher Interview

In ridge regression, what happens to the expected value of the estimate if you perturb your data matrix?

Quantitative Researcher Interview

What is the limit of x^x as x approaches 0?

Quantitative Researcher Interview

What is the difference between ordinary least squares (OLS) regression and Ridge regression?

Quantitative Researcher Interview

Implement a Python function to price an American option using backward induction.

Quantitative Researcher Interview

Design a random number generator that, given a list of probabilities corresponding to a discrete set of outcomes, outputs each outcome with its specified probability.

Software Engineer Interview

Implement a shared pointer class in C++ that manages shared ownership of a dynamically allocated object. Ensure proper reference counting and memory management.

Software Engineer Interview

Implement a shared_ptr smart pointer in C++ that manages reference counting and ensures proper deletion of the managed object when no references remain.

Software Engineer Interview

Given 25 horses and a race track where only 5 horses can race at a time, what is the minimum number of races required to determine the fastest horse?

Trader Interview

What is the square root of 800 rounded to two decimal places?

Trader Interview

Write code to generate the Fibonacci sequence.

Trader Interview

Given constraints on the correlations between three random variables, provide bounds for the correlation between one pair of them.

Trader Interview

Describe a program to determine whether a given integer is a prime number.

Trader Interview

What is the shape of the gamma curve for a call option?

Trader Interview

What is the house edge of a standard game of roulette?

Quantitative Trader Interview

What is the probability that a coin is biased (always lands heads), given that you observe five heads in a row?

Quantitative Trader Interview

Write Python code to determine if a given integer is a prime number.

Quantitative Trader Interview

Arrange the numbers 1 through 5 randomly around a circle. What is the expected number of numbers that are smaller than both of their neighbors?

Quantitative Trader Interview

What is the expected number of rolls needed to see each side of a fair six-sided die at least once?

Quant Trader Intern Interview

If you find a coin on the street, what is the probability that it is a biased (unfair) coin?

Quant Trader Intern Interview

Price an option where the underlying asset is the outcome of a fair six-sided die roll.

Fundamental Analyst Interview

Two tennis players, A and B, are at deuce. Player A wins a point with a 60% chance, and Player B wins a point with a 40% chance. What is the probability that A wins the game?

Fundamental Analyst Interview

You and a friend each independently receive a value drawn uniformly from (0,1). After seeing your value, you can choose to switch values with your friend or keep your original value. What is the probability that you end up with a higher value than your friend at the end of the game?

Fundamental Analyst Interview

James goes to the gym every Monday and Tuesday. On every other day of the week, he independently has a 30% chance of going to the gym. On a random day, James is spotted at the gym. What is the probability that it is Monday?

Quantitative Trading Intern Interview

Coin A lands heads with probability 2/3, and coin B lands heads with probability 1/3. If you flip the selected coin three times and observe heads each time, what is the probability that the coin is coin A?

Quantitative Trading Intern Interview

If N people each put their hat in a room and then each randomly pick one hat, what is the probability that a given person gets their own hat back?

Quant Research Intern Interview

Derive and explain the concept of put-call parity in options pricing.

Quant Research Intern Interview

Given n random variables from the same distribution, where the correlation between each pair of two variables is the same, what is the range of possible values for the correlation?

Quant Research Intern Interview

James goes to the gym every Monday and Tuesday. On every other day of the week, he independently has a 30% chance of going to the gym. On a random day, James is spotted at the gym. What is the probability that it is Monday?

Quantitative Trading Intern Interview

Coin A shows heads with probability 2/3, and coin B shows heads with probability 1/3. If a coin is chosen at random and flipped three times, each resulting in heads, what is the probability that the coin is coin A?

Quantitative Trading Intern Interview

How can you implement a rolling mean and variance of a data stream in O(1) time per update?

Quantitative Researcher Intern Interview

What is the expected value of the maximum among N normally distributed random variables?

Quantitative Researcher Intern Interview

Given a number, determine whether it can be represented as the sum of two or more consecutive integers.

Quantitative Researcher (Intern) Interview

What is the expected number of rolls of a fair six-sided die required until each face has appeared at least twice?

Semi-Systematic Trader Interview

Implement a ring buffer data structure. Write functions for both writing data to the buffer and reading data from it.

Software Engineer Intern Interview

How do you implement a rolling mean and variance for a data stream in O(1) time per update?

Quantitative Researcher Intern Interview

What is the expected value of the maximum of N independent normally distributed variables?

Quantitative Researcher Intern Interview

Given a number, determine whether it can be represented as the sum of two or more consecutive integers.

Quantitative Researcher (Intern) Interview

Three random variables X, Y, and Z are such that corr(X, Y) = corr(Y, Z) = corr(Z, X) = r. Give the upper and lower bounds for r.

Summer Internship Interview

What is the expected value of a fair 6-sided die?

Quantitative Trader Intern Interview

If you flip 8 fair coins, what is the probability that more than 4 of them are heads?

Fundamental Analyst Intern Interview

Estimate how many basketballs can fit in a standard school bus.

Analyst Interview

How is std::map implemented internally?

Quant Developer Interview

You have a scale and 18 balls. One of the balls is slightly heavier than the rest, which are identical. What is the minimum number of times you need to use the scale to determine which ball is the heavy ball?

Quant Developer Interview

Implement a service to maximize throughput by increasing the number of processes, given a fixed budget constraint.

Software Engineer New Grad Interview

Quant Interview Questions

You regress X1 against Y and obtain an R-squared value of 0.1. Then, you regress X2 against Y and obtain an R-squared value of 0.2. What is the maximum possible R-squared when regressing both X1 and X2 against Y together?

Given a string, return all possible permutations of its characters as a list. You may use recursive backtracking or any relevant Python packages.

Starting at the origin on a grid, you can move either up/down or left/right at each step. After N steps, how far can you be from the origin? Additionally, if there are two boundaries, what is the probability that you reach one boundary before the other?

If you toss a fair coin N times, what is the probability that you will get an even number of heads? Follow-up: Now, what if you have N - 1 fair coins, and the Nth coin is biased and lands on heads with probability p?

What happens if two features are highly collinear in Lasso regression?

Explain the basics of logistic regression.

What happens to the R-squared (R^2) value if you double the data in a regression analysis? What about the t-value?

We have n switches, each with probability p_i of being on and 1-p_i of being off. What is the probability that C or more switches are on? What if C is a uniform random variable between 0 and n?

Given an array representing the price of a stock on each day, and an integer k, determine the maximum profit you can achieve by completing at most k buy-sell transactions (you must sell before buying again).

Randomly generate points (x, y) on the circumference of a circle in two dimensions. What is Var(x)? What if you randomly generate points on the surface of a sphere instead?

In ridge regression, what happens to the expected value of the estimate if you perturb your data matrix?

What is the limit of x^x as x approaches 0?

What is the difference between ordinary least squares (OLS) regression and Ridge regression?

Implement a Python function to price an American option using backward induction.

What is the difference between concurrent and multicore systems?

How does std::unordered_map work in C++? Provide an example.

Implement a thread pool for sending messages.

Reverse the words in a sentence string in-place.

Design a random number generator that, given a list of probabilities corresponding to a discrete set of outcomes, outputs each outcome with its specified probability.

Implement a shared pointer class in C++ that manages shared ownership of a dynamically allocated object. Ensure proper reference counting and memory management.

Implement a shared_ptr smart pointer in C++ that manages reference counting and ensures proper deletion of the managed object when no references remain.

Given 25 horses and a race track where only 5 horses can race at a time, what is the minimum number of races required to determine the fastest horse?

What is the square root of 800 rounded to two decimal places?

Write code to generate the Fibonacci sequence.

Given constraints on the correlations between three random variables, provide bounds for the correlation between one pair of them.

Describe a program to determine whether a given integer is a prime number.

What is the shape of the gamma curve for a call option?

What is the house edge of a standard game of roulette?

What is the probability that a coin is biased (always lands heads), given that you observe five heads in a row?

Write Python code to determine if a given integer is a prime number.

Arrange the numbers 1 through 5 randomly around a circle. What is the expected number of numbers that are smaller than both of their neighbors?

What is the expected number of rolls needed to see each side of a fair six-sided die at least once?

If you find a coin on the street, what is the probability that it is a biased (unfair) coin?

Price an option where the underlying asset is the outcome of a fair six-sided die roll.

Two tennis players, A and B, are at deuce. Player A wins a point with a 60% chance, and Player B wins a point with a 40% chance. What is the probability that A wins the game?

You and a friend each independently receive a value drawn uniformly from (0,1). After seeing your value, you can choose to switch values with your friend or keep your original value. What is the probability that you end up with a higher value than your friend at the end of the game?

James goes to the gym every Monday and Tuesday. On every other day of the week, he independently has a 30% chance of going to the gym. On a random day, James is spotted at the gym. What is the probability that it is Monday?

Coin A lands heads with probability 2/3, and coin B lands heads with probability 1/3. If you flip the selected coin three times and observe heads each time, what is the probability that the coin is coin A?

If N people each put their hat in a room and then each randomly pick one hat, what is the probability that a given person gets their own hat back?

Derive and explain the concept of put-call parity in options pricing.

Given n random variables from the same distribution, where the correlation between each pair of two variables is the same, what is the range of possible values for the correlation?

James goes to the gym every Monday and Tuesday. On every other day of the week, he independently has a 30% chance of going to the gym. On a random day, James is spotted at the gym. What is the probability that it is Monday?

Coin A shows heads with probability 2/3, and coin B shows heads with probability 1/3. If a coin is chosen at random and flipped three times, each resulting in heads, what is the probability that the coin is coin A?

How can you implement a rolling mean and variance of a data stream in O(1) time per update?

What is the expected value of the maximum among N normally distributed random variables?

Given a number, determine whether it can be represented as the sum of two or more consecutive integers.

What is the expected number of rolls of a fair six-sided die required until each face has appeared at least twice?

Implement a ring buffer data structure. Write functions for both writing data to the buffer and reading data from it.

How do you implement a rolling mean and variance for a data stream in O(1) time per update?

What is the expected value of the maximum of N independent normally distributed variables?

Given a number, determine whether it can be represented as the sum of two or more consecutive integers.

Three random variables X, Y, and Z are such that corr(X, Y) = corr(Y, Z) = corr(Z, X) = r. Give the upper and lower bounds for r.

What is the expected value of a fair 6-sided die?

If you flip 8 fair coins, what is the probability that more than 4 of them are heads?

Estimate how many basketballs can fit in a standard school bus.

How is std::map implemented internally?

You have a scale and 18 balls. One of the balls is slightly heavier than the rest, which are identical. What is the minimum number of times you need to use the scale to determine which ball is the heavy ball?

Implement a service to maximize throughput by increasing the number of processes, given a fixed budget constraint.