Optiver Interview Questions - Complete List

Quant Interview Questions

125 questions

Evaluate: (5/16) divided by (12/3).

Trader Interview

If a fair coin is flipped until the first heads appears, what is the expected number of flips?

Trader Interview

It is 9-9 in a squash match. If I have a 30% chance to win any given point, what is the probability that I win the game?

Trader Interview

Given $100 to bet and the probabilities of each of 4 teams winning a bracket, how would you allocate your bet to maximize your expected return?

Trader Interview

What is the expected value of a card drawn at random from a standard deck of cards, where J=11, Q=12, K=13? Then, what would this expected value become if the value of all hearts cards is doubled?

Trader Interview

You start a tennis game at a score of 30-30. You have a probability of 0.6 of winning each point. What is the probability that you win the game?

Trader Interview

What is the expected value of a single six-sided dice roll if you are allowed to re-roll once, and keep the higher number from the two rolls?

Trader Interview

What is the value of 3/6 + 7/4?

Trader Interview

If the probability of a truck arriving in 10 minutes is 0.36, what is the probability that it arrives in 5 minutes?

Trader Interview

What is the formula to calculate the angle between the hour and minute hands of a clock at a given time?

Trader Interview

There are 25 horses and you want to identify the fastest 3 horses. You can race up to 5 horses at a time, and you do not have a watch to time the races. What is the minimum number of races needed to accomplish this?

Trader Interview

You have 3 cards, each labeled with consecutive integers n, n+1, and n+2, where n is unknown. All cards are placed face down. On your turn, you flip one card. After seeing its value, you can either 'stay' and keep that value, or 'flip' another card. If you choose to flip another card, you can again choose to 'stay' and keep that value, or flip the final card and keep its value. Design the optimal strategy for maximizing your expected value.

Trader Interview

Calculate the square roots of the following numbers: 600, 6000, 60000, and 24000.

Trader Interview

Given $1, $5, $20, and $100 dollar bills, what is the fewest number of bills needed to form amounts ranging from $1 to $100? For example, $4 would require four $1 bills. Bills can be reused, so the number is not cumulative. What is the minimum number of bills you must have to always provide exact change for both a transaction of $x and $(100-x), where x < 100?

Trader Interview

What is the smallest multiple of 77 that is greater than 70,000? What about 700,000?

Trader Interview

Find the largest 5-digit number such that the pairwise digit sums of its digits are all unique. Find the smallest 5-digit number with the same property. Explain why this property does not work for 7-digit numbers. Does it work for 6-digit numbers?

Trader Interview

What happens to the gamma of your position if you are short a straddle?

Trader Interview

What is the next number in the sequence: 1/2, 1/3, 1/6, 1/18, 1/108, ...?

Trader Interview

Let's play a game where we roll a die, and you receive a dollar amount equal to the number rolled. If you don't like the result, you may re-roll once and receive that result instead. How much would you pay to play this game?

Trader Interview

Assign A=1, B=2, ..., Z=26. Find an eight-letter English word such that the sum of the letter values is less than 50.

Trader Interview

How many levels can you build with 52 cards (a standard deck of cards)? What is the expected value (EV) of throwing a single die, and the EV of the sum of two dice rolled once and rolled 400 times? How would you create a market based on these expected values?

Trader Interview

What is the value of (3/21) divided by (1/7)?

Trader Interview

What are the six inputs in an equity option pricing model?

Trader Interview

If I roll a die and add each result to a running total, which die face value am I most likely to roll on the final roll that brings the running total to at least 1000?

Graduate Trader Interview

Suppose you flip 100 fair coins. What is the expected number of heads? What is the probability that exactly this many heads appear? What is the standard deviation of the number of heads?

Graduate Trader Interview

Complete the sequence: 2, 3, 5, 7, 11, 13.

Quant Trader Intern Interview

I throw two fair 6-sided dice. Which expected value is larger: the expected value of the smaller number, or the expected value of the difference between the two numbers?

Quant Trader Intern Interview

What is the next term in the sequence: 1/1, 1/2, 1/3, 1/5, 1/8?

Quant Trader Intern Interview

How many prime numbers are there from 1 to 10,000?

Quant Trader Intern Interview

What is the result of dividing 335 by 7?

Jr Trader Interview

Approximate the probability of getting exactly 50 heads in 100 tosses of a fair coin, not as a combinatorial expression, but as a percentage or a simplified fraction.

Quantitative Trading Intern Interview

What is the result of dividing 3/4 by 1/2? Show your calculations.

Quantitative Trading Intern Interview

What is the expected value of the sum of the values of three randomly selected cards from a standard deck of 52 playing cards?

Quantitative Trading Intern Interview

How much memory is required to store a linked list with 1,000 elements?

Software Engineer Interview

Design and implement a queue data structure.

Software Engineer Interview

Design a class diagram for a mock trading system.

Software Engineer Interview

Given an array of stock prices for n days, return the maximum profit achievable using at most k transactions. Complete the function int stock(int* prices, int k, int n).

Software Engineer Interview

Complete a function that takes in a seqId (int) and a character (from 'a' to 'z' or '-'), where seqIds may arrive out of order. When the collected characters, sorted by seqId, form a complete message in the format '-FullMessage-', the function should print the message immediately using the provided OnComplete(msg) function.

Software Engineer Interview

Implement a queue class without being able to resize memory.

Software Engineer Interview

Create a function in C++ to calculate the difference in days between two dates. Explain why your solution works.

Software Engineer Interview

Given a sentence s and a list of word pairs, where each pair consists of strings a and b indicating that word a can be converted to word b (but not necessarily vice versa), determine the minimum length sentence you can translate the original sentence to. If there is more than one shortest sentence, return the lexicographically smallest one.

Software Developer Interview

Write an algorithm that replaces words in a paragraph with their corresponding synonyms from a thesaurus.

Software Engineer Interview

Given the edges of a tree, output the pre-order traversal of the tree.

Software Engineer Interview

Given two sorted arrays, find the median of the combined arrays. Explain your algorithm and discuss how you would handle edge cases.

Software Engineer Interview

Given two dates, calculate the number of days between them.

Software Engineer Interview

Implement an algorithm to merge N sorted lists.

Software Engineer Interview

Why are the wait and notify methods declared in the Object class instead of the Thread class in Java?

Developer Interview

Given an array of integers, find the third largest number in the array.

Software Engineer Interview

Given two dates (for example, 12/21/2001 and 4/13/2007), calculate the number of days between them.

Software Engineer Interview

Given a list of tuples (AB, BC, AD, BE, CF), construct a binary tree and determine if it is a valid binary tree. You must check for the following five types of errors: (1) input is in the correct format; (2) no duplicate nodes; (3) the root has only two children; (4) every node is present in the tree; and (5) there are no cycles in the binary tree.

Software Engineer Interview

Given an encoded tree represented as a list of pairs of nodes, verify that the input is a valid tree and check whether there are any cycles in the structure.

Software Engineer Interview

Describe how you would store and sort the prices of items coming from multiple exchanges. Write code to demonstrate your approach.

Software Engineer Interview

Given a list containing all numbers from 1 to n except one missing number, find the missing number.

Software Developer Interview

How would you calculate the final price of an asset in a trading simulation given historical trading data?

Trading Intern Interview

Given four numbers, use arithmetic operations (+, -, ×, ÷) to make 24. Each number must be used exactly once.

Trading Intern Interview

What is the next number in the sequence: 1, 16, 7, 100, 13?

Trading Intern Interview

What is the expected value of a dice rolling game where you roll two standard six-sided dice, receive the sum of the two dice unless both show the same number, in which case you receive 0?

Trading Intern Interview

A family has children one by one until the number of daughters exceeds the number of sons by exactly one. What is the expected total number of children they will have?

Quantitative Trader Intern Interview

You roll three dice, remove any pair duplicates, and get paid the final sum. How much would you pay for a call option on this payout with a strike price of $14?

Quantitative Trader Intern Interview

What is the probability of getting 10 heads in a row when flipping a fair coin?

Quantitative Trader - Intern Interview

Given a bracket of teams and their relative strengths or ratings, determine how to allocate bets on the teams to maximize expected profit and loss (PnL).

Quantitative Trader Intern Interview

Calculate the value of 1.02 multiplied by (19.25 minus 0.36).

Quantitative Trader Intern Interview

In a perfectly efficient market, there are two companies: A with a success probability of 2/3 and B with a success probability of 1/2. Both companies are initially worth $50 each. If both companies either succeed or fail, nothing special happens. However, if one company succeeds and the other does not, then the company that does not succeed goes bankrupt. How would the price of each company change if both succeed?

Quantitative Trader Intern Interview

What is 3 plus 5 equal to?

Quantitative Trader - Intern Interview

How would you price an asset, given a list of bids and asks?

Trading Interview

How many cards are needed to build a 100-storey house of cards, assuming each storey is constructed in the standard way?

Trading Interview

Estimate how many digits are in 100 factorial (100!).

Internship Interview

Given a stream of stock price updates for both buy and sell orders, develop an efficient method to send and process the data to reduce overhead. Discuss considerations for using TCP versus UDP for transmitting this data.

Internship Interview

Out of 100 fair coin flips, what is the probability that you get at least 10 more heads than tails?

Quant Trading Intern Interview

Implement a linear regression algorithm from scratch, without using any machine learning libraries.

Quant Trading Intern Interview

Design a double-ended queue (deque) data structure that supports efficient insertion and deletion from both ends. Explain the operations and their time complexities.

Software Engineer Intern Interview

Describe how you would implement a queue data structure.

Software Engineer Intern Interview

How would you implement a queue? As a follow-up, actually build a queue using an array.

Software Engineer Intern Interview

What is the difference between UDP and TCP?

Software Engineer Intern Interview

Given a string representation of a binary tree, write a function to parse the string and construct the tree. Then, identify and return a list of errors found in the tree, such as nodes with more than two children, more than one root, or duplicate parent-child pairs.

Software Engineer Intern Interview

What is memory fragmentation, and how can it impact performance?

SWE Intern Interview

1. What is 73 × 87? Provide your answer without using a calculator and explain your reasoning. 2. What is the probability of drawing exactly two aces from a shuffled deck of cards?

Quantitative Analyst Interview

In a sequence of fair coin tosses with each outcome being either Heads (H) or Tails (T), what is the probability that the sequence HHH appears before HTH?

Quantitative Analyst Interview

How many nodes are there in a recombining binomial tree with n steps? How many paths lead to a particular node?

Quantitative Analyst Interview

You throw a standard die repeatedly, keeping a running total of the sums. What is the probability that when the running total exceeds 100 for the first time, the last die throw is a 2?

Quant Researcher Interview

Six dice are rolled simultaneously. What is the probability that the sum of the numbers shown is an even number?

Graduate Quantitative Researcher Interview

Provide a formal proof for the following: You have two groups of marbles—one group with 1 marble and the other with 2^n marbles. At each step, you can double the size of one group by taking marbles from the other group. Is it possible to reach a state where one group has 0 marbles and the other has 2^n + 1 marbles?

Graduate Quantitative Researcher Interview

What is the expected number of card draws before encountering an Ace in a standard 52-card deck, assuming cards are drawn one at a time without replacement?

Graduate Quantitative Researcher Interview

Given a list of exchange rates between different currency pairs and an amount in one currency, convert the amount to another specified currency.

Senior Software Engineer Interview

Create functions that simulate communication with exchanges and ensure that short-selling is avoided.

Senior Software Engineer Interview

Given a graph and two vertices, implement Dijkstra's algorithm to find the shortest distance between the pair. Additionally, return an error if there are multiple shortest paths with the same distance between the vertices.

Senior Software Engineer Interview

Given a mapping of 7 letters to their respective Morse code representations and a string of dots and dashes (with no spaces), find all valid interpretations of the Morse code string as sequences of those letters.

Senior Software Developer Interview

Given two very large integers represented as strings, implement multiplication of these numbers. You cannot use built-in big integer libraries. You must represent the digits as strings, implement addition of two strings, and then use that method as part of your multiplication function. How would you handle carries correctly in your implementation?

Senior Software Developer Interview

Given an exchange feed that provides up to five rows of depth information per second (five price points and their respective quantities, for both buy and sell sides of a specific stock), design and implement a more efficient solution on the server so that clients are only sent updates for price points whose quantity has changed, rather than sending all the data every time.

Senior Software Developer Interview

How many bits are required to represent a positive integer consisting of 16 digits, with no leading zeros, in binary representation?

Graduate Software Developer Interview

What is the use of atomic operations in multiprocessing?

Graduate Software Developer Interview

Given an array of integers, find the maximum possible sum of any non-empty contiguous subarray.

Graduate Software Engineer Interview

How many digits are there in 100! (100 factorial)?

Quantitative Researcher Intern Interview

How many bits are required to represent a positive integer with exactly 16 digits and no leading zeros, in binary representation?

Graduate Software Developer Interview

What is the use of atomic operations in multiprocessing?

Graduate Software Developer Interview

Make a market on the percentage of the Earth's Equator that passes through land.

Summer Intern Interview

100

Given randomly generated sequences of digits of increasing lengths, displayed one digit at a time, memorize each sequence and then recall and re-enter the sequence in sorted order.

Software Engineer Intern Interview

101

How much memory is required to store a linked list with 1,000 elements?

Software Engineer Interview

102

Design and implement a queue data structure.

Software Engineer Interview

103

Design a class diagram for a mock trading system.

Software Engineer Interview

104

Given an array of stock prices over n days, return the maximum profit that can be achieved with at most k transactions. Implement the function int stock(int* prices, int k, int n).

Software Engineer Interview

105

Complete a function that takes in a seqId (int) and a char (consisting of 'a'-'z' and '-'). The seqId values may be received out of order. A full message is formed when the characters, sorted by seqId, create a string in the format "-FullMessage-". The goal is to print out any full message exactly when it completes, using the provided OnComplete(msg) function.

Software Engineer Interview

106

Implement a queue data structure class without being able to resize the underlying memory allocation.

Software Engineer Interview

107

Create a function in C++ to calculate the difference in days between two dates. Explain why your solution works.

Software Engineer Interview

108

What is the expected number of moves required to exit the center of a 10x10 grid, assuming you can move one step in a random direction (up, down, left, right) at each move?

Quantitative Researcher Interview

109

What is the probability that, given a random number from 1 through 1,000,000, the digit 7 appears at least once in the selected number?

Quantitative Researcher Interview

110

You throw two dice and then may choose to reroll just one of them, and only once. What is the expected value?

Quantitative Researcher Interview

111

Identify any arbitrage opportunities, if any, in the given matrix of currency conversion rates.

Quantitative Researcher Interview

112

Can you derive the formula for ordinary linear regression?

Quantitative Researcher Interview

113

Someone has $100 to gamble. First, he will bet $1, and if he wins, he will leave. Otherwise, he will double his bet on each subsequent round until he wins. What is his profit from this game?

Quantitative Researcher Interview

114

Roll a die repeatedly until you obtain either a 5 or a 6. If you roll a 5, you cash out the sum of all previous rolls, excluding the 5. If you roll a 6, you receive no payout. You are allowed to cash out mid-game for the sum accrued up to that point. How can you maximize your expected payoff?

Quantitative Trader Interview

115

Given a simulated order book, how would you determine the fair value of a derivative?

Quantitative Trader Interview

116

What is the total number of ways to arrange 3 red balls, 7 blue balls, and 9 green balls in a line?

Quantitative Trader Interview

117

In a family, the parents continue having children until they have more girls than boys. What is the expected number of children in this family?

Quantitative Trader Interview

118

If you flip 100 fair coins, what is the variance of the number of Heads?

Quantitative Trader Interview

119

Divide 2 by 17 to four decimal places without using any calculator or external materials.

Quantitative Trader Interview

120

An interviewer will play Padel against me. The interviewer, being better, has a 70% chance to win a point against me. The game is played to 11 points, but a player must win by at least 2 points. The current score is 9 to 9. What is the probability that I win the game?

Quantitative Trader Interview

121

Estimate how many ways there are to add three positive integers to obtain a sum of 25.

Quantitative Trader Interview

122

You have 16 watermelons and a seesaw. What is the minimum number of tests required to determine the two heaviest watermelons?

Quantitative Trader Interview

123

Six people are sitting around a circle. What is the probability that they are seated in increasing order of age, either clockwise or counterclockwise?

Quantitative Trader Interview

124

A group of people wants to determine their average salary on the condition that no individual can find out anyone else's salary. Can they accomplish this, and if so, how?

Quantitative Trader Interview

125

Two people each flip a fair coin 4 times. What is the probability that both people end up with the same number of heads?

Quantitative Trader Interview

Quant Interview Questions

Evaluate: (5/16) divided by (12/3).

If a fair coin is flipped until the first heads appears, what is the expected number of flips?

It is 9-9 in a squash match. If I have a 30% chance to win any given point, what is the probability that I win the game?

What is the result of 2/5 + 11/4?

What are the first 5 decimal digits in the decimal expansion of 1/13?

What is the square root of 30,000 (calculated mentally)?

Given $100 to bet and the probabilities of each of 4 teams winning a bracket, how would you allocate your bet to maximize your expected return?

What is the expected value of a card drawn at random from a standard deck of cards, where J=11, Q=12, K=13? Then, what would this expected value become if the value of all hearts cards is doubled?

You start a tennis game at a score of 30-30. You have a probability of 0.6 of winning each point. What is the probability that you win the game?

What is the expected value of a single six-sided dice roll if you are allowed to re-roll once, and keep the higher number from the two rolls?

What is the value of 3/6 + 7/4?

If the probability of a truck arriving in 10 minutes is 0.36, what is the probability that it arrives in 5 minutes?

What is the formula to calculate the angle between the hour and minute hands of a clock at a given time?

There are 25 horses and you want to identify the fastest 3 horses. You can race up to 5 horses at a time, and you do not have a watch to time the races. What is the minimum number of races needed to accomplish this?

Calculate the square roots of the following numbers: 600, 6000, 60000, and 24000.

What is the smallest multiple of 77 that is greater than 70,000? What about 700,000?

Find the largest 5-digit number such that the pairwise digit sums of its digits are all unique. Find the smallest 5-digit number with the same property. Explain why this property does not work for 7-digit numbers. Does it work for 6-digit numbers?

What happens to the gamma of your position if you are short a straddle?

What is the next number in the sequence: 1/2, 1/3, 1/6, 1/18, 1/108, ...?

Let's play a game where we roll a die, and you receive a dollar amount equal to the number rolled. If you don't like the result, you may re-roll once and receive that result instead. How much would you pay to play this game?

Assign A=1, B=2, ..., Z=26. Find an eight-letter English word such that the sum of the letter values is less than 50.

How many levels can you build with 52 cards (a standard deck of cards)? What is the expected value (EV) of throwing a single die, and the EV of the sum of two dice rolled once and rolled 400 times? How would you create a market based on these expected values?

What is the value of (3/21) divided by (1/7)?

What are the six inputs in an equity option pricing model?

If I roll a die and add each result to a running total, which die face value am I most likely to roll on the final roll that brings the running total to at least 1000?

Suppose you flip 100 fair coins. What is the expected number of heads? What is the probability that exactly this many heads appear? What is the standard deviation of the number of heads?

Complete the sequence: 2, 3, 5, 7, 11, 13.

I throw two fair 6-sided dice. Which expected value is larger: the expected value of the smaller number, or the expected value of the difference between the two numbers?

What is the next term in the sequence: 1/1, 1/2, 1/3, 1/5, 1/8?

How many prime numbers are there from 1 to 10,000?

What is the result of dividing 335 by 7?

Approximate the probability of getting exactly 50 heads in 100 tosses of a fair coin, not as a combinatorial expression, but as a percentage or a simplified fraction.

What is the result of dividing 3/4 by 1/2? Show your calculations.

What is the expected value of the sum of the values of three randomly selected cards from a standard deck of 52 playing cards?

How much memory is required to store a linked list with 1,000 elements?

Design and implement a queue data structure.

Design a class diagram for a mock trading system.

Given an array of stock prices for n days, return the maximum profit achievable using at most k transactions. Complete the function int stock(int* prices, int k, int n).

Implement a queue class without being able to resize memory.

Create a function in C++ to calculate the difference in days between two dates. Explain why your solution works.

Write an algorithm that replaces words in a paragraph with their corresponding synonyms from a thesaurus.

Given the edges of a tree, output the pre-order traversal of the tree.

Given two sorted arrays, find the median of the combined arrays. Explain your algorithm and discuss how you would handle edge cases.

Given two dates, calculate the number of days between them.

Implement an algorithm to merge N sorted lists.

Why are the wait and notify methods declared in the Object class instead of the Thread class in Java?

Given an array of integers, find the third largest number in the array.

Given two dates (for example, 12/21/2001 and 4/13/2007), calculate the number of days between them.

Given an encoded tree represented as a list of pairs of nodes, verify that the input is a valid tree and check whether there are any cycles in the structure.

Describe how you would store and sort the prices of items coming from multiple exchanges. Write code to demonstrate your approach.

Given a list containing all numbers from 1 to n except one missing number, find the missing number.

How would you calculate the final price of an asset in a trading simulation given historical trading data?

Given four numbers, use arithmetic operations (+, -, ×, ÷) to make 24. Each number must be used exactly once.

What is the next number in the sequence: 1, 16, 7, 100, 13?

What is the expected value of a dice rolling game where you roll two standard six-sided dice, receive the sum of the two dice unless both show the same number, in which case you receive 0?

A family has children one by one until the number of daughters exceeds the number of sons by exactly one. What is the expected total number of children they will have?

You roll three dice, remove any pair duplicates, and get paid the final sum. How much would you pay for a call option on this payout with a strike price of $14?

What is the probability of getting 10 heads in a row when flipping a fair coin?

Given a bracket of teams and their relative strengths or ratings, determine how to allocate bets on the teams to maximize expected profit and loss (PnL).

Calculate the value of 1.02 multiplied by (19.25 minus 0.36).

What is 3 plus 5 equal to?

How would you price an asset, given a list of bids and asks?

How many cards are needed to build a 100-storey house of cards, assuming each storey is constructed in the standard way?

Estimate how many digits are in 100 factorial (100!).

Given a stream of stock price updates for both buy and sell orders, develop an efficient method to send and process the data to reduce overhead. Discuss considerations for using TCP versus UDP for transmitting this data.

Out of 100 fair coin flips, what is the probability that you get at least 10 more heads than tails?

Implement a linear regression algorithm from scratch, without using any machine learning libraries.

Design a double-ended queue (deque) data structure that supports efficient insertion and deletion from both ends. Explain the operations and their time complexities.

Describe how you would implement a queue data structure.

How would you implement a queue? As a follow-up, actually build a queue using an array.

What is the difference between UDP and TCP?

Given a string representation of a binary tree, write a function to parse the string and construct the tree. Then, identify and return a list of errors found in the tree, such as nodes with more than two children, more than one root, or duplicate parent-child pairs.

What is memory fragmentation, and how can it impact performance?

1. What is 73 × 87? Provide your answer without using a calculator and explain your reasoning. 2. What is the probability of drawing exactly two aces from a shuffled deck of cards?

In a sequence of fair coin tosses with each outcome being either Heads (H) or Tails (T), what is the probability that the sequence HHH appears before HTH?

How many nodes are there in a recombining binomial tree with n steps? How many paths lead to a particular node?

You throw a standard die repeatedly, keeping a running total of the sums. What is the probability that when the running total exceeds 100 for the first time, the last die throw is a 2?

Six dice are rolled simultaneously. What is the probability that the sum of the numbers shown is an even number?

What is the expected number of card draws before encountering an Ace in a standard 52-card deck, assuming cards are drawn one at a time without replacement?