2026 Optiver Interview Questions

Quant Interview Questions

90 Questions

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

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

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

121

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

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

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

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

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

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

115

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

Quantitative Trader 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

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

112

Can you derive the formula for ordinary linear regression?

Quantitative Researcher Interview

111

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

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

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

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

107

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

Software Engineer Interview

106

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

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

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

103

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

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

Summer Intern Interview

What is the use of atomic operations in multiprocessing?

Graduate Software Developer 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

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

Quantitative Researcher Intern Interview

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

Graduate Software Engineer Interview

What is the use of atomic operations in multiprocessing?

Graduate 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

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

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 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

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

Senior Software Engineer 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

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

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

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

Graduate Quantitative Researcher 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

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

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

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

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

SWE 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

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

Software Engineer Intern Interview

Describe how you would implement a queue data structure.

Software Engineer 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

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

Quant Trading Intern 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

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

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

Internship 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

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

Trading Interview

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

Quantitative Trader Intern Interview

What is 3 plus 5 equal to?

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

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

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

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

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

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

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

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

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

Trading Intern 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

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

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

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

Implement an algorithm to merge N sorted lists.

Software Engineer Interview

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

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

Software Engineer Interview

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

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

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

Software Engineer Interview

Implement a queue class without being able to resize memory.

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

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

Design and implement a queue data structure.

Software Engineer Interview

Design a class diagram for a mock trading system.

Software Engineer 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

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

Quantitative Trading Intern 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

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

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

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

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

Quant Trader Intern 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

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

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

Trader Interview

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Optiver

Interview Questions

Quant Interview Questions

How can you prevent gradient instability caused by the sigmoid function?

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

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?

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

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

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?

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

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

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

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?

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

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

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?

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?

Can you derive the formula for ordinary linear regression?

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

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

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?

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?

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

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

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).

Design a class diagram for a mock trading system.

Design and implement a queue data structure.

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

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.

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

What is the use of atomic operations in multiprocessing?

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

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

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

What is the use of atomic operations in multiprocessing?

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

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.

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.

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

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

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?

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

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?

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

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?

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?

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

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.

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

Describe how you would implement a queue data structure.

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

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

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

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.

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

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

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

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

What is 3 plus 5 equal to?

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).

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

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?

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

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?

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

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?

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

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.

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.

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

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.

Implement an algorithm to merge N sorted lists.

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

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

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

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

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