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Quant Research Interview Question - Susquehanna International Group

Quantitative research interviews at top trading firms like Susquehanna International Group (SIG) are renowned for their challenging, thought-provoking questions. These questions are designed to assess your mathematical intuition, problem-solving skills, and ability to apply core concepts under pressure. In this comprehensive guide, we’ll delve into two actual SIG quant research interview questions, providing step-by-step solutions, conceptual explanations, and strategic insights for candidates seeking to excel.

Quant Research Interview Questions Asked at SIG: Detailed Solutions & Concepts

Question 1: Best Guess for x Given z = x + y from a Joint Gaussian
Question 2: Rock, Paper, Scissors Against a Limited Opponent
SIG Quant Interview Preparation Tips

1. Drawing a Pair of (x, y) from a Joint Gaussian Distribution with 0 Covariance

1.1 The Question

Suppose you draw a pair of random variables $(x, y)$ from a joint Gaussian (normal) distribution with zero covariance. You know the standard deviations of $x$ and $y$, and you are told the value of $z = x + y$. What is your best guess for $x$?

1.2 Concepts Involved

Joint Gaussian Distribution: A multivariate normal distribution where each variable is normally distributed, and the joint behavior is described by means, variances, and covariances.
Covariance: A measure of how much two variables change together. Zero covariance implies the variables are uncorrelated (but not necessarily independent unless jointly normal).
Conditional Expectation: The best guess (minimum mean square error estimator) for a random variable given some information.

1.3 Mathematical Formulation

Let’s formalize:

$x \sim N(\mu_x, \sigma_x^2)$
$y \sim N(\mu_y, \sigma_y^2)$
Covariance: $\text{Cov}(x, y) = 0$
Given: $z = x + y$ observed (i.e., you know the value of $z$)
Question: What is the best guess for $x$ given $z$? In other words, find $\mathbb{E}[x \mid z]$.

1.4 Solution Approach

Since $x$ and $y$ are jointly normal with zero covariance, they are independent. But, once you know $z = x + y$, $x$ and $y$ become conditionally dependent.

Step 1: Express $x$ and $y$ in Terms of $z$

Given $z$, $y = z - x$.

Step 2: Compute the Conditional Distribution

Let’s use properties of the joint normal distribution. The conditional distribution of $x$ given $z$ is itself normal, with mean and variance:

\[ \mathbb{E}[x \mid z] = \mu_x + \text{Cov}(x, z) \frac{z - \mathbb{E}[z]}{\text{Var}(z)} \]

Let’s calculate the required quantities.

$\mu_z = \mathbb{E}[z] = \mu_x + \mu_y$
$\text{Var}(z) = \text{Var}(x + y) = \sigma_x^2 + \sigma_y^2$ (since covariance is 0)
$\text{Cov}(x, z) = \text{Cov}(x, x + y) = \text{Cov}(x, x) + \text{Cov}(x, y) = \sigma_x^2 + 0 = \sigma_x^2$

Plug these into the conditional mean formula:

\[ \mathbb{E}[x \mid z] = \mu_x + \sigma_x^2 \frac{z - (\mu_x + \mu_y)}{\sigma_x^2 + \sigma_y^2} \]

Step 3: Final Answer

Your best guess for $x$ given $z$ is:

\[ \boxed{ \mathbb{E}[x \mid z] = \mu_x + \frac{\sigma_x^2}{\sigma_x^2 + \sigma_y^2} \big(z - \mu_x - \mu_y\big) } \]

1.5 Intuitive Explanation

The formula above is a weighted average between the prior mean ($\mu_x$) and the observed deviation in $z$, weighted by the relative variance of $x$ to the total variance. If $x$ is very noisy (large $\sigma_x^2$), more of $z$ is attributed to $x$; if $y$ is noisy, less is attributed to $x$.

If $\sigma_x^2 \gg \sigma_y^2$: Most of $z$ is attributed to $x$.
If $\sigma_x^2 \ll \sigma_y^2$: Most of $z$ is attributed to $y$, so your best guess for $x$ is close to its mean.

1.6 Example Calculation

Suppose $\mu_x = 0$, $\mu_y = 0$, $\sigma_x = 2$, $\sigma_y = 3$, and $z = 5$.

$\sigma_x^2 = 4$, $\sigma_y^2 = 9$
$\mathbb{E}[x \mid z] = 0 + \frac{4}{13} \times 5 = \frac{20}{13} \approx 1.54$

So, your best guess for $x$ is approximately 1.54.

1.7 Python Code Example


def best_guess_x(z, mu_x, mu_y, sigma_x, sigma_y):
    var_x = sigma_x ** 2
    var_y = sigma_y ** 2
    return mu_x + (var_x / (var_x + var_y)) * (z - mu_x - mu_y)

# Example usage:
z = 5
mu_x = 0
mu_y = 0
sigma_x = 2
sigma_y = 3

guess = best_guess_x(z, mu_x, mu_y, sigma_x, sigma_y)
print(f"Best guess for x: {guess:.2f}")

1.8 Key Takeaways

This is a classic application of conditional expectation in multivariate normal distributions.
Recognize when variables are independent and when new information (like $z$) induces dependence.
The answer is a weighted mean, reflecting signal-to-noise ratios.

2. Rock, Paper, Scissors When the Opponent Cannot Play Rock

2.1 The Question

You play rock, paper, scissors with an opponent, but your opponent cannot play rock. Every time you win, you win $1; lose, you lose $1; draw, you win $0. What should you play to maximize your expected profit?

2.2 Concepts Involved

Game Theory: The study of optimal strategies in situations involving competing interests.
Payoff Matrix: A table showing the payoff for each possible action combination.
Mixed Strategies: Randomizing your choices to maximize expected payoff.
Opponent’s Constraints: The opponent cannot play rock, so their choices are limited to paper or scissors.

2.3 Payoff Matrix Construction

Let’s build the payoff matrix. The player (you) can play rock, paper, or scissors. The opponent can only play paper or scissors.

Your Move	Opponent: Paper	Opponent: Scissors
Rock	-1 (lose)	+1 (win)
Paper	0 (draw)	-1 (lose)
Scissors	+1 (win)	0 (draw)

2.4 Calculate Expected Payoff for Each Move

Let $p$ be the probability that the opponent plays paper, and $1-p$ that they play scissors.

If you play Rock:
- Win $1 if opponent plays scissors ($1-p$), lose $1 if opponent plays paper ($p$).
- Expected payoff: $-p + (1-p) = 1 - 2p$
If you play Paper:
- Draw if opponent plays paper ($p$), lose $1 if opponent plays scissors ($1-p$).
- Expected payoff: $0 \times p + (-1) \times (1-p) = p - 1$
If you play Scissors:
- Win $1 if opponent plays paper ($p$), draw if opponent plays scissors ($1-p$).
- Expected payoff: $1 \times p + 0 \times (1-p) = p$

2.5 Optimal Strategy

Now, you want to maximize your expected profit. Let’s compare the three expressions:

Rock: $1 - 2p$
Paper: $p - 1$
Scissors: $p$

Step 1: Find the Best Move Depending on $p$

For which $p$ is Rock best?
- Rock > Scissors: $1 - 2p > p \implies 1 > 3p \implies p < \frac{1}{3}$
- Rock > Paper: $1 - 2p > p - 1 \implies 2 > 3p \implies p < \frac{2}{3}$
So, if $p < \frac{1}{3}$, Rock is best.
For which $p$ is Scissors best?
- Scissors > Rock: $p > 1 - 2p \implies 3p > 1 \implies p > \frac{1}{3}$
- Scissors > Paper: $p > p - 1$ always true.
So, if $p > \frac{1}{3}$, Scissors is best.
At $p = \frac{1}{3}$, Rock and Scissors have the same expected payoff ($1 - 2p = p$).

Step 2: What Should You Actually Play?

If you know the opponent's distribution (i.e., you know $p$), play the best pure strategy as above.
If the opponent chooses randomly between paper and scissors (i.e., $p = 0.5$), then:
- Rock: $1 - 2 \times 0.5 = 0$
- Paper: $0.5 - 1 = -0.5$
- Scissors: $0.5$
So, always play scissors!
If you have no information about $p$, but assume the opponent plays paper and scissors at random, always play scissors.

2.6 General Insight

If your opponent cannot play rock, always play scissors to maximize your expected profit (assuming their choice between paper and scissors is not biased against you).

Scissors beats paper, and draws against scissors. Paper, by contrast, can only draw or lose; rock can only win or lose.
Your expected value per play is at least as good or better with scissors, unless you know the opponent almost always plays scissors (then you may want to randomize).

2.7 Python Simulation Example


import random

def expected_value(player_move, opponent_paper_prob):
    if player_move == 'rock':
        return 1 - 2 * opponent_paper_prob
    elif player_move == 'paper':
        return opponent_paper_prob - 1
    elif player_move == 'scissors':
        return opponent_paper_prob

# Simulate for p = 0.5 (opponent randomizes between paper and scissors)
for move in ['rock', 'paper', 'scissors']:
    ev = expected_value(move, 0.5)
    print(f"{move.capitalize()}: Expected Value = {ev:.2f}")

2.8 Key Takeaways

Understand the constraints on your opponent’s actions; exploit them for maximum gain.
Construct the payoff matrix and use algebraic reasoning for optimal strategy selection.

In this SIG interview variant, the correct move

is to always play scissors unless you have information about the opponent’s bias. This exploits their inability to play rock and gives you a positive expected value against a random opponent.

3. SIG Quant Interview Preparation Tips

3.1 Understanding SIG Interview Philosophy

Susquehanna International Group (SIG) is famous for its rigorous quant interviews. Their approach is rooted in assessing not just technical skills but also a candidate’s logical reasoning, game theory intuition, and ability to communicate complex ideas clearly. Questions often blend probability, statistics, market intuition, and game theory—mirroring the skills you’ll need on the trading floor.

3.2 Key Competencies SIG Looks For

Mathematical Rigor: Comfort with probability, statistics, and algebra under pressure.
Game Theory: Ability to analyze and solve games—not just classic ones, but also those with constraints or novel twists (like the RPS question above).
Statistical Intuition: Understanding of concepts like joint and conditional distributions, expectations, and variances.
Communication: Explaining your reasoning clearly and concisely under interview conditions.

3.3 How to Prepare for SIG Quant Interviews

Master Probability and Statistics:
- Review joint, marginal, and conditional distributions, especially for normal variables.
- Practice problems involving conditional expectation and Bayes’ theorem.
Practice Game Theory Puzzles:
- Solve classic and modified game theory problems (Prisoner’s Dilemma, RPS with constraints, etc.).
- Understand Nash Equilibria and mixed strategies.
Mock Interviews:
- Simulate interviews with peers or mentors, focusing on articulating your thought process.
- Be ready to justify every step and consider edge cases or alternate approaches when prompted.
Review Behavioral and Market Questions:
- While technical, SIG also values candidates who think like traders. Be ready for questions about risk, market intuition, and decision-making under uncertainty.

3.4 Common SIG Quant Interview Question Types

Probability Puzzles: Balls in bins, coin flips, dice games, etc.
Estimation Problems: Fermi problems (“How many piano tuners are there in New York?”).
Game Theory Scenarios: Modified games like the rock-paper-scissors variant discussed above.
Trading and Market Making: Questions about bid-ask spreads, arbitrage, and market dynamics.

3.5 Sample Probability Problem

Here’s an example of a probability puzzle you might encounter:

A box contains 3 red balls and 2 blue balls. You draw two balls at random without replacement. What is the probability that both balls are red?

Let’s compute:

Total ways to choose 2 balls: $C(5,2) = 10$
Ways to choose 2 reds: $C(3,2) = 3$
Probability: $3/10 = 0.3$

So the probability is 0.3.

4. Advanced Concepts Behind the SIG Questions

4.1 Deep Dive: Conditional Expectation in Gaussian Distributions

The first SIG question is a beautiful application of conditional expectation in joint Gaussian distributions. Let’s expand on why this is so powerful:

Linearity: For jointly normal variables, the conditional expectation is always linear in the observed variable.
Variance as “Weight”: The amount of information you gain about $x$ from knowing $z$ depends on how much of $z$’s variance is due to $x$ itself.
Bayesian Connection: This is fundamentally Bayesian updating—the observed sum $z$ shifts your belief about $x$ proportionally to its share of the uncertainty in $z$.

Such questions test your knowledge of both theory and real-world inference, vital for quantitative research.

4.2 Game Theory with Asymmetric Information

The rock-paper-scissors variant is a classic demonstration of exploiting an opponent’s constraints. In financial markets, recognizing and exploiting informational or behavioral biases is key. The skills tested include:

Payoff Matrix Construction: Translating a real-world scenario into a mathematical framework.
Optimal Pure/Mixed Strategy Calculation: Using algebra to maximize expected gain.
Adaptability: Adjusting your strategy as the opponent’s behavior changes.

5. Frequently Asked Questions About SIG Quant Interviews

Question	Answer
What level of math is expected?	Strong undergraduate-level probability, statistics, and linear algebra. Some knowledge of stochastic processes and optimization is a plus.
How much programming is required?	For research roles, Python/Matlab/R proficiency is helpful but not always central to the first-round interviews. Expect some coding or pseudo-code questions.
Are brainteasers common?	Yes, expect puzzles that test logical reasoning and creativity.
What is the interview format?	Usually a mix of phone/virtual screens, technical interviews, and behavioral/fit interviews.
What resources are recommended?	Heard on the Street by Timothy Crack Probability and Statistics by Morris DeGroot Game Theory 101 by William Spaniel Online sites: Glassdoor, QuantNet, SIG’s own recruiting pages

6. Conclusion

Quantitative research interviews at SIG are a blend of mathematical rigor, strategic intuition, and clear communication. By dissecting questions like the joint Gaussian conditional expectation and the rock-paper-scissors variant, you hone the core skills that SIG values most. Remember:

Always reduce the problem to its mathematical essence—identify variables, relationships, and constraints.
Construct payoff matrices or conditional expectations step by step.
Communicate your logic clearly, considering both technical details and practical intuition.
Exploit informational advantages, just as you would in markets.

Approach your preparation with curiosity and persistence. Practice a wide variety of problems, and you’ll be well-equipped to excel in SIG’s quant research interviews—and beyond.

7. References & Further Reading

SIG Interview Preparation Resources
Quantitative Finance Stack Exchange
Timothy Crack, Heard on the Street: Quantitative Questions from Wall Street Job Interviews
Morris DeGroot & Mark Schervish, Probability and Statistics
William Spaniel, Game Theory 101: The Complete Textbook

By mastering the concepts and strategies exemplified in these SIG quant research interview questions, you’ll be well-prepared to tackle even the toughest challenges in the world of quantitative finance.