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Quant Researcher Interview Questions from JP Morgan, Akuna Capital, and Others

1. What are the Standard Boundary Conditions for a Differential Equation ? (JP Morgan)

Differential equations are fundamental in quantitative finance, governing the evolution of asset prices, derivative pricing, and risk modeling. When solving a differential equation, we often require additional information in the form of boundary conditions to ensure a unique solution. Let's explore the standard types of boundary conditions encountered in both ordinary and partial differential equations.

Types of Boundary Conditions

Dirichlet Boundary Condition (First-type): Specifies the value of the function at the boundary.
Neumann Boundary Condition (Second-type): Specifies the derivative of the function at the boundary.
Robin Boundary Condition (Third-type or Mixed): Specifies a linear combination of the function and its derivative at the boundary.
Periodic Boundary Condition: Requires the function (and possibly its derivatives) to be periodic over the boundary.

Mathematical Formulation

Consider an ordinary differential equation (ODE):

$$ \frac{d^2u}{dx^2} = f(x), \quad a \leq x \leq b $$

Dirichlet:
$u(a) = \alpha$, $u(b) = \beta$
Neumann:
$u'(a) = \gamma$, $u'(b) = \delta$
Robin:
$a_1 u(a) + b_1 u'(a) = c_1$,
$a_2 u(b) + b_2 u'(b) = c_2$
Periodic:
$u(a) = u(b)$,
$u'(a) = u'(b)$

Applications in Quantitative Finance

A classic application is the Black-Scholes PDE for option pricing. For a European call option:

$$ \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} - rV = 0 $$

Typical boundary conditions are:

Terminal condition (Dirichlet): At expiry $t = T$, $V(S, T) = \max(S - K, 0)$
Boundary at $S = 0$ (Dirichlet): $V(0, t) = 0$
Boundary as $S \to \infty$ (Dirichlet): $V(S, t) \sim S - K e^{-r(T-t)}$

Boundary conditions are essential to ensure the well-posedness of the differential equation and to reflect the financial realities of the problem being modeled.

2. What is the Difference Between Uncorrelated vs. Independent Random Variables? (Akuna Capital)

Understanding the distinction between uncorrelated and independent random variables is vital in risk modeling and portfolio optimization. While these terms are related, they are not equivalent.

Definitions

Independence: Two random variables $X$ and $Y$ are independent if the occurrence of one does not affect the probability distribution of the other. Formally: $$ P(X \leq x, Y \leq y) = P(X \leq x) \cdot P(Y \leq y) \quad \forall x, y $$
Uncorrelated: Two random variables $X$ and $Y$ are uncorrelated if their covariance is zero: $$ \mathrm{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])] = 0 $$

Key Insight: Independence implies zero correlation, but zero correlation does not imply independence.

Example: Uncorrelated but Not Independent

Consider $X$ as a random variable uniformly distributed on $[-1, 1]$, and define $Y = X^2$.

$E[X] = 0$ (since the distribution is symmetric about 0)
$E[Y] = E[X^2] = \frac{1}{2} \int_{-1}^1 x^2 dx = \frac{1}{2} \cdot \frac{2}{3} = \frac{1}{3}$
$E[XY] = E[X^3] = \frac{1}{2} \int_{-1}^1 x^3 dx = 0$ (since $x^3$ is an odd function over symmetric limits)
$\mathrm{Cov}(X, Y) = E[XY] - E[X]E[Y] = 0 - 0 \cdot \frac{1}{3} = 0$

Thus, $X$ and $Y$ are uncorrelated. However, they are clearly not independent since $Y$ is a deterministic function of $X$.

Intuitive Explanation

Uncorrelatedness only captures the linear relationship between variables. Independence is a much stronger statement about their joint behavior, including all possible dependencies.

Practical Implications

In portfolio theory, assuming asset returns are independent when they are merely uncorrelated can lead to underestimated risk, particularly in the presence of nonlinear dependencies.

3. Distribution of the Sum of N Independent Uniform Random Variables (Akuna Capital)

Question: Let $X_1, X_2, ..., X_N$ be $N$ independent random variables, each uniformly distributed on $[0,1]$. Let $X = \sum_{i=1}^N X_i$. What is the distribution of $X$?

Irwin-Hall Distribution

The sum $X$ follows the Irwin-Hall distribution with parameter $N$.

The probability density function (pdf) is:

$$ f_X(x) = \frac{1}{(N-1)!} \sum_{k=0}^{\lfloor x \rfloor} (-1)^k \binom{N}{k} (x - k)^{N-1}, \quad x \in [0, N] $$

Properties

Mean: $E[X] = \frac{N}{2}$
Variance: $\mathrm{Var}[X] = \frac{N}{12}$
Support: $[0, N]$

Special Cases

N = 1: The distribution is uniform on $[0,1]$.
N = 2: The distribution is triangular on $[0,2]$, with pdf: $$ f_X(x) = \begin{cases} x, & 0 \leq x < 1 \\ 2 - x, & 1 \leq x \leq 2 \end{cases} $$

Asymptotic Behavior (Central Limit Theorem)

For large $N$, by the Central Limit Theorem (CLT), $X$ approaches a normal distribution:

$$ X \sim \mathcal{N}\left(\mu = \frac{N}{2}, \sigma^2 = \frac{N}{12}\right) $$

This approximation is often used in practice for $N \geq 10$.

Python Simulation Example


import numpy as np
import matplotlib.pyplot as plt

N = 5
num_samples = 100000
samples = np.sum(np.random.uniform(0, 1, (num_samples, N)), axis=1)

plt.hist(samples, bins=50, density=True, alpha=0.6, label='Simulation')
plt.title('Distribution of Sum of {} Uniform[0,1] RVs'.format(N))
plt.xlabel('Sum')
plt.ylabel('Density')
plt.show()

Practical Application in Quant Finance

This distribution is relevant when aggregating independent sources of risk or returns, such as when modeling the total P&L from multiple independent trading strategies.

4. What's the first quick test you would do to check a new options strategy? (Campbell & Company)

When evaluating a new trading strategy based on options, time and computational resources are limited. Thus, a quick initial test is essential to assess whether a strategy has potential before investing in deeper analysis.

Quick First Test: In-Sample P&L and Risk-Adjusted Return

A standard quick test is to backtest the strategy on historical data and compute the Sharpe ratio or another risk-adjusted metric. This gives a fast indication of whether the strategy has any "edge" or if its performance is due to randomness.

Sharpe Ratio: $$ \text{Sharpe Ratio} = \frac{E[R_p - R_f]}{\sigma_p} $$ where $R_p$ is the portfolio return, $R_f$ is the risk-free rate, and $\sigma_p$ is the standard deviation of returns.
Hit Ratio: Proportion of profitable trades.
Drawdown Analysis: Maximum drawdown during the test period.

Implementation Steps

Define Entry/Exit Rules: Clearly specify how the strategy generates buy/sell signals.
Collect Historical Option Price Data: Gather sufficiently granular data for backtesting.
Run Backtest: Simulate trades according to strategy rules.
Analyze Performance Metrics: Compute Sharpe ratio, average P&L, and drawdown.

Python Example: Simple Backtest


import pandas as pd
import numpy as np

# Assume df contains columns: 'option_price', 'signal'
df['returns'] = df['option_price'].pct_change().shift(-1) * df['signal']
strategy_pnl = df['returns'].cumsum()
sharpe = np.mean(df['returns']) / np.std(df['returns'])

print("Strategy Sharpe Ratio:", sharpe)

Other Quick Tests

Scatter Plot of Returns vs. Feature: Visual inspection for predictive power.
Correlation with Underlying Asset: Ensure new strategy is not just replicating the underlying.
Transaction Cost Sensitivity: Assess robustness to realistic bid-ask spreads and slippage.

Key Considerations

Overfitting Trap: Avoid reading too much into in-sample performance before out-of-sample validation.
Liquidity and Execution: Check if strategy signals can be executed at scale.

Conclusion

Quantitative researcher interviews at leading financial institutions probe both foundational knowledge and problem-solving skills. Mastery of boundary conditions in differential equations, a nuanced understanding of statistical relationships such as uncorrelated versus independent variables, familiarity with probability distributions, and the ability to rapidly evaluate trading strategies are all crucial for success. By practicing and deeply understanding the concepts behind these interview questions, candidates can distinguish themselves and build a robust toolkit for careers in quantitative research and trading.

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