Quant Researcher interview prep

Short answer: A p-value is the probability of observing a test statistic at least as extreme as the one you got, assuming the null hypothesis is true: P(data ≥ observed | H₀). It is a statement about the data under H₀ — not about the probability of the hypothesis itself.

In depth:

1. Direction of the conditioning — the p-value conditions on H₀. Getting P(H₀ | data) requires a prior and Bayes' theorem; you cannot extract it from a p-value alone.
2. Not the probability of the null — "p = 0.03 means H₀ is true with probability 3%" is the classic trap interviewers specifically listen for.
3. Not an effect size — with huge n, a tiny, economically meaningless effect yields p < 0.001. Statistical significance ≠ practical importance.

p-value  =  P(T ≥ t_obs | H₀ true)        ✓
p-value  ≠  P(H₀ true | data)             ✗
1 − p    ≠  P(the effect is real)         ✗

⚠️ Common mistake: "1 − p = 97% probability the effect is real." A p-value says nothing about the probability of hypotheses — only how surprising the data are if H₀ holds.

Short answer: β = Cov(x, y) / Var(x), intercept α = ȳ − β·x̄. β is how much y changes on average when x rises by one unit. CAPM beta is exactly this coefficient: a regression of asset returns on market returns.

In depth:

1. Derivation from the normal equations — minimize Σ(yᵢ − α − βxᵢ)², take derivatives with respect to α and β, set them to zero.
2. Via correlation — β = ρ(x, y) · σ_y / σ_x: the slope is the correlation scaled by the ratio of volatilities.
3. CAPM — β_market = Cov(rᵢ, r_m) / Var(r_m): a stock's "market beta" is literally the one-regressor OLS coefficient of asset returns against market returns. In an interview this must come out instantly, no scratch paper.

∂/∂α: Σ(yᵢ − α − βxᵢ) = 0    →  α = ȳ − β·x̄
∂/∂β: Σxᵢ(yᵢ − α − βxᵢ) = 0  →  β = Σ(xᵢ−x̄)(yᵢ−ȳ) / Σ(xᵢ−x̄)²
                                   = Cov(x, y) / Var(x)

⚠️ Common mistake: mixing up direction: the β from regressing y on x is not the β from regressing x on y (they coincide only when |ρ| = 1). And fumbling the formula — a QR is expected to write Cov/Var without thinking.

Short answer: A Wiener process (standard Brownian motion) W_t is a stochastic process with four defining properties: W₀ = 0; independent increments; W_t − W_s ~ N(0, t − s) for s < t; continuous sample paths almost surely. It is the basic source of randomness in continuous-time finance.

In depth:

1. W₀ = 0 — normalization of the starting point.
2. Independent increments — increments over disjoint intervals are independent: the past does not help predict the future (hence the Markov property).
3. Gaussian increments — W_t − W_s ~ N(0, t − s): variance grows linearly in time, so the typical range scales like √t.
4. Continuous paths — almost surely no jumps, yet the paths are maximally rough.

Bonus properties interviewers fish for:
• nowhere differentiable: |ΔW| ~ √Δt ≫ Δt as Δt → 0
• quadratic variation: Σ(ΔW)² → t  ⇒  (dW)² = dt
  (the seed from which Itô's lemma grows)
• self-similarity: W_ct ~ √c·W_t;  W_t = √t·Z with Z ~ N(0,1)

⚠️ Common mistake: offering "W_t ~ N(0, t)" as the definition. That is only a consequence: without independent increments and continuous paths, a process with those marginal distributions is not a Wiener process.

Short answer: L1 (lasso) penalizes the sum of |w| and drives some weights to exactly zero — you get a sparse model with built-in feature selection. L2 (ridge) penalizes the sum of w² and smoothly shrinks all weights toward zero without zeroing them, and it behaves better with correlated features.

In depth:

1. Geometry — the L1 ball has corners on the coordinate axes, and the optimum often lands exactly there → exact zeros. The L2 ball is smooth → shrinkage without zeros.
2. Bayesian view — L1 is equivalent to a Laplace prior (sharp peak at zero), L2 to a Gaussian prior.
3. Correlated features — L1 arbitrarily picks one representative of a group, L2 splits the weight among them → more stable.
4. Elastic net — a blend of both penalties when you want sparsity and stability.

⚠️ Common mistake: expecting "interpretability for free" from L1. The sparse feature set is unstable: refit on slightly different data and lasso will pick a different subset of the correlated factors — a hidden operational risk for production alpha.

Short answer: Monte Carlo "darts": throw N random points into the unit square and take the fraction landing inside the quarter circle (x² + y² ≤ 1). That fraction converges to π/4, so π ≈ 4 · (hit fraction).

In depth:

1. Geometry — the quarter circle of radius 1 has area π/4, the square has area 1; the hit probability is the ratio of areas.
2. Convergence — by the CLT the error decays as 1/√N: each extra significant digit costs ~100× more samples.
3. Vectorization — generate all points at once, no Python loop over individual points.

import numpy as np

rng = np.random.default_rng(42)
n = 10_000_000
x, y = rng.random(n), rng.random(n)
pi_hat = 4 * np.mean(x**2 + y**2 <= 1)
# error ~ 1/sqrt(n): at n = 10^7 roughly 3–4 digits are correct

⚠️ Common mistake: expecting Monte Carlo to reach precision quickly: going from 3 to 6 correct digits takes ~10⁶× more points. For the constant π itself this is the worst possible method — but the question tests whether you understand the 1/√N convergence rate shared by every MC estimate.