← Morgan Stanley Interview Insights
This is pretty standard if you've seen it before, but I fumbled the actual arbitrage construction under pressure.
Start by setting up a no-arbitrage argument: construct two portfolios—one with a long forward contract and a bond, and another with the underlying asset—and show that their payoffs are identical at maturity. Then, equate the present values to derive the forward price formula F = S * e^(rT).
Pro tip: Emphasize that the no-arbitrage argument assumes no transaction costs, no storage costs, and no dividends; mention that in practice, these factors adjust the formula, showing awareness of real-world complexities.
Assume no arbitrage opportunities exist. Consider two strategies: (1) buy the underlying asset at spot price S and hold until maturity T, and (2) enter a long forward contract at price F and invest the present value of F in a risk-free bond.
At maturity, both strategies yield the same payoff: the spot price of the asset at T. For strategy 1, you own the asset worth S_T. For strategy 2, the forward contract pays S_T - F, and the bond pays F, netting S_T.
Since the payoffs are identical, their initial costs must be equal to prevent arbitrage. The cost of strategy 1 is S. The cost of strategy 2 is the present value of F, which is F * e^(-rT). Thus, S = F * e^(-rT).
Rearrange the equation to get F = S * e^(rT). This shows that the theoretical fair forward price equals the spot price compounded at the risk-free rate over the contract's maturity.
Mention that this holds under no-arbitrage with no costs, dividends, or convenience yields. For assets with income or storage costs, the formula adjusts to F = (S - I) * e^(rT) or F = S * e^((r+u)T), etc.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Easier than the derivation question but I over-explained it.
Start by stating the core no-arbitrage principle: the futures price must equal the spot price compounded at the risk-free rate, adjusted for any carry costs or benefits. Then explain the intuition with a simple cash-and-carry example, and finally discuss how deviations would create arbitrage opportunities.
Pro tip: Emphasize that the formula is a theoretical fair value under no-arbitrage, not a prediction; in practice, factors like storage costs, dividends, and convenience yields cause deviations. This shows you understand both theory and real-world nuances.
Explain that in an efficient market, two portfolios with identical future payoffs must have the same current price, otherwise arbitrage exists.
Describe buying the asset spot with borrowed money and holding it to delivery, then compare with a long futures position. The futures price must equal the future value of the spot price.
Show that F = S * e^(rT) for no dividends, where r is the risk-free rate and T is time to maturity. Explain that the exponential reflects continuous compounding.
Mention that if the asset has storage costs, dividends, or convenience yield, the formula becomes F = S * e^((r + c - y)T), where c is cost and y is yield.
Explain that if F > S*e^(rT), one can arbitrage by shorting futures and buying spot; if F < S*e^(rT), reverse. In practice, transaction costs and market frictions limit arbitrage.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by defining replication as constructing a portfolio that matches the option's payoff, then invoke no-arbitrage to equate the option's price to the portfolio's cost. Explain that in a binomial setting, this yields a unique risk-neutral probability, and in continuous time, it leads to the Black-Scholes PDE. Conclude by noting that the risk-neutral expectation of the discounted payoff gives the price.
Pro tip: Emphasize that no-arbitrage ensures the replication portfolio is unique and its cost is the fair price; this avoids needing to estimate real-world drift or risk premia, which is a key insight for data scientists working with financial models.
Explain that replication means building a portfolio of the underlying asset and a risk-free bond that exactly matches the option's payoff at expiration.
State that if two assets have identical future payoffs, their current prices must be equal to prevent arbitrage; thus the option price equals the replication portfolio's cost.
Illustrate with a one-period binomial model: solve for the number of shares and bonds that replicate the option, then derive the risk-neutral probability.
Mention that in continuous time, dynamic replication leads to the Black-Scholes PDE, whose solution is the risk-neutral expected discounted payoff.
Conclude that the price is the expected discounted payoff under the risk-neutral measure, which is a direct consequence of no-arbitrage.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Blanked on the formal name for one of them mid-sentence.
Start by clearly listing the classical assumptions of OLS (linearity, exogeneity, homoskedasticity, no autocorrelation, no multicollinearity, and normality of errors). Then briefly explain the consequences if each assumption is violated, and mention how you would test or address violations in practice. Finally, connect the assumptions to the Gauss-Markov theorem to highlight why they matter for unbiasedness and efficiency.
Pro tip: Emphasize that while normality is not required for unbiasedness or efficiency (only for exact inference in small samples), it is often mistakenly listed as a core assumption. Showing this nuance demonstrates deeper understanding and practical maturity.
Enumerate the classical assumptions: linearity in parameters, random sampling, no perfect multicollinearity, zero conditional mean (exogeneity), homoskedasticity, and no autocorrelation. Optionally include normality of errors for inference.
For each assumption, state what it ensures: unbiasedness (exogeneity), efficiency (homoskedasticity and no autocorrelation), identifiability (no multicollinearity), and valid inference (normality).
Mention that under assumptions 1-5 (excluding normality), OLS is BLUE (Best Linear Unbiased Estimator). This shows you understand the theoretical foundation.
Briefly note common violations (e.g., heteroskedasticity, autocorrelation, multicollinearity) and how to detect (e.g., residual plots, VIF, Durbin-Watson) and address them (e.g., robust standard errors, transformations, regularization).
Tie the assumptions to real-world modeling: in large datasets, some assumptions matter less (e.g., normality for inference), but others (e.g., exogeneity) are critical for causal interpretation.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Taking the derivative, setting to zero, getting to the normal equations and then the hat matrix formula.
Start by setting up the OLS minimization problem in matrix notation, then derive the normal equations by taking the gradient of the sum of squared residuals and setting it to zero. Solve for the coefficient vector to obtain the closed-form estimator, and briefly discuss the assumptions and interpretation.
Pro tip: Emphasize the geometric interpretation: the OLS estimator projects the response vector onto the column space of the design matrix, making the residuals orthogonal to the predictors. This shows deeper understanding beyond algebraic manipulation.
State the linear model y = Xβ + ε and define the sum of squared residuals (SSR) as (y - Xβ)ᵀ(y - Xβ).
Expand the quadratic form: SSR = yᵀy - 2βᵀXᵀy + βᵀXᵀXβ.
Compute the derivative with respect to β: ∂SSR/∂β = -2Xᵀy + 2XᵀXβ, and set it to zero to obtain the normal equations XᵀXβ = Xᵀy.
Assuming XᵀX is invertible, solve the normal equations to get the closed-form OLS estimator: β̂ = (XᵀX)⁻¹Xᵀy.
Mention key assumptions (e.g., full rank of X, exogeneity) and properties (e.g., unbiasedness, BLUE under Gauss-Markov), and note the geometric projection interpretation.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
State the closed-form slope estimate as the ratio of the sample covariance between X and Y to the sample variance of X. Then briefly explain the intuition: the slope captures how much Y changes per unit change in X, scaled by the variability of X. Keep the answer concise and mathematically precise, as expected in a quantitative finance interview.
Pro tip: Mention that this formula is equivalent to the OLS estimator and that it highlights the importance of predictor variance—if X has little variance, the slope estimate becomes unstable. This shows you understand both the math and its practical implications for modeling.
State the simple linear regression model: Y = β0 + β1 X + ε, and clarify that you are estimating β1.
Present the formula: β1_hat = Cov(X, Y) / Var(X), where Cov and Var are sample statistics.
Define sample covariance as (1/(n-1)) Σ (Xi - X̄)(Yi - Ȳ) and sample variance as (1/(n-1)) Σ (Xi - X̄)^2, noting that the (n-1) cancels out.
Explain that the slope is the covariance normalized by the variance of X, measuring the linear association per unit of X variability.
Mention that this is the ordinary least squares (OLS) estimator, obtained by minimizing the sum of squared residuals.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
For BLUE you need the Gauss-Markov conditions, basically homoscedastic uncorrelated errors.
Start by distinguishing the Gauss-Markov assumptions needed for BLUE (linearity, exogeneity, no perfect collinearity, homoskedasticity, no autocorrelation) from the additional normality assumption required for exact t-tests and confidence intervals in small samples. Emphasize that under large samples, asymptotic normality can suffice via CLT, but for exact inference, normality of errors is key. Conclude by noting practical implications for model validation.
Pro tip: In finance, heteroskedasticity and autocorrelation are common; mention robust standard errors (e.g., White or Newey-West) as a practical fix when assumptions fail, showing you understand real-world data challenges.
List the five assumptions: linearity in parameters, random sampling, no perfect collinearity, zero conditional mean (exogeneity), and homoskedasticity (no autocorrelation if time series). Explain that these ensure OLS is BLUE.
Explain that for t-tests and confidence intervals to be exactly valid in small samples, we need the errors to be normally distributed (normality assumption). This is on top of Gauss-Markov.
Note that with large samples, the normality assumption can be relaxed due to the Central Limit Theorem, making t-tests and CIs asymptotically valid even if errors are not normal.
Mention how to check assumptions (residual plots, tests for heteroskedasticity/autocorrelation) and what to do if violated (robust standard errors, transformations, alternative estimators).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.