Quantitative Finance and Statistical Modeling Fundamentals

This interview assesses your foundational understanding of quantitative finance and statistical modeling, specifically focusing on derivatives pricing and linear regression. You will be asked to derive key formulas, explain underlying intuition, and discuss critical assumptions.

Part 1: Futures and Forwards Pricing

Assume an idealized, frictionless market characterized by:

• An underlying asset with a current spot price $S_0$.
• A continuously compounded risk-free interest rate $r$.
• A contract maturity $T$ (in years).
• No storage costs, no income or dividends from the asset, and no transaction costs or arbitrage opportunities.

Answer the following questions:

1. Derivation of Futures Price: Using a no-arbitrage argument, derive the theoretical fair forward/futures price $F_0$ and demonstrate that it satisfies the relationship:

   F_0 = S_0 * \exp(r * T)
   

2. Intuition: Explain the underlying intuition behind this formula. Why does the futures price relate to the spot price and the risk-free interest rate in this specific manner?
3. European Options Pricing (High-Level): Briefly explain, at a high level, how similar replication and no-arbitrage principles are applied to price simple European options on the same underlying asset. You are not required to derive a full closed-form solution like the Black-Scholes formula, but you should outline the overall approach (e.g., via constructing a replicating portfolio or using risk-neutral valuation).

Part 2: Linear Regression and Ordinary Least Squares (OLS)

Consider the standard linear regression model:

y = X * \beta + \epsilon

where:

• $y$ is an $n$-dimensional response vector.
• $X$ is an $n \times p$ design matrix with full column rank.
• $\beta$ is a $p$-dimensional vector of unknown parameters.
• $\epsilon$ is an $n$-dimensional error term.

Answer the following questions:

1. Classical OLS Assumptions: State the classical assumptions that underpin Ordinary Least Squares (OLS) linear regression. Examples of such assumptions include linearity, independence of errors, zero mean of errors, homoscedasticity, and absence of multicollinearity.
2. OLS Estimator Derivation: Derive the closed-form expression for the OLS estimator $\hat{\beta}$ by minimizing the sum of squared residuals:

   \text{minimize over } \beta: \quad ||y - X * \beta||^2
   

   Show the main steps of your derivation and provide the final matrix formula for $\hat{\beta}$.
3. Simple Linear Regression Slope: For a simple linear regression model with a single predictor ($y_i = \alpha + \beta * x_i + \epsilon_i$), write the closed-form expression for the slope coefficient $\hat{\beta}$ in terms of sample covariances and variances (e.g., using $\text{Cov}(x, y)$ and $\text{Var}(x)$).
4. Additional Assumptions for BLUE and Inference: Briefly state what additional assumptions are necessary for:
   • $\hat{\beta}$ to be the Best Linear Unbiased Estimator (BLUE).
   • Standard inference procedures (such as t-tests and confidence intervals) about $\beta$ to be valid.