Estimate Volatility from Gaussian and Brownian Samples
Company: Tudor
Role: Data Scientist
Category: Machine Learning
Difficulty: medium
Interview Round: Technical Screen
You are given two statistical estimation problems.
1. Let X_1, ..., X_100 be independent samples from N(0, sigma^2), where the mean is known to be 0 and sigma is unknown. Propose an estimator for sigma. Is the estimator unbiased? If not, how can it be debiased?
2. Let X_t = sigma W_t be a Brownian motion with X_0 = 0, observed at integer times t = 1, 2, ..., 100. Define H = max(X_1, ..., X_100), L = min(X_1, ..., X_100), C = X_100, and R = H - L. Consider calibrated versions of the following statistics for estimating volatility: (i) |C|, (ii) R - |C|/2, and (iii) R^2 + C^2. How would you compare these estimators, and which would you prefer under a mean-squared-error criterion? State whether you are estimating sigma or sigma^2.
Quick Answer: This question evaluates skills in statistical estimation and stochastic process modeling, specifically parameter estimation for Gaussian samples and volatility estimation from discrete Brownian motion observations, focusing on estimating the volatility parameter (sigma) or its variance (sigma^2).