MLE Fit for Student-t Distribution - Quant Researcher Interview Question

Problem Description

Asset returns frequently exhibit leptokurtosis, or heavy tails, rendering the Normal distribution insufficient for accurate risk modeling and Value-at-Risk calculations. The Student's t-distribution addresses this by incorporating a degrees-of-freedom parameter to control tail thickness, providing a more robust fit for financial time series. Maximum Likelihood Estimation (MLE) is the standard statistical method used to calibrate these parameters by maximizing the probability of observing the his