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TL;DR

Estimates parameters by maximizing the likelihood function: \( \hat{\theta}_{MLE} = \arg\max_\theta \prod_i f(x_i|\theta) \). Equivalently, maximize the log-likelihood \( \sum_i \log f(x_i|\theta) \).

By Valenke Exam Prep Team·Last updated 2026-06-03

Maximum Likelihood Estimation

Estimates parameters by maximizing the likelihood function: \( \hat{\theta}_{MLE} = \arg\max_\theta \prod_i f(x_i|\theta) \). Equivalently, maximize the log-likelihood \( \sum_i \log f(x_i|\theta) \).

Why it matters for interviews

MLE is the most widely used estimation method in quantitative finance: fitting distributions to returns, estimating GARCH parameters, calibrating option pricing models. Understanding its properties (consistency, asymptotic normality, efficiency) is essential.

Definition and Mathematical Foundation

Estimates parameters by maximizing the likelihood function: \( \hat{\theta}_{MLE} = \arg\max_\theta \prod_i f(x_i|\theta) \). Equivalently, maximize the log-likelihood \( \sum_i \log f(x_i|\theta) \).

Application in Quantitative Finance

MLE is the most widely used estimation method in quantitative finance: fitting distributions to returns, estimating GARCH parameters, calibrating option pricing models. Understanding its properties (consistency, asymptotic normality, efficiency) is essential.

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Frequently Asked Questions

What are the asymptotic properties of MLE?▼
Under regularity conditions: consistent (converges to true value), asymptotically normal, and asymptotically efficient (achieves the Cramer-Rao lower bound). These make MLE the default estimation method.
What is the Cramer-Rao lower bound?▼
The minimum variance achievable by any unbiased estimator: \( \text{Var}(\hat{\theta}) \geq 1/I(\theta) \) where \( I(\theta) \) is Fisher information. MLE achieves this bound asymptotically.
How is MLE used to fit return distributions?▼
Specify a parametric distribution (e.g., Student's t for fat tails), write the log-likelihood, and optimize numerically. The fitted parameters describe tail behavior, skewness, and kurtosis of returns.