Monte Carlo Sampling

Monte Carlo Sampling

Monte Carlo simulation uses random sampling to model uncertainty. In quantitative finance, it is used to price derivatives, estimate risk (VaR), and simulate portfolio paths.

A key building block is generating samples from a normal distribution using the Box-Muller transform. Given two independent uniform samples $U_1, U_2 \in (0, 1)$:

$Z_0 = \sqrt{-2 \ln U_1} \cos(2\pi U_2)$ $Z_1 = \sqrt{-2 \ln U_1} \sin(2\pi U_2)$

Both $Z_0$ and $Z_1$ are independent standard normal samples. Scale to mean $\mu$ and std $\sigma$: $X = \mu + \sigma Z$.

This transform is efficient because each pair $(U_1, U_2)$ yields two normal samples.

Your Task

Implement:

• mc_normal_samples(n, mu, sigma, seed) — generate n normal samples using Box-Muller and random.seed(seed)
• mc_mean(samples) — mean of a list of samples