Monte Carlo VaR

Monte Carlo VaR

Monte Carlo simulation estimates VaR by generating thousands of synthetic return scenarios from an assumed distribution, then taking the empirical percentile. It is more flexible than the parametric approach and can handle non-linear instruments.

Box-Muller Transform

To generate standard normal samples from uniform randoms U1, U2 ∈ (0,1):

Z1 = sqrt(-2 ln U1) · cos(2π U2)
Z2 = sqrt(-2 ln U1) · sin(2π U2)

Each pair (Z1, Z2) gives two independent standard normals.

Algorithm

1. Set random seed for reproducibility
2. Generate n_sim returns: r = μ + σ · Z
3. Sort simulated returns
4. VaR = −return at index int((1 − confidence) × n_sim)

Example

μ = 0.001, σ = 0.02, n = 10000, seed = 42, 95% confidence → VaR ≈ 0.0316