Monte Carlo Simulation

Estimating expectations by averaging random samples.

Monte Carlo simulation estimates an expectation by averaging i.i.d. samples:

E[g(X)] \approx \frac{1}{N} \sum_{i=1}^{N} g(X_i)

By the LLN this converges to the truth; by the CLT, the estimation error scales like $\sigma/\sqrt{N}$ . To halve the error you need $4\times$ the samples — Monte Carlo is robust but slow.

In quant finance, Monte Carlo prices any payoff you can simulate: simulate $N$ paths of an underlying under the risk-neutral measure, compute the payoff on each, and average. The method handles path-dependent options (Asian, barrier, lookback) where closed-form solutions don't exist, and high-dimensional baskets where lattice methods become impractical.

The big strength: dimension-independence. The convergence rate is $1/\sqrt{N}$ regardless of how many dimensions $X$ lives in. That's why Monte Carlo dominates lattice methods past about $4$ dimensions.