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Monte Carlo Simulation

When you cannot compute an expectation, sample it:

The Monte Carlo estimator
E[f(X)]≈1n∑i=1nf(Xi)E[f(X)] \approx \frac{1}{n}\sum_{i=1}^n f(X_i)

An average over draws, unbiased at any n, with an error that falls as one over root n.

Generate many scenarios, evaluate, average. The law of large numbers guarantees convergence and the CLT gives the error.

The rate, and why it is both bad and good

Standard error=σfn\text{Standard error} = \frac{\sigma_f}{\sqrt{n}}

Bad: one more digit of accuracy costs 100 times the samples. Monte Carlo is never precise cheaply.

Good: the rate does not depend on dimension. Numerical integration on a grid needs mdm^d points in dd dimensions and becomes impossible past a handful. Monte Carlo converges at 1n\frac{1}{\sqrt{n}} whether the problem has 2 dimensions or 200.

Key takeaway

Monte Carlo converges slowly but at a rate independent of dimension. That is the entire reason it dominates derivatives pricing and portfolio risk, where the dimension is the number of assets or time steps.

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