Monte Carlo simulation
Quantitative Finance glossary · Numerical Methods & Optimization
Definition
Numerical estimation of mathbbE[f(X)] ≈ 1/Nsum_i=1^N f(X^(i)) by sample averaging, with O(1/√(N)) convergence regardless of dimension — the property that makes MC the only practical pricer for high-dimensional path-dependent claims. Variance reduction (antithetic variates, control variates, importance sampling, stratified sampling) cuts the constant in front of 1/√(N) — often by an order of magnitude. Quasi-Monte Carlo (Sobol, Halton sequences) replaces pseudo-random with low-discrepancy points for O((log N)^d / N) convergence in smooth settings.
Monte Carlo simulation in plain English
When a calculation is too tangled to solve with a clean formula, you can just simulate it thousands of times and average the results — that's Monte Carlo, named after the casino. Want to know the average outcome of some messy process? Generate N random scenarios, compute the result in each, and take the mean; the more runs you do, the closer your estimate gets, with error shrinking like 1/√(N) (so 100× more runs cuts error 10× ). Its superpower is that this works even when there are dozens of moving parts at once, which is why banks use it to price complicated financial bets no formula can handle. Clever tricks (antithetic variates, Sobol sequences) squeeze more accuracy out of the same number of runs, like sampling a crowd more evenly instead of purely at random.
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