Quasi-Monte Carlo Method

Quasi-Monte Carlo (QMC) methods replace the pseudo-random sequences of standard Monte Carlo with deterministic low-discrepancy sequences — sequences designed to fill space more uniformly than random points. The canonical example is the Sobol sequence, which achieves a discrepancy of O((log N)^d / N) in d dimensions, compared to O(1/√N) for pure Monte Carlo. For smooth integrands in moderate dimensions, QMC can outperform Monte Carlo by orders of magnitude.

But QMC is not a universal replacement. It performs poorly when the integrand is highly irregular or when dimensionality is very high, because the (log N)^d factor eventually dominates. The choice between Monte Carlo and quasi-Monte Carlo is a trade-off between the insurance of randomness and the efficiency of structure — and the right choice, as always, depends on what you know about the problem. QMC methods are particularly effective in financial derivative pricing where the integrands are smooth and dimensions are moderate.

Quasi-Monte Carlo is the computational equivalent of obsessive orderliness: it works beautifully when the world cooperates, and fails catastrophically when the world is messier than expected. The deterministic perfection of low-discrepancy sequences is their strength and their fragility.