Quasi-Monte Carlo

Quasi-Monte Carlo (QMC) replaces pseudo-random points with deterministic low-discrepancy sequences (Sobol, Halton, Niederreiter). The points spread more evenly through the unit cube, reducing integration error.

For sufficiently smooth integrands, QMC converges at $O((\log N)^d / N)$ — much faster than Monte Carlo's $O(1/\sqrt{N})$ for moderate dimensions. The catch: the rate depends on dimension, and for very high $d$ the advantage disappears.

In practice, QMC dominates standard MC for option pricing in $5$–$50$ dimensions, common in basket options and yield-curve simulation. Above that, MC's dimension-independence usually wins back.

Practical tips: scramble Sobol sequences for randomized QMC, which gives both the deterministic-rate advantage and a usable confidence interval. Variance reduction techniques can be combined with QMC for further speedup.