Variance reduction

Variance reduction techniques shrink Monte Carlo standard errors without changing the number of paths. A 10× variance reduction = 100× fewer paths for the same accuracy = 100× speedup. The best techniques exploit problem-specific structure: control variates use a known-price related payoff; importance sampling shifts sampling toward where the payoff is non-zero; antithetic variates exploit symmetry.

Antithetic variates

For each random draw Z, also use -Z. The two estimates are negatively correlated for monotonic payoffs, reducing the average variance by up to 50% per pair.

V̂ = (1/(2N)) Σ [g(Z_i) + g(-Z_i)]
Var(V̂_antithetic) ≤ Var(V̂_standard) for symmetric or monotonic payoffs

Costs almost nothing (one extra payoff evaluation per sample). Standard in production MC engines. For non-monotonic payoffs (Asian, basket), benefit can be near zero.

Control variates

If Y has known expectation E[Y] = μ_Y and correlates with the payoff X you want to estimate:

V̂_CV = X̄ - c (Ȳ - μ_Y)
Optimal c* = Cov(X, Y) / Var(Y)
Var(V̂_CV) = Var(X)(1 - ρ_{X,Y}²)

With ρ_{X,Y} = 0.95, variance drops by 90% — a 10× reduction. For Asian options, use the geometric Asian (closed form) as a control for the arithmetic Asian. For barrier options, use the vanilla as the control. The art is finding a related instrument with known price and high correlation.

Importance sampling

When payoffs are zero on most paths (deep OTM options), most draws contribute nothing. Importance sampling biases the distribution toward 'interesting' regions and re-weights to remove the bias.

E_P[f(X)] = E_Q[f(X) dP/dQ]

Choose Q to have higher probability where f is large. The Radon-Nikodym derivative dP/dQ ('likelihood ratio') re-weights. For deep OTM call: shift the GBM drift to make the option finish in the money more often. Can give 100×-10000× variance reduction for deep tail events.

Stratified sampling

Partition the random variable's support into K strata; sample from each in proportion to expected payoff contribution. Reduces variance from cross-strata variance to within-strata variance only.

Quasi-Monte Carlo

Replace pseudo-random sequences with low-discrepancy sequences (Sobol, Halton, Niederreiter). For smooth d-dimensional integrands, convergence rate improves from O(1/√N) to O((log N)^d / N) — much faster in moderate dimensions. Industry-standard for option pricing in 5-30 dimensions.

Common random numbers (CRN)

When comparing two scenarios (e.g., computing a Greek by bump-and-reprice), use the same random sequence for both. CRN dramatically reduces variance of the difference; doubly important for finite-difference Greeks where the difference is small relative to either value alone.

Combining techniques

Multiple variance-reduction techniques compose: e.g., antithetic + control variate + quasi-MC. Production MC libraries (Quantlib, OpenGamma) implement all of the above and apply them automatically based on payoff structure.

Convergence rates

• Vanilla MC: O(1/√N).
• MC with antithetics: O(1/√N), constant reduced.
• MC with control variates: O(1/√N), constant proportional to √(1 - ρ²).
• Quasi-MC (smooth integrand): O((log N)^d / N) — superlinear improvement.
• Multilevel MC (Giles): O(1/N) for stochastic-SDE problems with discretisation bias.