What is quasi-Monte Carlo and when does it beat regular Monte Carlo?

Question

AcadiFi · Accepted Answer

Quasi-Monte Carlo (QMC) replaces pseudo-random numbers with deterministic low-discrepancy sequences that fill space more uniformly. Popular sequences: Halton, Sobol, Faure, Niederreiter.

Standard MC error converges at 1/sqrt(N). QMC can achieve up to 1/N * (log N)^d in smooth integrands with dimension d. For 5-dimensional problems and 100,000 samples, QMC is often 10-100x more accurate than MC.

Bogdana Takemura at Osaka Structured Products prices an Asian option — average over 252 daily fixings. 252-dimensional integration. Crude MC with 50,000 paths gives standard error 0.034. Sobol sequence gives standard error 0.008 with same sample count — 4x improvement, saving compute.

Why it works: pseudo-random numbers cluster and leave gaps. Low-discrepancy sequences minimize clustering by construction.

```mermaid
flowchart LR
  A[Pseudo-random] --> B[Clusters and Gaps]
  C[Sobol Sequence] --> D[Uniform Coverage]
  B --> E[Error 1/sqrt N]
  D --> F[Error log^d N / N]
```

Practical considerations:
1. Effective dimension — QMC convergence degrades with high dimension. Brownian bridge construction reduces effective dimension for path-dependent options.
2. Scrambling — randomized QMC adds controlled randomness, enabling unbiased error estimates while preserving low-discrepancy.
3. Sobol is best for most finance applications; direction numbers from Joe-Kuo provide quality up to thousands of dimensions.
4. Not useful for MCMC or when samples need to be i.i.d.

Limits: discontinuous payoffs (barrier options, digitals) break smoothness assumption. For those, combine QMC with smoothing tricks. Also, high-dimensional rough problems may not benefit — fall back to crude MC with variance reduction.

Bogdana's team uses Sobol-with-Brownian-bridge for path-dependent Asians and cliquets. For discontinuous barriers, they use crude MC with importance sampling. Choosing the right tool depends on payoff smoothness and effective dimension.

What is quasi-Monte Carlo and when does it beat regular Monte Carlo?

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