Decision brief · Numerical methods
How many simulations are enough?
Simulation quality depends on measured sampling error, discretization bias, and variance reduction—not path count alone.
Depth 1 · Answer
“Enough” means the reported error is small enough for the decision—not that a round path count was reached.
Independent replications recovered the theoretical N⁻¹ᐟ² error law within replication noise. Exact GBM transitions remained unbiased across step counts, Euler bias contracted as the time step shrank, and antithetic pairing reduced empirical standard error by about 30% at the same 20,000-path budget.
Published 8 August 2026 · Underlying investigation last updated 6 July 2026
- Error rate
- N⁻¹ᐟ²
- 10× less error
- ≈100× paths
- Antithetic SE reduction
- ≈30%
- Primary seed
- 42
Why it matters
Pricing engineers, quantitative developers, and model reviewers
The result matters whenever simulation supports a valuation, risk threshold, capital estimate, stress result, or software acceptance test—especially for path-dependent problems.
Decision affected
When to stop computing and trust the numerical estimate
Set the error tolerance from the economic decision, measure estimator variability, test discretization separately, and stop only when the combined numerical budget is visible and acceptable.
Depth 2 · Evidence
Sampling error and scheme bias behaved as theory requires
The experiments were fully synthetic so known analytical moments and prices served as benchmarks. Independent replications at increasing path counts produced a log–log RMSE slope within noise of −0.5. Reducing error by a factor of ten therefore requires roughly one hundred times as many independent paths.
Exact transition schemes for GBM avoided time-step bias at observation points, while Euler error shrank with Δt. This distinction is essential: adding paths reduces sampling error but cannot repair a biased discretization scheme.
At an equal 20,000-path budget, antithetic normal draws reduced empirical standard error by about 30% across 30 independent replications. The gain is payoff-dependent; it is not a universal multiplier. The notebook then applies simulation where path dependence matters: a discretely monitored barrier option whose value depends on monitoring frequency.
Depth 3 · Application
Make the error budget part of the output contract
- Report a standard error or confidence interval beside every Monte Carlo point estimate.
- Test convergence across independent replications, not only with a single seeded run.
- Budget discretization bias by step refinement separately from statistical error by path count.
- Use exact transitions where available, and evaluate variance reduction before buying more compute.
- Match the method to the contract: closed forms where they apply, simulation where path details change value.
Depth 4 · Limits
A reproduced convergence law does not validate every simulation
- Every Monte Carlo estimate remains a random draw; reproducibility by seed does not eliminate uncertainty.
- CIR and Heston discretization bias was not fully budgeted by step-halving in this study.
- Antithetics can be weak for strongly convex or non-monotone payoffs.
- Regime-switching parameters were chosen rather than empirically estimated.
- All experiments used the PCG64 pseudorandom-generator family.
Depth 5 · Method and code
See the processes, derive the error law, and rerun the study
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