Numerical methods · Simulation errorNumerical validation

Monte Carlo convergence, discretization bias, and variance reduction

Sampling error followed the N⁻¹ᐟ² law; enough paths is defined by the decision's tolerance.

Path count alone does not settle simulation quality. Measured sampling error, discretization bias, and variance reduction do.

Research by Kyle WisniewskiQuantitative Markets & Institutions Lab8 Aug 2026Revised 10 Aug 20264 figuresLedger entry

Commit 4806df9Evidence ad24c49Data 2 Jan 2015 – 6 Jul 2026 · 2,892 rowsUniverse 15 instrumentsTests 239 / 239 passed

N⁻¹ᐟ²Error rate of the Monte Carlo estimator
≈100×paths needed for 10× less error
−0.55Fitted log–log RMSE slope across 16 replications
≈30%Antithetic SE reduction at the same 20,000-path budget
1.30×Empirical SE, independent ÷ antithetic
42Primary seed (PCG64)

“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 stayed unbiased across step counts. Euler bias contracted as the time step shrank. Antithetic pairing cut empirical standard error by about 30% at the same 20,000-path budget.

Monte Carlo error decays like 1/√N

RMSE of the price estimate · 16 replications per N · seeds 7000 + rep · notebook 04

Lines of root-mean-square pricing error against path count on a logarithmic axis for independent and antithetic sampling, against the inverse-square-root reference line.
number of paths N (log scale)	RMSE, independent draws	RMSE, antithetic pairs	N⁻¹ᐟ² reference
250	0.847309	0.854915	0.847309
1,000	0.523811	0.362206	0.423654
4,000	0.241325	0.178972	0.211827
16,000	0.0881235	0.0583958	0.105914
64,000	0.0473331	0.0415612	0.0529568
256,000	0.0208947	0.0211394	0.0264784

Discretisation bias: exact scheme versus Euler–Maruyama

price − closed form · 400,000 common random numbers · seed 42 · notebook 04

Lines of the pricing bias of the exact GBM scheme and of Euler–Maruyama against the number of time steps; the exact scheme sits inside its standard-error band while Euler bias shrinks linearly with the step.
time steps per year	exact scheme bias	exact scheme bias · low	exact scheme bias · high	Euler bias
1	0.00344949	-0.023333	0.023333	-0.247628
2	0.0000788349	-0.0233198	0.0233198	-0.0936805
4	0.0200917	-0.0233585	0.0233585	-0.019163
12	0.0382653	-0.0233389	0.0233389	0.0276325
52	-0.0005839	-0.0232543	0.0232543	-0.00265872

Antithetic variates at an equal 20,000-path budget

empirical SE across 30 replications · seeds 1000 + rep · notebook 04

Two columns comparing the empirical standard error of the price estimate with independent draws and with antithetic pairs.
sampling	empirical SE
independent	0.119786
antithetic	0.0921096

Relative sampling error · N⁻¹ᐟ² law, normalised at 10³ paths

relative error · theoretical law · finding page

Line of the theoretical inverse-square-root law normalised to one at one thousand paths, falling to 0.32 at ten thousand and 0.10 at one hundred thousand paths.
paths (log scale)	relative error
1,000	1
10,000	0.316228
100,000	0.1

Exact versus Euler GBM against the closed-form call (S₀ = K = 100, r = 5%, σ = 20%, T = 1)

Steps	Exact scheme	Exact bias	Euler	Euler bias	MC std err
1	10.454	0.00344949	10.203	-0.247628	0.023333
2	10.4507	0.0000788349	10.3569	-0.0936805	0.0233198
4	10.4707	0.0200917	10.4314	-0.019163	0.0233585
12	10.4888	0.0382653	10.4782	0.0276325	0.0233389
52	10.45	-0.0005839	10.4479	-0.00265872	0.0232543

Antithetic variates across 30 independent replications

Antithetic	Mean estimate	Empirical SE	Bias vs BS
no	10.4638	0.119786	0.0132019
yes	10.4278	0.0921096	-0.0227393

SE reduction factor 1.30× at equal path count.

RMSE by path count

N	RMSE (independent)	RMSE (antithetic)	N⁻¹ᐟ² reference
250	0.847309	0.854915	0.847309
1,000	0.523811	0.362206	0.423654
4,000	0.241325	0.178972	0.211827
16,000	0.0881235	0.0583958	0.105914
64,000	0.0473331	0.0415612	0.0529568
256,000	0.0208947	0.0211394	0.0264784

Fitted log–log slope -0.551; replication-noise SE 0.031, OLS SE 0.023.

Process checks

Check	Value	Reference
OU κ recovered from AR(1) fit	2.01059	2
OU half-life recovered (years)	0.344748	0.346574
Regime model: share of time in crisis	0.140493	0.142857
Regime model: terminal log-return skew	-0.306574	0
Regime model: terminal log-return excess kurtosis	0.311882	0

CIR boundary behaviour (2,000 paths, 5 years, full-truncation Euler)

Case	κ	θ	σ	2κθ	σ²	Fraction of paths touching zero
Feller satisfied	2	0.04	0.3	0.16	0.09	0.0875
Feller violated	0.5	0.04	0.45	0.04	0.2025	0.9535

A payoff with no closed form: the discretely monitored up-and-out call (40,000 antithetic paths, 252 steps)

Contract	Price	Std err	P(survive)
vanilla call (BS closed form)	10.4506	—	1
vanilla call (MC, same paths)	10.5121	0.0742345	1
up-and-out call, B=130, daily monitoring (252/yr)	3.55065	0.0318834	0.7831
up-and-out call, B=130, monthly monitoring (12/yr)	4.28538	0.0356831	0.823325

Audience and decision

Research significance

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 context

Numerical stopping criteria and economic error tolerance

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.

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. Cutting error tenfold therefore takes roughly one hundred times as many independent paths.

Exact transition schemes for GBM avoided time-step bias at observation points. Euler error shrank with Δt. That distinction matters. Adding paths reduces sampling error. It cannot repair a biased discretization scheme.

At an equal 20,000-path budget, antithetic normal draws cut empirical standard error by about 30%. The reduction held across 30 independent replications. The gain depends on the payoff and is not a universal multiplier. The notebook then applies simulation where path dependence matters. Its example is a discretely monitored barrier option, whose value moves with monitoring frequency.

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 refining steps; budget statistical error by adding paths.
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 paths change value.

See the processes, derive the error law, and rerun the study

Run stochastic simulations → Read the derivation → Inspect the research record → Reproduce from the executed notebook ↗

Geometric Brownian motion

$$dS_t = \mu S_t\,dt + \sigma S_t\,dW_t \qquad\Longrightarrow\qquad S_t = S_0 \exp\!\left[\left(\mu - \tfrac{1}{2}\sigma^2\right)t + \sigma W_t\right]$$

Applications. The default model for equity prices. It is the substrate of Black-Scholes and of most textbook portfolio mathematics.

Captured dynamics. Proportional randomness (returns, not price changes, are the natural unit), positivity of prices, and compounding. That includes the subtle $-\tfrac{1}{2}\sigma^2$ drag separating median from mean growth.

Omitted dynamics. Everything the rest of this page exists for: volatility is constant, returns are Gaussian and independent, and nothing ever jumps. Real markets violate all three, persistently.

Ornstein-Uhlenbeck

$$dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t$$

Applications. Short-rate models (Vasicek), commodity convenience yields, volatility itself, and the spread dynamics behind statistical arbitrage.

Captured dynamics. Mean reversion — the pull toward $\theta$ at speed $\kappa$, with a stationary distribution $\mathcal{N}(\theta, \sigma^2/2\kappa)$. The half-life of a shock is $\ln 2 / \kappa$, a genuinely useful number in trading.

Omitted dynamics. Gaussian increments allow negative values, fatal for nominal rates near zero and tolerable for spreads. The reversion speed is assumed constant. The level it reverts to must actually exist. Many "mean-reverting" spreads are regimes in disguise, and OU is silent about the regime change that ends the trade.

Merton jump-diffusion

$$\frac{dS_t}{S_{t^-}} = (\mu - \lambda \bar{k})\,dt + \sigma\,dW_t + (J-1)\,dN_t, \qquad \ln J \sim \mathcal{N}(\mu_J, \sigma_J^2)$$

Applications. Pricing short-dated options where crash risk dominates; credit and event-risk modeling; any setting where the smile is steep at short maturities.

Captured dynamics. Discontinuity. A Poisson process $N_t$ with intensity $\lambda$ fires jumps of lognormal size $J$. That produces genuine fat tails, and with $\mu_J < 0$ the negative skew that equity markets exhibit. It explains why deep out-of-the-money puts cost "too much" under Black-Scholes.

Omitted dynamics. Jump risk here is diversifiable and independent of the diffusion. In reality crashes arrive precisely when volatility is already elevated, and jump intensity is anything but constant. Volatility between jumps is still flat. Merton fixes the tails and leaves clustering unexplained.

Heston stochastic volatility

$$dS_t = \mu S_t\,dt + \sqrt{v_t}\,S_t\,dW_t^S, \qquad dv_t = \kappa(\theta - v_t)\,dt + \xi\sqrt{v_t}\,dW_t^v, \qquad d\langle W^S, W^v\rangle_t = \rho\,dt$$

Applications. The workhorse stochastic-volatility model for equity index options. Its characteristic function is known in closed form, so whole smiles calibrate quickly.

Captured dynamics. Variance as a mean-reverting random process (CIR dynamics), volatility clustering, and the leverage effect through $\rho < 0$. Prices fall as volatility rises, which generates the implied-vol skew. Simulation uses a full-truncation Euler scheme. That scheme floors $v_t$ at zero inside the square roots, so the discretized variance cannot go negative.

Omitted dynamics. No jumps, so very short-dated smiles are too flat. One volatility factor, so the term structure of skew is too rigid. Calibrated parameters also drift over time, which indicates residual misspecification. The case study treats this in depth.

Regime-switching GBM

$$dS_t = \mu_{Z_t} S_t\,dt + \sigma_{Z_t} S_t\,dW_t, \qquad Z_t \in \{\text{calm}, \text{stress}\}\ \text{a 2-state Markov chain}$$

Applications. Tactical allocation, risk-regime dashboards, and econometric business-cycle work in the Hamilton (1989) tradition.

Captured dynamics. Long low-volatility intervals with positive drift, interrupted by high-volatility intervals with weaker returns. The two-state model reproduces fat tails and clustering as an emergent mixture, with no exotic distributions required. Each path carries its own hidden regime sequence.

Omitted dynamics. The chain's transition probabilities are fixed, and the regimes are labeled in hindsight. Real regime identification is only obvious after the fact. The model represents regime-dependent return behavior without identifying a causal mechanism.

Numbers

Statement	Value	As stated	Note
Error rate of the Monte Carlo estimator	N⁻¹ᐟ²
paths needed for 10× less error	≈100×	≈100× paths
Fitted log–log RMSE slope across 16 replications	−0.55	within noise of −0.5	Replication-noise SE of the slope 0.031 (OLS residual SE 0.023); \|slope + 0.5\| = 0.051 = 1.62 SE.
Antithetic SE reduction at the same 20,000-path budget	≈30%	≈30%	SE ratio independent / antithetic = 1.30 across 30 replications (equivalent to 1.7× more paths); the notebook reports this ratio as 'about 30%'.
Empirical SE, independent ÷ antithetic	1.30×	1.30x
Primary seed (PCG64)	42	42
Paths per replication in the antithetic test	20,000	20,000
Independent replications in the antithetic test	30	30
Relative error at 10³ paths (N⁻¹ᐟ² law, normalised)	1.00	1.00
Relative error at 10⁴ paths	0.32	0.32
Relative error at 10⁵ paths	0.10	0.10
OU κ recovered from the AR(1) fit (true 2.0)	2.011	2.011
Time spent in the crisis regime	14.0%	14.0%

Notes

Monte Carlo Error Scales as

O(1/\sqrt{N})

4 min · Prerequisites: variance, sample means, and the central limit theorem

Estimate $\theta = \mathbb{E}[f(X)]$ by the sample mean $\hat\theta_N = \frac{1}{N}\sum_{i=1}^N f(X_i)$ over independent draws. The estimator is unbiased, and its standard error follows from nothing deeper than the variance of a mean:¹

$\operatorname{se}(\hat\theta_N) = \frac{\sigma_f}{\sqrt{N}}, \qquad \sigma_f^2 = \operatorname{Var}[f(X)],$

with the central limit theorem supplying asymptotic normality and hence confidence intervals. Three consequences structure all practical Monte Carlo work:

1. Computational scaling. One more decimal digit of accuracy costs a factor of 100 in samples. Monte Carlo is the method of choice not because it converges fast but because its rate does not deteriorate with dimension — a 100-dimensional basket option converges at the same $N^{-1/2}$ as a one-dimensional one, while grid methods collapse under $O(h^{-d})$ nodes.

2. Variance reduction. Since the rate is fixed, accuracy can improve by shrinking $\sigma_f$ : antithetic variates, control variates (price the arithmetic Asian against the geometric one, which is known in closed form), importance sampling for tail events, and quasi-random sequences that trade independence for low discrepancy and near- $N^{-1}$ behavior.

3. Tail estimates inherit the worst constants. Estimating $\mathbb{P}[L > \ell] = p$ by naive simulation has relative standard error $\approx \sqrt{(1-p)/(pN)}$ — for a $p = 0.1\%$ event, ten percent relative accuracy needs on the order of $10^7$ paths. Rare-event estimation without importance sampling is statistically unstable unless the estimator changes, which links this note directly to the next one.

Glasserman, P. (2003), Monte Carlo Methods in Financial Engineering , Springer, pp. 1–38 and 185–337 (foundations, variance reduction, and quasi-Monte Carlo). doi:10.1007/978-0-387-21617-1 ↩

Limitations

Every Monte Carlo estimate is a random draw. A fixed seed reproduces it without removing 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.

Cite this

Wisniewski, K. (2026, August 8). Monte Carlo convergence, discretization bias, and variance reduction. Quantitative Markets & Institutions Lab. https://www.kylewisniewski.com/lab/monte-carlo

@misc{wisniewski2026monte,
  author = {Wisniewski, Kyle},
  title = {Monte Carlo convergence, discretization bias, and variance reduction},
  year = {2026},
  month = {aug},
  howpublished = {\url{https://www.kylewisniewski.com/lab/monte-carlo}},
  note = {Numerical validation · Quantitative Markets & Institutions Lab · commit 4806df9}
}