Statistics / Random Sampling / Inference

Monte Carlo Methods Studio

This playground turns repeated random sampling into numerical estimation. Switch between π estimation, hit-or-miss integration, and risk-neutral option pricing to see how Monte Carlo estimates converge, how confidence intervals shrink, and why variance matters as much as sample count.

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Live State

Estimate 0

Reference 0

Absolute Error 0

95% CI Half-Width 0

Samples 0

Samples / Sec 0

What You Are Seeing

Modes

Monte Carlo Problem Families

Each mode uses random sampling differently: geometry for π, area counting for integrals, and risk-neutral path sampling for derivative pricing.

Sampling

Simulation Controls

Samples Per Frame 300 Higher throughput speeds convergence but can make the animation jump instead of breathe.

Retained Visual Samples 2400 The estimator always uses every sample; this slider only limits how many recent samples stay visible on the canvas.

Problem Setup

Mode-Specific Parameters

The classic π experiment samples uniformly in the unit square and checks whether each point falls inside the quarter unit disk.

Estimator. If the hit probability is p, then π = 4p.

Variance source. Every point is a Bernoulli trial, so uncertainty shrinks like 1 / √n, not linearly.

Hit-or-miss integration estimates area by sampling inside a bounding rectangle and counting how many points fall below the chosen function.

Integrand sin(πx) All three examples live on [0, 1] and stay within a unit-height bounding box for clean comparison.

This mode prices a European call under the Black-Scholes model by sampling lognormal terminal prices under the risk-neutral measure.

Spot S₀ 100

Strike K 100

Volatility σ 0.20

Rate r 0.05

Maturity T 1.00

Readout

Convergence Diagnostics

Current Modeπ Estimation

Hit Ratio / ITM Ratio0.0000

Sample Variance Proxy0.000000

Reference Gap0.000000

FPS0.0

Retained Samples0

Estimate History

Method

Estimator Equations

Unbiasedness. Each estimator targets an expectation: hit probability, average function value, or discounted payoff. Sampling noise stays random, not systematically shifted.

Square-root law. Doubling accuracy typically requires about four times as many samples, which is why variance reduction is a core Monte Carlo design problem.

Reference solution. Each mode compares the random estimate against either a closed-form truth or a known analytic benchmark so the convergence story stays grounded.

Accuracy note: option pricing here assumes geometric Brownian motion and a frictionless Black-Scholes setting, so the Monte Carlo result is being compared against the matching model's exact price, not against real market behavior.