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European option under geometric Brownian motion
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Terminal price sampled exactly: S_T = S₀·exp((r − σ²/2)T + σ√T·z). Payoffs discounted at e^(−rT). Benchmark: Black–Scholes with Hart's Φ.
Terminal price distribution
Call estimate as paths accumulate (dashed = Black–Scholes)
Estimates against the benchmark
Call (Monte Carlo)
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Call (Black–Scholes)
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|error| and SE units
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95% CI (call)
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Put (Monte Carlo)
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Put (Black–Scholes)
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Mean terminal price
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Std deviation
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Min
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5th percentile
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Median
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95th percentile
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Max
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π from points in a circle
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π ≈ 4 · (points inside the unit circle) / (points in the square). Pseudo-random standard error: 4·√(p(1−p)/n), p = π/4.
Sample (first 20,000 points)
Result
Estimated π
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|error|
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Pseudo-random SE at this n
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Points in circle
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Points in square
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Geometric Brownian motion paths
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S₁ from S₀ = 1 has mean e^μ and standard deviation e^μ·√(e^(σ²) − 1); both printed beside the simulated values. Paths always use the seeded pseudo-random source: a path of n steps is an n-dimensional integral, and a one-dimensional Sobol sequence would correlate the steps.
Paths (first 60 drawn)
Terminal value distribution
Terminal statistics
Mean
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Std deviation
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Min
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5th percentile
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25th percentile
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Median
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75th percentile
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95th percentile
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Max
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Three-asset portfolio Value-at-Risk
Weight asset 3
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Returns jointly normal via a Cholesky factor of the correlation matrix. VaR and expected shortfall reported as positive losses; parametric VaR = −(μ + σ·Φ⁻¹(0.05))·V.
One-period P&L distribution
Sorted outcomes (loss tail on the left)
Risk metrics
VaR 95% (Monte Carlo)
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VaR 95% (parametric)
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VaR 99% (Monte Carlo)
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Expected shortfall 95%
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Expected return
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Portfolio volatility
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Return / volatility
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Mean P&L
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Best outcome
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Worst outcome
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