Monte-Carlo replication error of a discretely-rehedged Black-Scholes hedge, benchmarked live against Derman & Kamal (Goldman Sachs) — a browser port of the project's NumPy/SciPy engine. Every path is simulated on this page.
std(P&L) ≈ √(π/4n) · Vega · σAn option writer sells the option, collects the premium, and rehedges n times. Continuous rehedging replicates exactly; discrete rehedging leaves a residual error that decays like 1/√n.
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Option type
Vol mismatch
P&L mean—≈ 0 for a fair, self-financing hedge
P&L std — empirical—Monte-Carlo replication error
Derman-Kamal std—closed-form prediction
Formula gap—empirical vs analytic
5% VaR—loss not exceeded 95% of the time
P&L distribution
Where the writer's final P&L lands across every simulated path.
Discrete hedging leaves fat, slightly left-skewed tails — the hedger's residual gap risk.Table view
Statistic
Value
Replication error vs rehedge frequency
The headline result: Monte-Carlo error against the analytic 1/√n law.
On log-log axes the error falls as a straight 1/√n line — the Monte-Carlo dots sit on the analytic curve.Table view
Rehedges
Monte-Carlo std
Derman-Kamal std
Gap
Sample hedged price paths
Forty simulated underlying paths, coloured by the P&L the hedge actually realised on each.
lossgain· neutral grey marks break-even, arms clip at ±2σ
Paths that whip around the strike near expiry are the hardest to hedge discretely — the delta flips between 0 and 1 faster than the rehedge clock can follow.Table view