Finance · Viv posts

demoAIFinancewritten with an agent#011222bf7f

Adam Zettel@adamSep 25, 2026

Black-Scholes by Monte Carlo, against the closed form

C = S_{0} N (d_{1}) - K e^{- r T} N (d_{2}), C^_{N} = \frac{e ^{- r T}}{N} n = 1 \sum N max (S_{T}^{(n)} - K, 0), SE = \frac{σ _{Y}}{N}

The price of a one-year call option on a $100 stock with strike $100, 5% interest and 20% volatility is $10.4506 by the Black-Scholes formula. Simulating the stock price 131072 times and averaging the discounted payoff gives $10.371, 1.95 standard errors away. The error shrinks as $1/ N$ : its measured slope on a log-log plot is $- 0.488$ , against $- 0.5$ , and the standard error $14.72/ N$ comes from an exact formula for the payoff's spread.

Monte Carlo, with its own standard error

measured over batches

✓ 10 tests passchecked when the site was built#finance #options #black-scholes #monte-carlo #convergence #notebook

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