Price an option
Simulate thousands of stock paths and watch the Monte Carlo price close in on Black-Scholes.
The Monte Carlo estimate will close in on the dashed Black-Scholes line
The stock trades at $100 today.
- Black-Scholes
- $10.45
- Monte Carlo
- –
What’s going on
A call pays max(S_T − K, 0) at expiry; a put pays max(K − S_T, 0). To price one, simulate the stock under risk-neutral rules, where it grows at the interest rate: S_T = S₀ · exp((r − σ²/2)T + σ√T · Z) with Z a standard normal draw. Average the payoffs, discount by e^(−rT), and you have a Monte Carlo price.
Black-Scholes does the same average in closed form: C = S₀N(d₁) − Ke^(−rT)N(d₂), with d₁ = [ln(S₀/K) + (r + σ²/2)T] / (σ√T) and d₂ = d₁ − σ√T. N is the normal CDF from the normal curve topic. At S₀ = K = 100, one year, 5% and σ = 20%, the call is worth $10.45.
The Monte Carlo error shrinks like 1/√n: four times the paths, half the error. That is why the shaded band narrows slowly, and why banks use Monte Carlo for options too complex for a formula.
A textbook model (constant volatility, no dividends). For learning, not investment advice.