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Black–Scholes Model

Closed-form price for European options.

Black–Scholes prices an option from five inputs: spot, strike, time, rate, and volatility. Volatility is the only unobservable input, so traders invert the formula to quote 'implied volatility'. It earned a Nobel Prize and remains the lingua franca of options.

C = S·N(d₁) − K·e^(−rT)·N(d₂)
d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T),  d₂ = d₁ − σ√T

from scipy.stats import norm
import numpy as np
d1 = (np.log(S/K) + (r + 0.5*sigma**2)*T) / (sigma*np.sqrt(T))
d2 = d1 - sigma*np.sqrt(T)
call = S*norm.cdf(d1) - K*np.exp(-r*T)*norm.cdf(d2)