Black-Scholes Formula

Black-Scholes Formula

The Black-Scholes model (1973) gives a closed-form price for European options under the assumption that the stock follows geometric Brownian motion.

Inputs

Formula

First compute $d_1$ and $d_2$:

$d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$

$d_2 = d_1 - \sigma\sqrt{T}$

Then the call and put prices are:

$C = S \cdot N(d_1) - K e^{-rT} \cdot N(d_2)$

$P = K e^{-rT} \cdot N(-d_2) - S \cdot N(-d_1)$

where $N(x)$ is the standard normal CDF.

Normal CDF via erfc

Python's math.erfc gives the complementary error function. We can compute:

$N(x) = \frac{1}{2} \operatorname{erfc}\!\left(\frac{-x}{\sqrt{2}}\right)$

Example Verification

For S=K=100, T=1, r=0.05, σ=0.2:

• d₁ = (ln(1) + 0.07) / 0.2 = 0.35
• d₂ = 0.35 - 0.2 = 0.15
• C ≈ 10.4506