Black-Scholes Model
The Black-Scholes model prices European options under geometric Brownian motion. It derives a PDE whose solution gives the famous Black-Scholes formula.
Key Points
- Assumptions include constant volatility, no arbitrage, and risk-free hedging.
- The Black-Scholes PDE is a backward parabolic equation.
- The Greeks measure sensitivity to parameters.
Formulas
Black-Scholes PDE
$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$$
Call option price
$$C = S_0 N(d_1) - K e^{-rT} N(d_2)$$
d1, d2
$$d_{1,2} = \frac{\log(S_0/K) + (r \pm \sigma^2/2)T}{\sigma\sqrt{T}}$$
Code Example
from scipy.stats import norm
def black_scholes_call(S, K, T, r, sigma):
d1 = (np.log(S/K) + (r + 0.5*sigma**2)*T) / (sigma*np.sqrt(T))
d2 = d1 - sigma*np.sqrt(T)
return S*norm.cdf(d1) - K*np.exp(-r*T)*norm.cdf(d2)
print(black_scholes_call(100, 100, 1, 0.05, 0.2))